an elementary approach to primesrichard/msp/talks/katie.pdf · 2019. 5. 31. · primes professor...
TRANSCRIPT
![Page 1: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/1.jpg)
An Elementary Approach toPrimes
Professor Richard BlecksmithDept. of Mathematical SciencesNorthern Illinois University
http://math.niu.edu/∼richard/Math230
– p. 1
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Primes and Composites
A positive integer greater than 1 is prime if its onlydivisors are 1 and itself.
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Primes and Composites
A positive integer greater than 1 is prime if its onlydivisors are 1 and itself.The List of Primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, etc
![Page 4: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/4.jpg)
Primes and Composites
A positive integer greater than 1 is prime if its onlydivisors are 1 and itself.The List of Primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, etcBut not 4 = 2× 2, 6 = 2× 3, etc.
![Page 5: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/5.jpg)
Primes and Composites
A positive integer greater than 1 is prime if its onlydivisors are 1 and itself.The List of Primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, etcBut not 4 = 2× 2, 6 = 2× 3, etc.A positive integer greater than 1 which factors iscalled a composite number.
![Page 6: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/6.jpg)
Primes and Composites
A positive integer greater than 1 is prime if its onlydivisors are 1 and itself.The List of Primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, etcBut not 4 = 2× 2, 6 = 2× 3, etc.A positive integer greater than 1 which factors iscalled a composite number.The number 1 is called a unit. It is neither prime norcomposite.
![Page 7: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/7.jpg)
Question
What are the primes good for?
![Page 8: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/8.jpg)
Question
What are the primes good for?They are useful in many arithmetic problems.
![Page 9: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/9.jpg)
Grade School Problem
Add the fractions1
6+
1
9
![Page 10: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/10.jpg)
Grade School Problem
Add the fractions1
6+
1
9Solution: Find a common denominator.
![Page 11: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/11.jpg)
Grade School Problem
Add the fractions1
6+
1
9Solution: Find a common denominator.You could use 6× 9 = 54, but a better way is torecognize that both 6 and 9 share the factor 3.
![Page 12: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/12.jpg)
Grade School Problem
Add the fractions1
6+
1
9Solution: Find a common denominator.You could use 6× 9 = 54, but a better way is torecognize that both 6 and 9 share the factor 3.So we can use 18 as a common denominator:
![Page 13: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/13.jpg)
Grade School Problem
Add the fractions1
6+
1
9Solution: Find a common denominator.You could use 6× 9 = 54, but a better way is torecognize that both 6 and 9 share the factor 3.So we can use 18 as a common denominator:
1
6+
1
9=
3
18+
2
18=
5
18
![Page 14: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/14.jpg)
Least Common Multiple
The smallest number that is divisible by both a and bis called the least common multiple of a and b or
LCM[a, b].
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Least Common Multiple
The smallest number that is divisible by both a and bis called the least common multiple of a and b or
LCM[a, b].Examples:
![Page 16: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/16.jpg)
Least Common Multiple
The smallest number that is divisible by both a and bis called the least common multiple of a and b or
LCM[a, b].Examples:1. LCM [6, 18] = 18
![Page 17: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/17.jpg)
Least Common Multiple
The smallest number that is divisible by both a and bis called the least common multiple of a and b or
LCM[a, b].Examples:1. LCM [6, 18] = 182. LCM [25, 35] = 175
![Page 18: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/18.jpg)
Least Common Multiple
The smallest number that is divisible by both a and bis called the least common multiple of a and b or
LCM[a, b].Examples:1. LCM [6, 18] = 182. LCM [25, 35] = 1753. LCM [100, 210] = 2100
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Greatest Common DivisorThe greatest common divisor is the largest numberthat divides both a and b.
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Greatest Common DivisorThe greatest common divisor is the largest numberthat divides both a and b.The least common multiple is related to the greatestcommon divisor by the following rule:
![Page 21: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/21.jpg)
Greatest Common DivisorThe greatest common divisor is the largest numberthat divides both a and b.The least common multiple is related to the greatestcommon divisor by the following rule:Rule
LCM[a, b] =a · b
GCD(a, b)
![Page 22: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/22.jpg)
Greatest Common DivisorThe greatest common divisor is the largest numberthat divides both a and b.The least common multiple is related to the greatestcommon divisor by the following rule:Rule
LCM[a, b] =a · b
GCD(a, b)
Examples:
![Page 23: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/23.jpg)
Greatest Common DivisorThe greatest common divisor is the largest numberthat divides both a and b.The least common multiple is related to the greatestcommon divisor by the following rule:Rule
LCM[a, b] =a · b
GCD(a, b)
Examples:
LCM [6, 18] =6 · 18
GCD(6, 18)=
6 · 18
6= 18.
![Page 24: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/24.jpg)
Greatest Common DivisorThe greatest common divisor is the largest numberthat divides both a and b.The least common multiple is related to the greatestcommon divisor by the following rule:Rule
LCM[a, b] =a · b
GCD(a, b)
Examples:
LCM [6, 18] =6 · 18
GCD(6, 18)=
6 · 18
6= 18.
LCM [25, 35] =25 · 35
GCD(25, 35)=
25 · 35
5= 175.
![Page 25: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/25.jpg)
Finding GCD
The greatest common divisor can be readily found byfactoring the numbers a and b into primes.
![Page 26: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/26.jpg)
Finding GCD
The greatest common divisor can be readily found byfactoring the numbers a and b into primes.Find GCD [30, 18]
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Finding GCD
The greatest common divisor can be readily found byfactoring the numbers a and b into primes.Find GCD [30, 18]6 = 2× 3× 5 and 18 = 2× 3× 3.
![Page 28: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/28.jpg)
Finding GCD
The greatest common divisor can be readily found byfactoring the numbers a and b into primes.Find GCD [30, 18]6 = 2× 3× 5 and 18 = 2× 3× 3.So GCD(30, 18) = 2× 3
![Page 29: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/29.jpg)
Finding GCD
The greatest common divisor can be readily found byfactoring the numbers a and b into primes.Find GCD [30, 18]6 = 2× 3× 5 and 18 = 2× 3× 3.So GCD(30, 18) = 2× 3Find GCD [25, 35]
![Page 30: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/30.jpg)
Finding GCD
The greatest common divisor can be readily found byfactoring the numbers a and b into primes.Find GCD [30, 18]6 = 2× 3× 5 and 18 = 2× 3× 3.So GCD(30, 18) = 2× 3Find GCD [25, 35]25 = 5× 5 and 35 = 5× 7.
![Page 31: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/31.jpg)
Finding GCD
The greatest common divisor can be readily found byfactoring the numbers a and b into primes.Find GCD [30, 18]6 = 2× 3× 5 and 18 = 2× 3× 3.So GCD(30, 18) = 2× 3Find GCD [25, 35]25 = 5× 5 and 35 = 5× 7.So GCD(25, 35) = 5
![Page 32: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/32.jpg)
Katie’s ProblemFind the GCD (99, 100).
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Katie’s ProblemFind the GCD (99, 100).Fifth Grade Textbook Solution: Factor 99 and 100into primes:
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Katie’s ProblemFind the GCD (99, 100).Fifth Grade Textbook Solution: Factor 99 and 100into primes:99 = 9× 11 = 3× 3× 11
![Page 35: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/35.jpg)
Katie’s ProblemFind the GCD (99, 100).Fifth Grade Textbook Solution: Factor 99 and 100into primes:99 = 9× 11 = 3× 3× 11100 = 10× 10 = 2× 2× 5× 5
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Katie’s ProblemFind the GCD (99, 100).Fifth Grade Textbook Solution: Factor 99 and 100into primes:99 = 9× 11 = 3× 3× 11100 = 10× 10 = 2× 2× 5× 5Since none of the primes factors of 99 (3 and 11)overlap with the prime factors of 100 (2 and 5),
![Page 37: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/37.jpg)
Katie’s ProblemFind the GCD (99, 100).Fifth Grade Textbook Solution: Factor 99 and 100into primes:99 = 9× 11 = 3× 3× 11100 = 10× 10 = 2× 2× 5× 5Since none of the primes factors of 99 (3 and 11)overlap with the prime factors of 100 (2 and 5),
GCD (99, 100) = 1
![Page 38: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/38.jpg)
Katie’s Dad’s Solution:Any number which divides both 99 and 100 mustdivide their difference
100− 99 = 1.
![Page 39: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/39.jpg)
Katie’s Dad’s Solution:Any number which divides both 99 and 100 mustdivide their difference
100− 99 = 1.
But the only number which divides 1 is 1 itself.
![Page 40: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/40.jpg)
Katie’s Dad’s Solution:Any number which divides both 99 and 100 mustdivide their difference
100− 99 = 1.
But the only number which divides 1 is 1 itself.So
GCD (99, 100) = 1
![Page 41: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/41.jpg)
Another example:
Find GCD (17463287493, 17463287494) = 1
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Another example:
Find GCD (17463287493, 17463287494) = 1
![Page 43: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/43.jpg)
Another example:
Find GCD (17463287493, 17463287494) = 1
![Page 44: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/44.jpg)
Another example:
Find GCD (17463287493, 17463287494) = 1
![Page 45: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/45.jpg)
Another example:
Find GCD (17463287493, 17463287494) = 1
![Page 46: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/46.jpg)
Katie’s Theorem:Two consecutive integers
x and x+ 1
have no factor in common (except 1).
![Page 47: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/47.jpg)
Katie’s Theorem:Two consecutive integers
x and x+ 1
have no factor in common (except 1).Numbers with no common factor (except 1) are calledrelatively prime.
![Page 48: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/48.jpg)
Are primes plentiful or scarce?
What do you think?
![Page 49: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/49.jpg)
Are primes plentiful or scarce?
What do you think?We will consider both points of view:
![Page 50: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/50.jpg)
Are primes plentiful or scarce?
What do you think?We will consider both points of view:1. The primes are scarce.
![Page 51: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/51.jpg)
Are primes plentiful or scarce?
What do you think?We will consider both points of view:1. The primes are scarce.2. The primes are plentiful.
![Page 52: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/52.jpg)
Handling Complaints
Whenever a student complains:
![Page 53: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/53.jpg)
Handling Complaints
Whenever a student complains:“What good is all this math we’re learning?”
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Handling Complaints
Whenever a student complains:“What good is all this math we’re learning?”my standard response is:
![Page 55: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/55.jpg)
Handling Complaints
Whenever a student complains:“What good is all this math we’re learning?”my standard response is:“Good question. Let me tell you my favorite wordproblem.”
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Handling Complaints
Whenever a student complains:“What good is all this math we’re learning?”my standard response is:“Good question. Let me tell you my favorite wordproblem.”Invariably I hear “Never mind!”
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Handling Complaints
Whenever a student complains:“What good is all this math we’re learning?”my standard response is:“Good question. Let me tell you my favorite wordproblem.”Invariably I hear “Never mind!”We are going to take a little sojourn into the world ofword problems.
![Page 58: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/58.jpg)
Solving Word Problems
• Create Variables
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Solving Word Problems
• Create Variables
• Organize Data
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Solving Word Problems
• Create Variables
• Organize Data
• Draw a diagram
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Solving Word Problems
• Create Variables
• Organize Data
• Draw a diagram
• Translate Words → Equation (English to Math)
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Solving Word Problems
• Create Variables
• Organize Data
• Draw a diagram
• Translate Words → Equation (English to Math)
• Solve the equation.
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Building Bookcases
You wish to build a bookcase.
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Building Bookcases
You wish to build a bookcase.You are using lumber which is 3/4 inch thick.
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Building Bookcases
You wish to build a bookcase.You are using lumber which is 3/4 inch thick.The dimensions are 36 inches wide, 48 inches high.
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Building Bookcases
You wish to build a bookcase.You are using lumber which is 3/4 inch thick.The dimensions are 36 inches wide, 48 inches high.You want to install 2 interior shelves so that theshelves are equally spaced.
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Building Bookcases
You wish to build a bookcase.You are using lumber which is 3/4 inch thick.The dimensions are 36 inches wide, 48 inches high.You want to install 2 interior shelves so that theshelves are equally spaced.What is the spacing between the shelves?
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Bookcase Picture
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Bookcase Picture
✻
❄
48
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Bookcase Picture
✻
❄
48
✲✛ 36
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Bookcase Picture
✻
❄
48
✲✛ 36
x
x
x
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Bookcase Picture
✻
❄
48
✲✛ 36
x
x
x
3x+ 4(.75) = 48
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Bookcase Solution3x+ 4(.75) = 48
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Bookcase Solution3x+ 4(.75) = 483x+ 3 = 48
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Bookcase Solution3x+ 4(.75) = 483x+ 3 = 483x = 45
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Bookcase Solution3x+ 4(.75) = 483x+ 3 = 483x = 45x = 15
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Bookcase Solution3x+ 4(.75) = 483x+ 3 = 483x = 45x = 15Make the shelves 15 inches apart.
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Professors and StudentsConsider the sentence:
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Professors and StudentsConsider the sentence:For every professor at a small liberal arts college,there are two students.
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Professors and StudentsConsider the sentence:For every professor at a small liberal arts college,there are two students.If P = the number of professors and
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Professors and StudentsConsider the sentence:For every professor at a small liberal arts college,there are two students.If P = the number of professors andS = the number of students
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Professors and StudentsConsider the sentence:For every professor at a small liberal arts college,there are two students.If P = the number of professors andS = the number of studentstranslated the above sentence as a mathematicalequation.
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Professors and StudentsConsider the sentence:For every professor at a small liberal arts college,there are two students.If P = the number of professors andS = the number of studentstranslated the above sentence as a mathematicalequation.(a) P = 2S(b) S = 2P(c) Some other answer
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Word Problems - YuckAre the following word problems solved in the sameway:
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Word Problems - YuckAre the following word problems solved in the sameway:Painting Problem. Bob can paint a room in one hour.His brother Bill can paint the room in 45 minutes.How long will it take them, working together, to paintthe room?
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Word Problems - YuckAre the following word problems solved in the sameway:Painting Problem. Bob can paint a room in one hour.His brother Bill can paint the room in 45 minutes.How long will it take them, working together, to paintthe room?Laundry Problem. Joan can wash a load of laundry inone hour. Her mother Mary can do the laundary in 45minutes. How long will it take them, workingtogether, to do a load of laundry?
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Word Problems - YuckAre the following word problems solved in the sameway:Painting Problem. Bob can paint a room in one hour.His brother Bill can paint the room in 45 minutes.How long will it take them, working together, to paintthe room?Laundry Problem. Joan can wash a load of laundry inone hour. Her mother Mary can do the laundary in 45minutes. How long will it take them, workingtogether, to do a load of laundry?Explain your answer.
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Mary Jane
Lucy and Lottie Hill are 90 years old.
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Mary Jane
Lucy and Lottie Hill are 90 years old. Mary Jane, onthe other hand, is half again as old
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Mary Jane
Lucy and Lottie Hill are 90 years old. Mary Jane, onthe other hand, is half again as old as when she washalf again as old as when
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Mary Jane
Lucy and Lottie Hill are 90 years old. Mary Jane, onthe other hand, is half again as old as when she washalf again as old as when she lacked 5 years of beinghalf as old as
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Mary Jane
Lucy and Lottie Hill are 90 years old. Mary Jane, onthe other hand, is half again as old as when she washalf again as old as when she lacked 5 years of beinghalf as old as was on her last birthday.
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Mary Jane
Lucy and Lottie Hill are 90 years old. Mary Jane, onthe other hand, is half again as old as when she washalf again as old as when she lacked 5 years of beinghalf as old as was on her last birthday.How old is Mary Jane?
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Mary Jane
Lucy and Lottie Hill are 90 years old. Mary Jane, onthe other hand, is half again as old as when she washalf again as old as when she lacked 5 years of beinghalf as old as was on her last birthday.How old is Mary Jane?This is why people hate math.
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Mary Jane Cont’d
Let x = Mary Jane’s age.
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Mary Jane Cont’d
Let x = Mary Jane’s age.How old was she on her last birthday?
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Mary Jane Cont’d
Let x = Mary Jane’s age.How old was she on her last birthday? x
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Mary Jane Cont’d
Let x = Mary Jane’s age.How old was she on her last birthday? xHalf as old as she was on her last birthday is
![Page 99: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/99.jpg)
Mary Jane Cont’d
Let x = Mary Jane’s age.How old was she on her last birthday? xHalf as old as she was on her last birthday is x/2
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Mary Jane Cont’d
Let x = Mary Jane’s age.How old was she on her last birthday? xHalf as old as she was on her last birthday is x/2Lacking 5 years of being half as old as she was on her
last birthday is
![Page 101: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/101.jpg)
Mary Jane Cont’d
Let x = Mary Jane’s age.How old was she on her last birthday? xHalf as old as she was on her last birthday is x/2Lacking 5 years of being half as old as she was on her
last birthday is x/2− 5
![Page 102: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/102.jpg)
Mary Jane Cont’d
Let x = Mary Jane’s age.How old was she on her last birthday? xHalf as old as she was on her last birthday is x/2Lacking 5 years of being half as old as she was on her
last birthday is x/2− 5
Half again means 50% more,
![Page 103: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/103.jpg)
Mary Jane Cont’d
Let x = Mary Jane’s age.How old was she on her last birthday? xHalf as old as she was on her last birthday is x/2Lacking 5 years of being half as old as she was on her
last birthday is x/2− 5
Half again means 50% more, or 11
2times,
![Page 104: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/104.jpg)
Mary Jane Cont’d
Let x = Mary Jane’s age.How old was she on her last birthday? xHalf as old as she was on her last birthday is x/2Lacking 5 years of being half as old as she was on her
last birthday is x/2− 5
Half again means 50% more, or 11
2times, or 3
2times.
![Page 105: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/105.jpg)
Mary Jane Cont’d
Let x = Mary Jane’s age.How old was she on her last birthday? xHalf as old as she was on her last birthday is x/2Lacking 5 years of being half as old as she was on her
last birthday is x/2− 5
Half again means 50% more, or 11
2times, or 3
2times.
For example, if gas costs 2 dollars and next year itcosts half again as much as this year, then next year a
gallon of gas will cost
![Page 106: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/106.jpg)
Mary Jane Cont’d
Let x = Mary Jane’s age.How old was she on her last birthday? xHalf as old as she was on her last birthday is x/2Lacking 5 years of being half as old as she was on her
last birthday is x/2− 5
Half again means 50% more, or 11
2times, or 3
2times.
For example, if gas costs 2 dollars and next year itcosts half again as much as this year, then next year a
gallon of gas will cost 3
2× 2 = 3 dollars
![Page 107: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/107.jpg)
Mary Jane Equation
Lucy and Lottie Hill are 90 years old.
![Page 108: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/108.jpg)
Mary Jane Equation
Lucy and Lottie Hill are 90 years old. Mary Jane, on
the other hand, is half again as old as
![Page 109: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/109.jpg)
Mary Jane Equation
Lucy and Lottie Hill are 90 years old. Mary Jane, on
the other hand, is half again as old as (32
times)
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Mary Jane Equation
Lucy and Lottie Hill are 90 years old. Mary Jane, on
the other hand, is half again as old as (32
times) when
she was half again as old as
![Page 111: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/111.jpg)
Mary Jane Equation
Lucy and Lottie Hill are 90 years old. Mary Jane, on
the other hand, is half again as old as (32
times) when
she was half again as old as (another 3
2times)
![Page 112: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/112.jpg)
Mary Jane Equation
Lucy and Lottie Hill are 90 years old. Mary Jane, on
the other hand, is half again as old as (32
times) when
she was half again as old as (another 3
2times) when
she lacked 5 years of being half as old as was on herlast birthday
![Page 113: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/113.jpg)
Mary Jane Equation
Lucy and Lottie Hill are 90 years old. Mary Jane, on
the other hand, is half again as old as (32
times) when
she was half again as old as (another 3
2times) when
she lacked 5 years of being half as old as was on herlast birthday (x
2− 5).
![Page 114: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/114.jpg)
Mary Jane Equation
Lucy and Lottie Hill are 90 years old. Mary Jane, on
the other hand, is half again as old as (32
times) when
she was half again as old as (another 3
2times) when
she lacked 5 years of being half as old as was on herlast birthday (x
2− 5).
Equation:
![Page 115: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/115.jpg)
Mary Jane Equation
Lucy and Lottie Hill are 90 years old. Mary Jane, on
the other hand, is half again as old as (32
times) when
she was half again as old as (another 3
2times) when
she lacked 5 years of being half as old as was on herlast birthday (x
2− 5).
Equation: x = 3
2
![Page 116: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/116.jpg)
Mary Jane Equation
Lucy and Lottie Hill are 90 years old. Mary Jane, on
the other hand, is half again as old as (32
times) when
she was half again as old as (another 3
2times) when
she lacked 5 years of being half as old as was on herlast birthday (x
2− 5).
Equation: x = 3
2
3
2
![Page 117: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/117.jpg)
Mary Jane Equation
Lucy and Lottie Hill are 90 years old. Mary Jane, on
the other hand, is half again as old as (32
times) when
she was half again as old as (another 3
2times) when
she lacked 5 years of being half as old as was on herlast birthday (x
2− 5).
Equation: x = 3
2
3
2
(
x
2− 5
)
![Page 118: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/118.jpg)
Mary Jane Equation
Lucy and Lottie Hill are 90 years old. Mary Jane, on
the other hand, is half again as old as (32
times) when
she was half again as old as (another 3
2times) when
she lacked 5 years of being half as old as was on herlast birthday (x
2− 5).
Equation: x = 3
2
3
2
(
x
2− 5
)
Solve this equation for x.
![Page 119: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/119.jpg)
Mary Jane Equation
Lucy and Lottie Hill are 90 years old. Mary Jane, on
the other hand, is half again as old as (32
times) when
she was half again as old as (another 3
2times) when
she lacked 5 years of being half as old as was on herlast birthday (x
2− 5).
Equation: x = 3
2
3
2
(
x
2− 5
)
Solve this equation for x.
Why does the problem begin with Lucy and LottieHill?
![Page 120: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/120.jpg)
Charise’s Problem:For the first five consecutive birthdays after she wasborn, Mary’s age exactly divided her grandfather’sage.
![Page 121: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/121.jpg)
Charise’s Problem:For the first five consecutive birthdays after she wasborn, Mary’s age exactly divided her grandfather’sage.How old was Mary’s grandfather when she was born?
![Page 122: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/122.jpg)
Charise’s Problem:For the first five consecutive birthdays after she wasborn, Mary’s age exactly divided her grandfather’sage.How old was Mary’s grandfather when she was born?Can you solve this problem by coming up with anequation to solve for x?
![Page 123: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/123.jpg)
Answer for ChariseHow old was Mary’s grandfather when she was born
![Page 124: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/124.jpg)
Answer for ChariseHow old was Mary’s grandfather when she was bornif for her first five consecutive birthdays, Mary’s ageexactly divided her grandfather’s age?
![Page 125: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/125.jpg)
Answer for ChariseHow old was Mary’s grandfather when she was bornif for her first five consecutive birthdays, Mary’s ageexactly divided her grandfather’s age?Answer: Gramps was 60 when Mary was born.
![Page 126: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/126.jpg)
Answer for ChariseHow old was Mary’s grandfather when she was bornif for her first five consecutive birthdays, Mary’s ageexactly divided her grandfather’s age?Answer: Gramps was 60 when Mary was born.Verification
![Page 127: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/127.jpg)
Answer for ChariseHow old was Mary’s grandfather when she was bornif for her first five consecutive birthdays, Mary’s ageexactly divided her grandfather’s age?Answer: Gramps was 60 when Mary was born.VerificationMary’s age Gramps’ age
![Page 128: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/128.jpg)
Answer for ChariseHow old was Mary’s grandfather when she was bornif for her first five consecutive birthdays, Mary’s ageexactly divided her grandfather’s age?Answer: Gramps was 60 when Mary was born.VerificationMary’s age Gramps’ age
1
![Page 129: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/129.jpg)
Answer for ChariseHow old was Mary’s grandfather when she was bornif for her first five consecutive birthdays, Mary’s ageexactly divided her grandfather’s age?Answer: Gramps was 60 when Mary was born.VerificationMary’s age Gramps’ age
1 61
![Page 130: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/130.jpg)
Answer for ChariseHow old was Mary’s grandfather when she was bornif for her first five consecutive birthdays, Mary’s ageexactly divided her grandfather’s age?Answer: Gramps was 60 when Mary was born.VerificationMary’s age Gramps’ age
1 61
3
![Page 131: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/131.jpg)
Answer for ChariseHow old was Mary’s grandfather when she was bornif for her first five consecutive birthdays, Mary’s ageexactly divided her grandfather’s age?Answer: Gramps was 60 when Mary was born.VerificationMary’s age Gramps’ age
1 61
3
4
![Page 132: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/132.jpg)
Answer for ChariseHow old was Mary’s grandfather when she was bornif for her first five consecutive birthdays, Mary’s ageexactly divided her grandfather’s age?Answer: Gramps was 60 when Mary was born.VerificationMary’s age Gramps’ age
1 61
3
4
5
![Page 133: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/133.jpg)
Answer for ChariseHow old was Mary’s grandfather when she was bornif for her first five consecutive birthdays, Mary’s ageexactly divided her grandfather’s age?Answer: Gramps was 60 when Mary was born.VerificationMary’s age Gramps’ age
1 61
2
3
4
5
![Page 134: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/134.jpg)
Answer for ChariseHow old was Mary’s grandfather when she was bornif for her first five consecutive birthdays, Mary’s ageexactly divided her grandfather’s age?Answer: Gramps was 60 when Mary was born.VerificationMary’s age Gramps’ age
1 61
2 62
3
4
5
![Page 135: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/135.jpg)
Answer for ChariseHow old was Mary’s grandfather when she was bornif for her first five consecutive birthdays, Mary’s ageexactly divided her grandfather’s age?Answer: Gramps was 60 when Mary was born.VerificationMary’s age Gramps’ age
1 61
2 62
3 63
4
5
![Page 136: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/136.jpg)
Answer for ChariseHow old was Mary’s grandfather when she was bornif for her first five consecutive birthdays, Mary’s ageexactly divided her grandfather’s age?Answer: Gramps was 60 when Mary was born.VerificationMary’s age Gramps’ age
1 61
2 62
3 63
4 64
5
![Page 137: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/137.jpg)
Answer for ChariseHow old was Mary’s grandfather when she was bornif for her first five consecutive birthdays, Mary’s ageexactly divided her grandfather’s age?Answer: Gramps was 60 when Mary was born.VerificationMary’s age Gramps’ age
1 61
2 62
3 63
4 64
5 65
![Page 138: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/138.jpg)
Answer for ChariseHow old was Mary’s grandfather when she was bornif for her first five consecutive birthdays, Mary’s ageexactly divided her grandfather’s age?Answer: Gramps was 60 when Mary was born.VerificationMary’s age Gramps’ age
1 61
2 62
3 63
4 64
5 65
Note that 60 = LCM [1, 3, 4, 5]
![Page 139: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/139.jpg)
Another solutionAnother solution to Charise’s problem is the number
![Page 140: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/140.jpg)
Another solutionAnother solution to Charise’s problem is the number
x = 5! = 1× 2× 3× 4× 5 = 120
![Page 141: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/141.jpg)
Another solutionAnother solution to Charise’s problem is the number
x = 5! = 1× 2× 3× 4× 5 = 120
This would be a very old Grandpaw.
![Page 142: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/142.jpg)
Prime DesertsA group of n consecutive composite integers is calleda prime desert of length n.
![Page 143: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/143.jpg)
Prime DesertsA group of n consecutive composite integers is calleda prime desert of length n.Show that
24, 25, 26, 27, 28
is a prime desert of length 5.
![Page 144: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/144.jpg)
Prime DesertsA group of n consecutive composite integers is calleda prime desert of length n.Show that
24, 25, 26, 27, 28
is a prime desert of length 5.Show that
90, 91, 92, 93, 94, 95, 96
is a prime desert of length 7.
![Page 145: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/145.jpg)
Prime DesertsA group of n consecutive composite integers is calleda prime desert of length n.Show that
24, 25, 26, 27, 28
is a prime desert of length 5.Show that
90, 91, 92, 93, 94, 95, 96
is a prime desert of length 7.How long can prime deserts be?
![Page 146: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/146.jpg)
Shoot for 100Problem Find 100 consecutive composite numbers.
![Page 147: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/147.jpg)
Shoot for 100Problem Find 100 consecutive composite numbers.Answer: Let
x = 1× 2× 3× 4× 5× · · · × 100× 101.
![Page 148: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/148.jpg)
Shoot for 100Problem Find 100 consecutive composite numbers.Answer: Let
x = 1× 2× 3× 4× 5× · · · × 100× 101.
Then the numbers
x+ 2, x+ 3, x+ 4, . . . , x+ 101
are all composite.
![Page 149: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/149.jpg)
How do you know?
To see this, simply observe that
![Page 150: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/150.jpg)
How do you know?
To see this, simply observe that
2 divides x+ 2
![Page 151: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/151.jpg)
How do you know?
To see this, simply observe that
2 divides x+ 2
3 divides x+ 3
![Page 152: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/152.jpg)
How do you know?
To see this, simply observe that
2 divides x+ 2
3 divides x+ 3
4 divides x+ 4
![Page 153: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/153.jpg)
How do you know?
To see this, simply observe that
2 divides x+ 2
3 divides x+ 3
4 divides x+ 4
......
...
![Page 154: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/154.jpg)
How do you know?
To see this, simply observe that
2 divides x+ 2
3 divides x+ 3
4 divides x+ 4
......
...
101 divides x+ 101
![Page 155: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/155.jpg)
How do you know?
To see this, simply observe that
2 divides x+ 2
3 divides x+ 3
4 divides x+ 4
......
...
101 divides x+ 101
What is the length of this list?
![Page 156: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/156.jpg)
Arbitrarily Long Prime Deserts
For any integer n, prove there exists n consecutivecompositive numbers.
![Page 157: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/157.jpg)
Arbitrarily Long Prime Deserts
For any integer n, prove there exists n consecutivecompositive numbers.Consequently, there exist arbitrarily long primedeserts.
![Page 158: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/158.jpg)
Arbitrarily Long Prime Deserts
For any integer n, prove there exists n consecutivecompositive numbers.Consequently, there exist arbitrarily long primedeserts.What does “arbitrarily long” mean?
![Page 159: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/159.jpg)
Arbitrarily Long Prime Deserts
For any integer n, prove there exists n consecutivecompositive numbers.Consequently, there exist arbitrarily long primedeserts.What does “arbitrarily long” mean?Conclusion: Primes are scarce.
![Page 160: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/160.jpg)
Euclid’s TheoremOver two thousand years ago Euclid (the geomtrydude) showed that we never run out of primes.
![Page 161: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/161.jpg)
Euclid’s TheoremOver two thousand years ago Euclid (the geomtrydude) showed that we never run out of primes.Euclid’s Theorem Given any positive integer n, thereis always a prime p larger than n.
![Page 162: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/162.jpg)
Euclid’s TheoremOver two thousand years ago Euclid (the geomtrydude) showed that we never run out of primes.Euclid’s Theorem Given any positive integer n, thereis always a prime p larger than n.To see why this Theorem is true, use Katie’s Theoremwith
![Page 163: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/163.jpg)
Euclid’s TheoremOver two thousand years ago Euclid (the geomtrydude) showed that we never run out of primes.Euclid’s Theorem Given any positive integer n, thereis always a prime p larger than n.To see why this Theorem is true, use Katie’s Theoremwith
x = n! = 1× 2× 3× 4× 5× · · · × n.
![Page 164: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/164.jpg)
Euclid’s TheoremOver two thousand years ago Euclid (the geomtrydude) showed that we never run out of primes.Euclid’s Theorem Given any positive integer n, thereis always a prime p larger than n.To see why this Theorem is true, use Katie’s Theoremwith
x = n! = 1× 2× 3× 4× 5× · · · × n.
Explain how Katie’s Theorem shows that none of theprimes less than n divide x+ 1.
![Page 165: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/165.jpg)
An Infinity of Primes
Because, no matter how large n is, there is a primenumber bigger than n, you never run out of primes,the following theorem is true.
![Page 166: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/166.jpg)
An Infinity of Primes
Because, no matter how large n is, there is a primenumber bigger than n, you never run out of primes,the following theorem is true.
Theorem There are infinitely many primes.
![Page 167: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/167.jpg)
An Infinity of Primes
Because, no matter how large n is, there is a primenumber bigger than n, you never run out of primes,the following theorem is true.
Theorem There are infinitely many primes.
Since there are infinitely many primes, they must beplentiful.
![Page 168: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/168.jpg)
Second thought
Wait a minute, there are infinitely many powers of 10(10, 100, 1000, 10000, etc.) but these do not seemvery plentiful.
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Second thought
Wait a minute, there are infinitely many powers of 10(10, 100, 1000, 10000, etc.) but these do not seemvery plentiful.There are only 12 powers of 10 up to one trillion.
![Page 170: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/170.jpg)
Second thought
Wait a minute, there are infinitely many powers of 10(10, 100, 1000, 10000, etc.) but these do not seemvery plentiful.There are only 12 powers of 10 up to one trillion.We want to know: Are big primes as rare as finding aparking space on campus?
![Page 171: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/171.jpg)
Second thought
Wait a minute, there are infinitely many powers of 10(10, 100, 1000, 10000, etc.) but these do not seemvery plentiful.There are only 12 powers of 10 up to one trillion.We want to know: Are big primes as rare as finding aparking space on campus?or as winning the lottery?
![Page 172: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/172.jpg)
Enter Probability
The question “Are the primes plentiful or scarce?”really boils down to:
![Page 173: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/173.jpg)
Enter Probability
The question “Are the primes plentiful or scarce?”really boils down to:If I pick a large integer (at random) what is theprobability it will be prime?
![Page 174: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/174.jpg)
Enter Probability
The question “Are the primes plentiful or scarce?”really boils down to:If I pick a large integer (at random) what is theprobability it will be prime?This question has an answer.
![Page 175: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/175.jpg)
Enter Probability
The question “Are the primes plentiful or scarce?”really boils down to:If I pick a large integer (at random) what is theprobability it will be prime?This question has an answer.
![Page 176: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/176.jpg)
The Prime Number TheoremPick a number at random between 1 and some largebound N .
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The Prime Number TheoremPick a number at random between 1 and some largebound N .Then the probability the number you chose is prime isapproximately 1 out of 2.3 times the number of digitsof N .
![Page 178: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/178.jpg)
The Prime Number TheoremPick a number at random between 1 and some largebound N .Then the probability the number you chose is prime isapproximately 1 out of 2.3 times the number of digitsof N .Example: The probability a one hundred digit numberchosen at random is prime is roughly
1/230
![Page 179: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/179.jpg)
Application to Encryption
RSA and many security systems and secret codesrequire large primes around 100 digits long.
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Application to Encryption
RSA and many security systems and secret codesrequire large primes around 100 digits long.In order to implement these codes, we need to knowhow likely it is that a 100 digit number chosen atrandom will be prime.
![Page 181: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/181.jpg)
Application to Encryption
RSA and many security systems and secret codesrequire large primes around 100 digits long.In order to implement these codes, we need to knowhow likely it is that a 100 digit number chosen atrandom will be prime.By the Prime Number Theorem, the answer is roughly
1/230.
![Page 182: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/182.jpg)
Application to Encryption
RSA and many security systems and secret codesrequire large primes around 100 digits long.In order to implement these codes, we need to knowhow likely it is that a 100 digit number chosen atrandom will be prime.By the Prime Number Theorem, the answer is roughly
1/230.If you avoid even numbers, the probability increases
to 1/115.
![Page 183: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/183.jpg)
Application to Encryption
RSA and many security systems and secret codesrequire large primes around 100 digits long.In order to implement these codes, we need to knowhow likely it is that a 100 digit number chosen atrandom will be prime.By the Prime Number Theorem, the answer is roughly
1/230.If you avoid even numbers, the probability increases
to 1/115.Why?
![Page 184: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/184.jpg)
Primes of form 4x + 1
![Page 185: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/185.jpg)
Primes of form 4x + 1
x 4x+ 1 Prime or Composite?
![Page 186: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/186.jpg)
Primes of form 4x + 1
x 4x+ 1 Prime or Composite?
1
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Primes of form 4x + 1
x 4x+ 1 Prime or Composite?
1 5
![Page 188: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/188.jpg)
Primes of form 4x + 1
x 4x+ 1 Prime or Composite?
1 5 Prime
![Page 189: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/189.jpg)
Primes of form 4x + 1
x 4x+ 1 Prime or Composite?
1 5 Prime
2
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Primes of form 4x + 1
x 4x+ 1 Prime or Composite?
1 5 Prime
2 9
![Page 191: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/191.jpg)
Primes of form 4x + 1
x 4x+ 1 Prime or Composite?
1 5 Prime
2 9 Composite
![Page 192: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/192.jpg)
Primes of form 4x + 1
x 4x+ 1 Prime or Composite?
1 5 Prime
2 9 Composite
3
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Primes of form 4x + 1
x 4x+ 1 Prime or Composite?
1 5 Prime
2 9 Composite
3 13
![Page 194: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/194.jpg)
Primes of form 4x + 1
x 4x+ 1 Prime or Composite?
1 5 Prime
2 9 Composite
3 13 Prime
![Page 195: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/195.jpg)
Primes of form 4x + 1
x 4x+ 1 Prime or Composite?
1 5 Prime
2 9 Composite
3 13 Prime
4
![Page 196: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/196.jpg)
Primes of form 4x + 1
x 4x+ 1 Prime or Composite?
1 5 Prime
2 9 Composite
3 13 Prime
4 17
![Page 197: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/197.jpg)
Primes of form 4x + 1
x 4x+ 1 Prime or Composite?
1 5 Prime
2 9 Composite
3 13 Prime
4 17 Prime
![Page 198: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/198.jpg)
Primes of form 4x + 1
x 4x+ 1 Prime or Composite?
1 5 Prime
2 9 Composite
3 13 Prime
4 17 Prime
5
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Primes of form 4x + 1
x 4x+ 1 Prime or Composite?
1 5 Prime
2 9 Composite
3 13 Prime
4 17 Prime
5 21
![Page 200: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/200.jpg)
Primes of form 4x + 1
x 4x+ 1 Prime or Composite?
1 5 Prime
2 9 Composite
3 13 Prime
4 17 Prime
5 21 Composite
![Page 201: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/201.jpg)
Primes of form 4x + 1
x 4x+ 1 Prime or Composite?
1 5 Prime
2 9 Composite
3 13 Prime
4 17 Prime
5 21 Composite
6
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Primes of form 4x + 1
x 4x+ 1 Prime or Composite?
1 5 Prime
2 9 Composite
3 13 Prime
4 17 Prime
5 21 Composite
6 25
![Page 203: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/203.jpg)
Primes of form 4x + 1
x 4x+ 1 Prime or Composite?
1 5 Prime
2 9 Composite
3 13 Prime
4 17 Prime
5 21 Composite
6 25 Composite
![Page 204: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/204.jpg)
Primes of form 4x + 1
x 4x+ 1 Prime or Composite?
1 5 Prime
2 9 Composite
3 13 Prime
4 17 Prime
5 21 Composite
6 25 Composite
7
![Page 205: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/205.jpg)
Primes of form 4x + 1
x 4x+ 1 Prime or Composite?
1 5 Prime
2 9 Composite
3 13 Prime
4 17 Prime
5 21 Composite
6 25 Composite
7 29
![Page 206: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/206.jpg)
Primes of form 4x + 1
x 4x+ 1 Prime or Composite?
1 5 Prime
2 9 Composite
3 13 Prime
4 17 Prime
5 21 Composite
6 25 Composite
7 29 Prime
![Page 207: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/207.jpg)
Primes of form 4x + 1
x 4x+ 1 Prime or Composite?
1 5 Prime
2 9 Composite
3 13 Prime
4 17 Prime
5 21 Composite
6 25 Composite
7 29 Prime
8
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Primes of form 4x + 1
x 4x+ 1 Prime or Composite?
1 5 Prime
2 9 Composite
3 13 Prime
4 17 Prime
5 21 Composite
6 25 Composite
7 29 Prime
8 33
![Page 209: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/209.jpg)
Primes of form 4x + 1
x 4x+ 1 Prime or Composite?
1 5 Prime
2 9 Composite
3 13 Prime
4 17 Prime
5 21 Composite
6 25 Composite
7 29 Prime
8 33 Composite
![Page 210: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/210.jpg)
Primes of form 4x + 1
x 4x+ 1 Prime or Composite?
1 5 Prime
2 9 Composite
3 13 Prime
4 17 Prime
5 21 Composite
6 25 Composite
7 29 Prime
8 33 Composite
9
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Primes of form 4x + 1
x 4x+ 1 Prime or Composite?
1 5 Prime
2 9 Composite
3 13 Prime
4 17 Prime
5 21 Composite
6 25 Composite
7 29 Prime
8 33 Composite
9 37
![Page 212: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/212.jpg)
Primes of form 4x + 1
x 4x+ 1 Prime or Composite?
1 5 Prime
2 9 Composite
3 13 Prime
4 17 Prime
5 21 Composite
6 25 Composite
7 29 Prime
8 33 Composite
9 37 Prime
![Page 213: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/213.jpg)
Primes of form 4x + 1
x 4x+ 1 Prime or Composite?
1 5 Prime
2 9 Composite
3 13 Prime
4 17 Prime
5 21 Composite
6 25 Composite
7 29 Prime
8 33 Composite
9 37 Prime
It has been proven that this list consists of infinitelymany primes.
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Twin PrimesPrimes sometimes come in pairs, two apart:
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Twin PrimesPrimes sometimes come in pairs, two apart:3 and 5
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Twin PrimesPrimes sometimes come in pairs, two apart:3 and 55 and 7
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Twin PrimesPrimes sometimes come in pairs, two apart:3 and 55 and 711 and 13
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Twin PrimesPrimes sometimes come in pairs, two apart:3 and 55 and 711 and 1317 and 19
![Page 219: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/219.jpg)
Twin PrimesPrimes sometimes come in pairs, two apart:3 and 55 and 711 and 1317 and 1933218925× 2169690 − 1 and 33218925× 2169690 + 1
![Page 220: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/220.jpg)
Twin PrimesPrimes sometimes come in pairs, two apart:3 and 55 and 711 and 1317 and 1933218925× 2169690 − 1 and 33218925× 2169690 + 1This last example has more than 51,000 digits!
![Page 221: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/221.jpg)
Twin PrimesPrimes sometimes come in pairs, two apart:3 and 55 and 711 and 1317 and 1933218925× 2169690 − 1 and 33218925× 2169690 + 1This last example has more than 51,000 digits!Such numbers are called twin primes.
![Page 222: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/222.jpg)
Twin PrimesPrimes sometimes come in pairs, two apart:3 and 55 and 711 and 1317 and 1933218925× 2169690 − 1 and 33218925× 2169690 + 1This last example has more than 51,000 digits!Such numbers are called twin primes.It is unknown (that is, unproven) whether there areinfinitely many twin primes.
![Page 223: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/223.jpg)
Primes of form x2 + 1
If you examine the table for y = x2 + 1, you will findmany primes.
![Page 224: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/224.jpg)
Primes of form x2 + 1
If you examine the table for y = x2 + 1, you will findmany primes.You need only consider when x is even.
![Page 225: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/225.jpg)
Primes of form x2 + 1
If you examine the table for y = x2 + 1, you will findmany primes.You need only consider when x is even. Why?
![Page 226: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/226.jpg)
Primes of form x2 + 1
If you examine the table for y = x2 + 1, you will findmany primes.You need only consider when x is even. Why?
x x2 + 1 Prime or Composite?
![Page 227: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/227.jpg)
Primes of form x2 + 1
If you examine the table for y = x2 + 1, you will findmany primes.You need only consider when x is even. Why?
x x2 + 1 Prime or Composite?
2
![Page 228: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/228.jpg)
Primes of form x2 + 1
If you examine the table for y = x2 + 1, you will findmany primes.You need only consider when x is even. Why?
x x2 + 1 Prime or Composite?
2 5
![Page 229: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/229.jpg)
Primes of form x2 + 1
If you examine the table for y = x2 + 1, you will findmany primes.You need only consider when x is even. Why?
x x2 + 1 Prime or Composite?
2 5 Prime
![Page 230: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/230.jpg)
Primes of form x2 + 1
If you examine the table for y = x2 + 1, you will findmany primes.You need only consider when x is even. Why?
x x2 + 1 Prime or Composite?
2 5 Prime
4
![Page 231: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/231.jpg)
Primes of form x2 + 1
If you examine the table for y = x2 + 1, you will findmany primes.You need only consider when x is even. Why?
x x2 + 1 Prime or Composite?
2 5 Prime
4 17
![Page 232: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/232.jpg)
Primes of form x2 + 1
If you examine the table for y = x2 + 1, you will findmany primes.You need only consider when x is even. Why?
x x2 + 1 Prime or Composite?
2 5 Prime
4 17 Prime
![Page 233: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/233.jpg)
Primes of form x2 + 1
If you examine the table for y = x2 + 1, you will findmany primes.You need only consider when x is even. Why?
x x2 + 1 Prime or Composite?
2 5 Prime
4 17 Prime
6
![Page 234: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/234.jpg)
Primes of form x2 + 1
If you examine the table for y = x2 + 1, you will findmany primes.You need only consider when x is even. Why?
x x2 + 1 Prime or Composite?
2 5 Prime
4 17 Prime
6 37
![Page 235: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/235.jpg)
Primes of form x2 + 1
If you examine the table for y = x2 + 1, you will findmany primes.You need only consider when x is even. Why?
x x2 + 1 Prime or Composite?
2 5 Prime
4 17 Prime
6 37 Prime
![Page 236: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/236.jpg)
Primes of form x2 + 1
If you examine the table for y = x2 + 1, you will findmany primes.You need only consider when x is even. Why?
x x2 + 1 Prime or Composite?
2 5 Prime
4 17 Prime
6 37 Prime
8
![Page 237: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/237.jpg)
Primes of form x2 + 1
If you examine the table for y = x2 + 1, you will findmany primes.You need only consider when x is even. Why?
x x2 + 1 Prime or Composite?
2 5 Prime
4 17 Prime
6 37 Prime
8 65
![Page 238: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/238.jpg)
Primes of form x2 + 1
If you examine the table for y = x2 + 1, you will findmany primes.You need only consider when x is even. Why?
x x2 + 1 Prime or Composite?
2 5 Prime
4 17 Prime
6 37 Prime
8 65 Composite
![Page 239: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/239.jpg)
Primes of form x2 + 1
If you examine the table for y = x2 + 1, you will findmany primes.You need only consider when x is even. Why?
x x2 + 1 Prime or Composite?
2 5 Prime
4 17 Prime
6 37 Prime
8 65 Composite
10
![Page 240: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/240.jpg)
Primes of form x2 + 1
If you examine the table for y = x2 + 1, you will findmany primes.You need only consider when x is even. Why?
x x2 + 1 Prime or Composite?
2 5 Prime
4 17 Prime
6 37 Prime
8 65 Composite
10 101
![Page 241: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/241.jpg)
Primes of form x2 + 1
If you examine the table for y = x2 + 1, you will findmany primes.You need only consider when x is even. Why?
x x2 + 1 Prime or Composite?
2 5 Prime
4 17 Prime
6 37 Prime
8 65 Composite
10 101 Prime
![Page 242: An Elementary Approach to Primesrichard/MSP/talks/katie.pdf · 2019. 5. 31. · Primes Professor Richard Blecksmith Dept. of Mathematical Sciences ... Solution: Find a common denominator](https://reader035.vdocuments.us/reader035/viewer/2022071413/610aaeb1f31d65727316e761/html5/thumbnails/242.jpg)
Primes of form x2 + 1
If you examine the table for y = x2 + 1, you will findmany primes.You need only consider when x is even. Why?
x x2 + 1 Prime or Composite?
2 5 Prime
4 17 Prime
6 37 Prime
8 65 Composite
10 101 Prime
No one knows whether there are infinitely manyprimes in this list.