© t madas. a prime number or simply a prime, is a number with exactly two factors. these two...
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© T Madas
© T Madas
A prime number or simply a prime, is a number with exactly two factors.These two factors are always the number 1 and the prime number itself
All prime numbers are odd except number 2
1 is not a prime number.
2 is the smallest prime
There is no largest prime.
There are infinite prime numbers
© T Madas
The Sieve of Eratosthenes can be used to find the prime numbers up to 100 very quickly
100999897969594939291
90898887868584838281
80797877767574737271
70696867666564636261
60595857565554535251
50494847464544434241
40393837363534333231
30292827262524232221
20191817161514131211
10987654321
© T Madas
The Sieve of Eratosthenes can be used to find the prime numbers up to 100 very quickly
100999897969594939291
90898887868584838281
80797877767574737271
70696867666564636261
60595857565554535251
50494847464544434241
40393837363534333231
30292827262524232221
20191817161514131211
10987654321
Cross off the number 1
© T Madas
100999897969594939291
90898887868584838281
80797877767574737271
70696867666564636261
60595857565554535251
50494847464544434241
40393837363534333231
30292827262524232221
20191817161514131211
1098765432
The Sieve of Eratosthenes can be used to find the prime numbers up to 100 very quickly
Cross off all the multiples of 2 except 2
© T Madas
9997959391
8987858381
7977757371
6967656361
5957555351
4947454341
3937353331
2927252321
1917151311
9753
The Sieve of Eratosthenes can be used to find the prime numbers up to 100 very quickly
Cross off all the multiples of 3 except 3
2
© T Madas
979591
898583
79777371
676561
595553
49474341
373531
292523
19171311
75
The Sieve of Eratosthenes can be used to find the prime numbers up to 100 very quickly
Cross off all the multiples of 5 except 5
2 3
© T Madas
9791
8983
79777371
6761
5953
49474341
3731
2923
19171311
7
The Sieve of Eratosthenes can be used to find the prime numbers up to 100 very quickly
Cross off all the multiples of 7 except 7
2 3 5
© T Madas
97
8983
797371
6761
5953
474341
3731
2923
19171311
72 3 5
The Sieve of Eratosthenes can be used to find the prime numbers up to 100 very quickly
These are the prime numbers up to 100
© T Madas
ThePrime Numbersup to 200
© T Madas200199198197196195194193192191
190189188187186185184183182181
180179178177176175174173172171
170169168167166165164163162161
160159158157156155154153152151
150149148147146145144143142141
140139138137136135134133132131
130129128127126125124123122121
120119118117116115114113112111
110109108107106105104103102101
100999897969594939291
90898887868584838281
80797877767574737271
70696867666564636261
60595857565554535251
50494847464544434241
40393837363534333231
30292827262524232221
20191817161514131211
10987654321
Primes up to 200
© T Madas200199198197196195194193192191
190189188187186185184183182181
180179178177176175174173172171
170169168167166165164163162161
160159158157156155154153152151
150149148147146145144143142141
140139138137136135134133132131
130129128127126125124123122121
120119118117116115114113112111
110109108107106105104103102101
100999897969594939291
90898887868584838281
80797877767574737271
70696867666564636261
60595857565554535251
50494847464544434241
40393837363534333231
30292827262524232221
20191817161514131211
10987654321
Cross off:number 1multiples of 2 except 2multiples of 3 except 3multiples of 5 except 5multiples of 7 except 7multiples of 11 except 11multiples of 13 except 13
Primes up to 200
© T Madas200199198197196195194193192191
190189188187186185184183182181
180179178177176175174173172171
170169168167166165164163162161
160159158157156155154153152151
150149148147146145144143142141
140139138137136135134133132131
130129128127126125124123122121
120119118117116115114113112111
110109108107106105104103102101
100999897969594939291
90898887868584838281
80797877767574737271
70696867666564636261
60595857565554535251
50494847464544434241
40393837363534333231
30292827262524232221
20191817161514131211
10987654321
Primes up to 200Cross off:number 1multiples of 2 except 2multiples of 3 except 3multiples of 5 except 5multiples of 7 except 7multiples of 11 except 11multiples of 13 except 13
© T Madas
Interesting Facts Involving Primes
© T Madas
Every even number other than 2, can be written as the sum of two primes
16 = 3 + 13= 5 + 11
22 = 3 + 19= 11 + 11
40 = 3 + 37= 11 + 29
52 = 5 + 47= 11 + 41
= 17 + 23
Write these even numbers as the sum of two primes, at least three different ways
50 = 3 + 47= 7 + 43
100 = 3 + 97 = 11 + 89
150
= 11 + 139= 13 + 137
200
= 3 + 197= 7 + 193
= 19 + 131
= 13 + 37
= 17 + 83
= 13 + 187
© T Madas
Every even number other than 2, can be written as the sum of two primes
This statement is known as the Goldbach conjecture.In 1742 Christian Goldbach requested from Leonhard Euler, the most prolific mathematician of all times, for a proof for his conjecture.Euler could not prove this statement, nor has anyone else to this day, although no counter example can be found.
C Goldbach1690 - 1764
L Euler1707 - 1783
© T Madas
Every odd number other than 1, can be written as the sum of a prime and a power of 23 = 2 + 20
17 = 13 + 22
35 = 31 + 22= 19 + 24
81 = 79 + 21= 17 + 26
= 3 + 25
Write these odd numbers as the sum of a prime and a power of 2
25 = 23 + 21= 17 + 23
75 = 73 + 21= 71 + 22
125
= 109 + 24= 64 + 26
175
= 173 + 21= 167 + 23
= 67 + 23
= 47 + 27
© T Madas
Every even number can be written as the difference of 2 consecutive primes
2 = 5 – 3
4 = 11 – 7
6 = 29 – 23= 37 – 31
8 = 97 – 89
= 59 – 53
Write these even numbers as the difference of 2 consecutive primes
10 = 149 – 139
12 = 211 – 199
14 = 127 – 113
= 7 – 5
= 17 – 13
© T Madas
Every prime number greater than 3 is of the form 6n ± 1, where n is a natural number5 = 6 x 1 – 1
7 = 6 x 1 + 1
11 = 6 x 2 – 1
13= 6 x 2 + 1
17 = 6 x 3 – 1
19 = 6 x 3 + 1
23 = 6 x 4 – 1
29= 6 x 5 – 1
Careful because the converse statement is not true:Every number of the form 6n ± 1 is not a prime number
© T Madas
Every prime number of the form 4n + 1, where n is a natural number, can be written as the sum of 2 square numbers
5 = 4 x 1 + 1
13 = 4 x 3 + 1
17 = 4 x 4 + 1
29= 4 x 7 + 1
37 = 4 x 9 + 1
41 = 4 x 10 + 1
53 = 4 x 13 + 1
61= 4 x 15 + 1
= 4 + 1
= 9 + 4
= 16 + 1
= 25 + 4
= 36 + 1
= 25 + 16
= 49 + 4
= 36 + 25
© T Madas
Prime numbers which are of the form 2 n – 1,
where n is a natural number, are called Mersenne Primes1st Mersenne: 22 – 1 = 32nd Mersenne: 23 – 1 = 73rd Mersenne: 25 – 1 = 314th Mersenne: 27 – 1 = 1275th Mersenne: 213 – 1 = 81916th Mersenne: 217 – 1 = 1310717th Mersenne: 219 – 1 = 5242878th Mersenne: 231 – 1 = 21474836479th Mersenne: 261 – 1 = 230584300921369395110th Mersenne: 289 – 1 = 618970019642690137449562111On May 15, 2004, Josh Findley discovered the 41st known Mersenne Prime, 224,036,583 – 1. The number has 6 320 430 digits and is now the largest known prime number!
© T Madas
Perfect Numbers
© T Madas
A perfect number is a number which is equal to the sum of its factors, other than the number itself.
6 is perfect because: 1 + 2 + 3 = 6
A deficient number is a number which is more than the sum of its factors, other than the number itself
8 is deficient because: 1 + 2 + 4 = 7
An abundant, or excessive number is a number which is less than the sum of its factors, other than the number itself
12 is abundant because: 1 + 2 + 3 + 4 + 6 = 16
Classify the numbers from 3 to 30 according to these categories
© T Madas
A1+2+3+5+6+10+15=4230D1+2+4+8=1516
D129D1+3+5=915
P1+2+4+7+14=2828D1+2+7=1014
D1+3+9=1327D113
D1+2+13=1626A1+2+3+4+6=1612
D1+5=625D111
A1+2+3+4+6+12=2824D1+2+5=810
D123D1+3=49
D1+2+11=1422D1+2+4=78
D1+3+7=1121D17
A1+2+4+5+10=2220P1+2+3=66
D119D15
A1+2+3+6+9=2118D1+2=34
D117D13
© T Madas
The definition of a perfect number dates back to the ancient Greeks.
It was in fact Euclid that proved that a number of the form (2n – 1)2n – 1 will be a perfect number provided that:
2n – 1 is a prime, which is known as Mersenne Prime
Since the perfect numbers are connected to the Mersenne Primes, there are very few perfect numbers that we are aware of, given we only know 41 Mersenne Primes
© T Madas
The definition of a perfect number dates back to the ancient Greeks.
It was in fact Euclid that proved that a number of the form (2n – 1)2n – 1 will be a perfect number provided that:
2n – 1 is a prime, which is known as Mersenne Prime
1st Perfect: (22 – 1)22 – 1 = 3 x 2= 61st Mersenne: 22 – 1,
2nd Mersenne: 23 – 1,
3rd Mersenne: 25 – 1,
4th Mersenne: 27 – 1,
5th Mersenne: 213 – 1,
2nd Perfect: (23 – 1)23 – 1= 7 x 4= 28
3rd Perfect: (25 – 1)25 – 1 = 31 x 16 = 496
4th Perfect: (27 – 1)27 – 1 = 127 x 64 = 8128
5th Perfect: (213 – 1)213 – 1 = 33550336
© T Madas
© T Madas
Worksheets
© T Madas
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The S
ieve o
f Era
tost
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Cro
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Th
e S
ieve o
f Era
tost
henes
can b
e u
sed t
o fi
nd t
he p
rim
e n
um
bers
up t
o 1
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very
q
uic
kly
Cro
ss o
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nu
mb
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1m
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s o
f 2
excep
t 2
mu
ltip
les o
f 3
excep
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mu
ltip
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excep
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excep
t 7
Pri
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p t
o 1
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© T Madas
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Cro
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nu
mb
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1m
ult
iple
s o
f 2
excep
t 2
mu
ltip
les o
f 3
excep
t 3
mu
ltip
les o
f 5
excep
t 5
mu
ltip
les o
f 7
excep
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les o
f 1
1 e
xcep
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s o
f 1
3 e
xcep
t 1
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Pri
mes u
p t
o 2
00
The S
ieve o
f Era
tost
henes
can b
e u
sed t
o fi
nd
the p
rim
e n
um
bers
up t
o 1
00
very
quic
kly
© T Madas