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TRANSCRIPT
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Chapter 2Pythagorean Triples
September 14, 2017
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
The Babylonians
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
The Beginnings
Pythagorean triples pre-date Pythagoras by over 1000 years.
This is Plimpton-322 and dates to 1800 BCE. It is a piece inthe G.A. Plimpton collection of Columbia University.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
The Beginnings
Pythagorean triples pre-date Pythagoras by over 1000 years.
This is Plimpton-322 and dates to 1800 BCE. It is a piece inthe G.A. Plimpton collection of Columbia University.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Plimpton-322
No one is sure why the Babylonians have these triples onthis cuneiform tablet. It is possible that they were the firstto come up with the theorem or that these were calculationsinvolving the teacher/student instruction.
The most likelypractical explanation is that they were measurements of landareas made by surveyors.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Plimpton-322
No one is sure why the Babylonians have these triples onthis cuneiform tablet. It is possible that they were the firstto come up with the theorem or that these were calculationsinvolving the teacher/student instruction. The most likelypractical explanation is that they were measurements of landareas made by surveyors.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Pythagoras of Samos
I c. 569 BC - (500-475) BC
I Settled in Crotona (Greek colony in southern Italy)where he founded a philosophical and religious school
I All things are numbers. Mathematics is the basis foreverything, and geometry is the highest form ofmathematical studies. The physical world can beunderstood through mathematics.
I Numbers have personalities, characteristics, strengthsand weaknesses.
I Certain symbols have a mystical significanceI Not sure how he died
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Pythagoras of Samos
I c. 569 BC - (500-475) BCI Settled in Crotona (Greek colony in southern Italy)
where he founded a philosophical and religious school
I All things are numbers. Mathematics is the basis foreverything, and geometry is the highest form ofmathematical studies. The physical world can beunderstood through mathematics.
I Numbers have personalities, characteristics, strengthsand weaknesses.
I Certain symbols have a mystical significanceI Not sure how he died
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Pythagoras of Samos
I c. 569 BC - (500-475) BCI Settled in Crotona (Greek colony in southern Italy)
where he founded a philosophical and religious schoolI All things are numbers. Mathematics is the basis for
everything, and geometry is the highest form ofmathematical studies. The physical world can beunderstood through mathematics.
I Numbers have personalities, characteristics, strengthsand weaknesses.
I Certain symbols have a mystical significanceI Not sure how he died
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Pythagoras of Samos
I c. 569 BC - (500-475) BCI Settled in Crotona (Greek colony in southern Italy)
where he founded a philosophical and religious schoolI All things are numbers. Mathematics is the basis for
everything, and geometry is the highest form ofmathematical studies. The physical world can beunderstood through mathematics.
I Numbers have personalities, characteristics, strengthsand weaknesses.
I Certain symbols have a mystical significanceI Not sure how he died
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Pythagoras of Samos
I c. 569 BC - (500-475) BCI Settled in Crotona (Greek colony in southern Italy)
where he founded a philosophical and religious schoolI All things are numbers. Mathematics is the basis for
everything, and geometry is the highest form ofmathematical studies. The physical world can beunderstood through mathematics.
I Numbers have personalities, characteristics, strengthsand weaknesses.
I Certain symbols have a mystical significance
I Not sure how he died
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Pythagoras of Samos
I c. 569 BC - (500-475) BCI Settled in Crotona (Greek colony in southern Italy)
where he founded a philosophical and religious schoolI All things are numbers. Mathematics is the basis for
everything, and geometry is the highest form ofmathematical studies. The physical world can beunderstood through mathematics.
I Numbers have personalities, characteristics, strengthsand weaknesses.
I Certain symbols have a mystical significanceI Not sure how he died
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Attributed to Pythagoras
I The sum of the angles of a triangle is equal to two rightangles.
I The theorem of Pythagoras - for a right-angled trianglethe square on the hypotenuse is equal to the sum of thesquares on the other two sides.
I Constructing figures of a given area and geometricalalgebra.
I The discovery of irrational numbers is attributed to thePythagoreans, but seems unlikely to have been the ideaof Pythagoras.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Attributed to Pythagoras
I The sum of the angles of a triangle is equal to two rightangles.
I The theorem of Pythagoras - for a right-angled trianglethe square on the hypotenuse is equal to the sum of thesquares on the other two sides.
I Constructing figures of a given area and geometricalalgebra.
I The discovery of irrational numbers is attributed to thePythagoreans, but seems unlikely to have been the ideaof Pythagoras.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Attributed to Pythagoras
I The sum of the angles of a triangle is equal to two rightangles.
I The theorem of Pythagoras - for a right-angled trianglethe square on the hypotenuse is equal to the sum of thesquares on the other two sides.
I Constructing figures of a given area and geometricalalgebra.
I The discovery of irrational numbers is attributed to thePythagoreans, but seems unlikely to have been the ideaof Pythagoras.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Attributed to Pythagoras
I The sum of the angles of a triangle is equal to two rightangles.
I The theorem of Pythagoras - for a right-angled trianglethe square on the hypotenuse is equal to the sum of thesquares on the other two sides.
I Constructing figures of a given area and geometricalalgebra.
I The discovery of irrational numbers is attributed to thePythagoreans, but seems unlikely to have been the ideaof Pythagoras.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
More of Pythagoras
There is no way to know if Pythagoras actually proved thistheorem.
Part of the reason is because they were so secretive. Most oftheir ideas were never made public during their lifetime.
They even supposedly killed members for talking about ideasinvolving irrational numbers in public.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
More of Pythagoras
There is no way to know if Pythagoras actually proved thistheorem.
Part of the reason is because they were so secretive. Most oftheir ideas were never made public during their lifetime.
They even supposedly killed members for talking about ideasinvolving irrational numbers in public.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
More of Pythagoras
There is no way to know if Pythagoras actually proved thistheorem.
Part of the reason is because they were so secretive. Most oftheir ideas were never made public during their lifetime.
They even supposedly killed members for talking about ideasinvolving irrational numbers in public.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
More on Pythagoras
Because of this, it took more than 200 more years forirrational numbers to be ‘discovered’.
They discovered that if you had a triangle with legs 1 and 1in a right triangle, the hypotenuse length of
√2 cannot be
expressed with a ruler with fractional parts.
This deeply disturbed Pythagoras, who believed that ‘All isNumber’ and they called such numbers a logon(unutterable).
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
More on Pythagoras
Because of this, it took more than 200 more years forirrational numbers to be ‘discovered’.
They discovered that if you had a triangle with legs 1 and 1in a right triangle, the hypotenuse length of
√2 cannot be
expressed with a ruler with fractional parts.
This deeply disturbed Pythagoras, who believed that ‘All isNumber’ and they called such numbers a logon(unutterable).
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
More on Pythagoras
Because of this, it took more than 200 more years forirrational numbers to be ‘discovered’.
They discovered that if you had a triangle with legs 1 and 1in a right triangle, the hypotenuse length of
√2 cannot be
expressed with a ruler with fractional parts.
This deeply disturbed Pythagoras, who believed that ‘All isNumber’ and they called such numbers a logon(unutterable).
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
The GreeksThe Greeks, among others were the ones 200 years later whodeveloped a way to deal with these numbers.
The Greeks found practical uses based on not havingmeasuring tools, like rulers. They used notched rope tomake 3-4-5 triangles to make right angles.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
The GreeksThe Greeks, among others were the ones 200 years later whodeveloped a way to deal with these numbers.
The Greeks found practical uses based on not havingmeasuring tools, like rulers. They used notched rope tomake 3-4-5 triangles to make right angles.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Primitive Pythagorean Triples
For us, the Pythagorean Theorem is not nearly as practical.
We all know the statement - which is ...
TheoremIn a right triangle, the lengths of the sides are in therelationship a2 + b2 = c2 where the legs are of lengths a andb and the hypotenuse is of length c.
What is of more interest to us are primitive Pythagoreantriples which means that (a, b, c) = 1.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Primitive Pythagorean Triples
For us, the Pythagorean Theorem is not nearly as practical.
We all know the statement - which is ...
TheoremIn a right triangle, the lengths of the sides are in therelationship a2 + b2 = c2 where the legs are of lengths a andb and the hypotenuse is of length c.
What is of more interest to us are primitive Pythagoreantriples which means that (a, b, c) = 1.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Primitive Pythagorean Triples
For us, the Pythagorean Theorem is not nearly as practical.
We all know the statement - which is ...
TheoremIn a right triangle, the lengths of the sides are in therelationship a2 + b2 = c2 where the legs are of lengths a andb and the hypotenuse is of length c.
What is of more interest to us are primitive Pythagoreantriples which means that (a, b, c) = 1.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Primitive Pythagorean Triples
For us, the Pythagorean Theorem is not nearly as practical.
We all know the statement - which is ...
TheoremIn a right triangle, the lengths of the sides are in therelationship a2 + b2 = c2 where the legs are of lengths a andb and the hypotenuse is of length c.
What is of more interest to us are primitive Pythagoreantriples which means that (a, b, c) = 1.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Primitive Pythagorean Triples
First, let’s prove that we have to have (a, b, c) = 1 to have aprimitive triple.
If a = 2n, b = 2m, and c = 2k , then we have (a, b, c) = 2.So, necessarily this is not the case and we need something tobe odd.
Can a = 2n and b = 2m? No because the sum of two evenscannot be odd.
Similarly, if we have one of a and b be even and the otherodd, then c must be odd, so we cannot have two of a, b andc be even.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Primitive Pythagorean Triples
First, let’s prove that we have to have (a, b, c) = 1 to have aprimitive triple.
If a = 2n, b = 2m, and c = 2k , then we have (a, b, c) = 2.So, necessarily this is not the case and we need something tobe odd.
Can a = 2n and b = 2m? No because the sum of two evenscannot be odd.
Similarly, if we have one of a and b be even and the otherodd, then c must be odd, so we cannot have two of a, b andc be even.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Primitive Pythagorean Triples
First, let’s prove that we have to have (a, b, c) = 1 to have aprimitive triple.
If a = 2n, b = 2m, and c = 2k , then we have (a, b, c) = 2.So, necessarily this is not the case and we need something tobe odd.
Can a = 2n and b = 2m?
No because the sum of two evenscannot be odd.
Similarly, if we have one of a and b be even and the otherodd, then c must be odd, so we cannot have two of a, b andc be even.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Primitive Pythagorean Triples
First, let’s prove that we have to have (a, b, c) = 1 to have aprimitive triple.
If a = 2n, b = 2m, and c = 2k , then we have (a, b, c) = 2.So, necessarily this is not the case and we need something tobe odd.
Can a = 2n and b = 2m? No because the sum of two evenscannot be odd.
Similarly, if we have one of a and b be even and the otherodd, then c must be odd, so we cannot have two of a, b andc be even.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Primitive Pythagorean Triples
First, let’s prove that we have to have (a, b, c) = 1 to have aprimitive triple.
If a = 2n, b = 2m, and c = 2k , then we have (a, b, c) = 2.So, necessarily this is not the case and we need something tobe odd.
Can a = 2n and b = 2m? No because the sum of two evenscannot be odd.
Similarly, if we have one of a and b be even and the otherodd, then c must be odd, so we cannot have two of a, b andc be even.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Primitive Pythagorean Triples
This leaves two options:
I a = 2n + 1, b = 2m + 1, c = 2k
I a = 2n + 1, b = 2m, c = 2k + 1
Let a = 2n + 1, b = 2m + 1, c = 2k . How can we decide ifthis scenario works?
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Primitive Pythagorean Triples
This leaves two options:
I a = 2n + 1, b = 2m + 1, c = 2k
I a = 2n + 1, b = 2m, c = 2k + 1
Let a = 2n + 1, b = 2m + 1, c = 2k . How can we decide ifthis scenario works?
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Primitive Pythagorean Triples
This leaves two options:
I a = 2n + 1, b = 2m + 1, c = 2k
I a = 2n + 1, b = 2m, c = 2k + 1
Let a = 2n + 1, b = 2m + 1, c = 2k . How can we decide ifthis scenario works?
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Primitive Pythagorean Triples
(2n + 1)2 + (2m + 1)2 = (2k)2
4n2 + 4n + 1 + 4m2 + 4m + 1 = 4k2
2(2n2 + 2n + 2m2 + 2m + 1) = 2(2k2)
This means that we have a common factor and so we do nothave (a, b, c) = 1.
So it must be the case that exactly one of a and b is evenand c must be odd.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Primitive Pythagorean Triples
(2n + 1)2 + (2m + 1)2 = (2k)2
4n2 + 4n + 1 + 4m2 + 4m + 1 = 4k2
2(2n2 + 2n + 2m2 + 2m + 1) = 2(2k2)
This means that we have a common factor and so we do nothave (a, b, c) = 1.
So it must be the case that exactly one of a and b is evenand c must be odd.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Primitive Pythagorean Triples
(2n + 1)2 + (2m + 1)2 = (2k)2
4n2 + 4n + 1 + 4m2 + 4m + 1 = 4k2
2(2n2 + 2n + 2m2 + 2m + 1) = 2(2k2)
This means that we have a common factor and so we do nothave (a, b, c) = 1.
So it must be the case that exactly one of a and b is evenand c must be odd.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Primitive Pythagorean Triples
(2n + 1)2 + (2m + 1)2 = (2k)2
4n2 + 4n + 1 + 4m2 + 4m + 1 = 4k2
2(2n2 + 2n + 2m2 + 2m + 1) = 2(2k2)
This means that we have a common factor and so we do nothave (a, b, c) = 1.
So it must be the case that exactly one of a and b is evenand c must be odd.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Primitive Pythagorean Triples
(2n + 1)2 + (2m + 1)2 = (2k)2
4n2 + 4n + 1 + 4m2 + 4m + 1 = 4k2
2(2n2 + 2n + 2m2 + 2m + 1) = 2(2k2)
This means that we have a common factor and so we do nothave (a, b, c) = 1.
So it must be the case that exactly one of a and b is evenand c must be odd.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Other Properties
I Exactly one of a, b is divisible by 3.
I Exactly one of a, b is divisible by 4.
I Exactly one of a, b, c is divisible by 5.
We know more than one can’t be, but why does one have tobe?
You will prove the first and third ones for homework, so wewill prove the second one a little later on.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Other Properties
I Exactly one of a, b is divisible by 3.
I Exactly one of a, b is divisible by 4.
I Exactly one of a, b, c is divisible by 5.
We know more than one can’t be, but why does one have tobe?
You will prove the first and third ones for homework, so wewill prove the second one a little later on.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Other Properties
I Exactly one of a, b is divisible by 3.
I Exactly one of a, b is divisible by 4.
I Exactly one of a, b, c is divisible by 5.
We know more than one can’t be, but why does one have tobe?
You will prove the first and third ones for homework, so wewill prove the second one a little later on.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Other Properties
I Exactly one of a, b, (b + a) or (b − a) is divisible by 7.
I (3, 4, 5) ⇒ 7|(3 + 4)
I (8, 15, 17) ⇒ 7|(15− 8)
I (16, 63, 65) ⇒ 7|63
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Other Properties
I Exactly one of a, b, (b + a) or (b − a) is divisible by 7.
I (3, 4, 5)
⇒ 7|(3 + 4)
I (8, 15, 17) ⇒ 7|(15− 8)
I (16, 63, 65) ⇒ 7|63
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Other Properties
I Exactly one of a, b, (b + a) or (b − a) is divisible by 7.
I (3, 4, 5) ⇒ 7|(3 + 4)
I (8, 15, 17) ⇒ 7|(15− 8)
I (16, 63, 65) ⇒ 7|63
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Other Properties
I Exactly one of a, b, (b + a) or (b − a) is divisible by 7.
I (3, 4, 5) ⇒ 7|(3 + 4)
I (8, 15, 17)
⇒ 7|(15− 8)
I (16, 63, 65) ⇒ 7|63
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Other Properties
I Exactly one of a, b, (b + a) or (b − a) is divisible by 7.
I (3, 4, 5) ⇒ 7|(3 + 4)
I (8, 15, 17) ⇒ 7|(15− 8)
I (16, 63, 65) ⇒ 7|63
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Other Properties
I Exactly one of a, b, (b + a) or (b − a) is divisible by 7.
I (3, 4, 5) ⇒ 7|(3 + 4)
I (8, 15, 17) ⇒ 7|(15− 8)
I (16, 63, 65)
⇒ 7|63
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Other Properties
I Exactly one of a, b, (b + a) or (b − a) is divisible by 7.
I (3, 4, 5) ⇒ 7|(3 + 4)
I (8, 15, 17) ⇒ 7|(15− 8)
I (16, 63, 65) ⇒ 7|63
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Other Properties
I There are no primitive Pythagorean triples where thehypotenuse and a leg differ by a prime power greaterthan 2.
I (3, 4, 5)
I (5, 12, 13)
I (7, 24, 25)
I (9, 40, 41)
I (8, 15, 17)
I (20, 21, 29)
I (12, 35, 37)
I (28, 45, 53)
I (16, 63, 65)
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Other Properties
I There are no primitive Pythagorean triples where thehypotenuse and a leg differ by a prime power greaterthan 2.
I (3, 4, 5)
I (5, 12, 13)
I (7, 24, 25)
I (9, 40, 41)
I (8, 15, 17)
I (20, 21, 29)
I (12, 35, 37)
I (28, 45, 53)
I (16, 63, 65)
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Other Properties
I There are no primitive Pythagorean triples where thehypotenuse and a leg differ by a prime power greaterthan 2.
I (3, 4, 5)
I (5, 12, 13)
I (7, 24, 25)
I (9, 40, 41)
I (8, 15, 17)
I (20, 21, 29)
I (12, 35, 37)
I (28, 45, 53)
I (16, 63, 65)
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Other Properties
I There are no primitive Pythagorean triples where thehypotenuse and a leg differ by a prime power greaterthan 2.
I (3, 4, 5)
I (5, 12, 13)
I (7, 24, 25)
I (9, 40, 41)
I (8, 15, 17)
I (20, 21, 29)
I (12, 35, 37)
I (28, 45, 53)
I (16, 63, 65)
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Other Properties
I There are no primitive Pythagorean triples where thehypotenuse and a leg differ by a prime power greaterthan 2.
I (3, 4, 5)
I (5, 12, 13)
I (7, 24, 25)
I (9, 40, 41)
I (8, 15, 17)
I (20, 21, 29)
I (12, 35, 37)
I (28, 45, 53)
I (16, 63, 65)
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Other Properties
I There are no primitive Pythagorean triples where thehypotenuse and a leg differ by a prime power greaterthan 2.
I (3, 4, 5)
I (5, 12, 13)
I (7, 24, 25)
I (9, 40, 41)
I (8, 15, 17)
I (20, 21, 29)
I (12, 35, 37)
I (28, 45, 53)
I (16, 63, 65)
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Other Properties
I There are no primitive Pythagorean triples where thehypotenuse and a leg differ by a prime power greaterthan 2.
I (3, 4, 5)
I (5, 12, 13)
I (7, 24, 25)
I (9, 40, 41)
I (8, 15, 17)
I (20, 21, 29)
I (12, 35, 37)
I (28, 45, 53)
I (16, 63, 65)
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Other Properties
I There are no primitive Pythagorean triples where thehypotenuse and a leg differ by a prime power greaterthan 2.
I (3, 4, 5)
I (5, 12, 13)
I (7, 24, 25)
I (9, 40, 41)
I (8, 15, 17)
I (20, 21, 29)
I (12, 35, 37)
I (28, 45, 53)
I (16, 63, 65)
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Other Properties
I There are no primitive Pythagorean triples where thehypotenuse and a leg differ by a prime power greaterthan 2.
I (3, 4, 5)
I (5, 12, 13)
I (7, 24, 25)
I (9, 40, 41)
I (8, 15, 17)
I (20, 21, 29)
I (12, 35, 37)
I (28, 45, 53)
I (16, 63, 65)
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Other Properties
I There are no primitive Pythagorean triples where thehypotenuse and a leg differ by a prime power greaterthan 2.
I (3, 4, 5)
I (5, 12, 13)
I (7, 24, 25)
I (9, 40, 41)
I (8, 15, 17)
I (20, 21, 29)
I (12, 35, 37)
I (28, 45, 53)
I (16, 63, 65)
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Other Properties
I The sum of the hypotenuse and the even leg is thesquare of an odd number.
I 5 + 4 = 9 = 32
I 41 + 40 = 81 = 92
I 20 + 29 = 49 = 72
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Other Properties
I The sum of the hypotenuse and the even leg is thesquare of an odd number.
I 5 + 4 = 9 = 32
I 41 + 40 = 81 = 92
I 20 + 29 = 49 = 72
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Other Properties
I The sum of the hypotenuse and the even leg is thesquare of an odd number.
I 5 + 4 = 9 = 32
I 41 + 40 = 81 = 92
I 20 + 29 = 49 = 72
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Other Properties
I The sum of the hypotenuse and the even leg is thesquare of an odd number.
I 5 + 4 = 9 = 32
I 41 + 40 = 81 = 92
I 20 + 29 = 49 = 72
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Other Properties
I The average of the hypotenuse and the odd leg is aperfect square.
I 3+52 = 4 = 22
I 25+72 = 16 = 42
I 35+372 = 36 = 62
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Other Properties
I The average of the hypotenuse and the odd leg is aperfect square.
I 3+52 = 4 = 22
I 25+72 = 16 = 42
I 35+372 = 36 = 62
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Other Properties
I The average of the hypotenuse and the odd leg is aperfect square.
I 3+52 = 4 = 22
I 25+72 = 16 = 42
I 35+372 = 36 = 62
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Other Properties
I The average of the hypotenuse and the odd leg is aperfect square.
I 3+52 = 4 = 22
I 25+72 = 16 = 42
I 35+372 = 36 = 62
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Finding Primitive Triples
So just how do we go about finding these triples?
The textbook uses Euclid’s Formula (Theorem 2.1) whichthey justify pretty well, but it seems to me like they aretaking too much credit for it ...
TheoremYou will get every primitive Pythagorean triple (a, b, c) withodd a and even b using
a = st b = s2−t2
2 c = s2+t2
2
where s > t ≥ 1 are chosen to be any odd integers with(s, t) = 1.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Finding Primitive Triples
So just how do we go about finding these triples?
The textbook uses Euclid’s Formula (Theorem 2.1) whichthey justify pretty well, but it seems to me like they aretaking too much credit for it ...
TheoremYou will get every primitive Pythagorean triple (a, b, c) withodd a and even b using
a = st b = s2−t2
2 c = s2+t2
2
where s > t ≥ 1 are chosen to be any odd integers with(s, t) = 1.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Finding Primitive Triples
So just how do we go about finding these triples?
The textbook uses Euclid’s Formula (Theorem 2.1) whichthey justify pretty well, but it seems to me like they aretaking too much credit for it ...
TheoremYou will get every primitive Pythagorean triple (a, b, c) withodd a and even b using
a = st b = s2−t2
2 c = s2+t2
2
where s > t ≥ 1 are chosen to be any odd integers with(s, t) = 1.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Proof of Euclid’s Formula
Proof.Consider
a2 + b2 = c2 ⇒ a2 = c2 − b2
= (c + b)(c − b)
Assertion 1: (c + b) and (c − b) share no common factors.
Suppose d |(c + b) and d |(c − b). Then, we have
d |((c + b) + (c − b))⇒ d |2cd |((c + b)− (c − b))⇒ d |2b
Implication? Since (b, c) = 1, d = 1 or d = 2.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Proof of Euclid’s Formula
Proof.Consider
a2 + b2 = c2 ⇒ a2 = c2 − b2
= (c + b)(c − b)
Assertion 1: (c + b) and (c − b) share no common factors.
Suppose d |(c + b) and d |(c − b). Then, we have
d |((c + b) + (c − b))⇒ d |2cd |((c + b)− (c − b))⇒ d |2b
Implication? Since (b, c) = 1, d = 1 or d = 2.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Proof of Euclid’s Formula
Proof.Consider
a2 + b2 = c2 ⇒ a2 = c2 − b2
= (c + b)(c − b)
Assertion 1: (c + b) and (c − b) share no common factors.
Suppose d |(c + b) and d |(c − b). Then, we have
d |((c + b) + (c − b))⇒ d |2cd |((c + b)− (c − b))⇒ d |2b
Implication? Since (b, c) = 1, d = 1 or d = 2.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Proof of Euclid’s Formula
Proof.Consider
a2 + b2 = c2 ⇒ a2 = c2 − b2
= (c + b)(c − b)
Assertion 1: (c + b) and (c − b) share no common factors.
Suppose d |(c + b) and d |(c − b).
Then, we have
d |((c + b) + (c − b))⇒ d |2cd |((c + b)− (c − b))⇒ d |2b
Implication? Since (b, c) = 1, d = 1 or d = 2.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Proof of Euclid’s Formula
Proof.Consider
a2 + b2 = c2 ⇒ a2 = c2 − b2
= (c + b)(c − b)
Assertion 1: (c + b) and (c − b) share no common factors.
Suppose d |(c + b) and d |(c − b). Then, we have
d |((c + b) + (c − b))⇒ d |2c
d |((c + b)− (c − b))⇒ d |2b
Implication? Since (b, c) = 1, d = 1 or d = 2.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Proof of Euclid’s Formula
Proof.Consider
a2 + b2 = c2 ⇒ a2 = c2 − b2
= (c + b)(c − b)
Assertion 1: (c + b) and (c − b) share no common factors.
Suppose d |(c + b) and d |(c − b). Then, we have
d |((c + b) + (c − b))⇒ d |2cd |((c + b)− (c − b))⇒ d |2b
Implication? Since (b, c) = 1, d = 1 or d = 2.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Proof of Euclid’s Formula
Proof.Consider
a2 + b2 = c2 ⇒ a2 = c2 − b2
= (c + b)(c − b)
Assertion 1: (c + b) and (c − b) share no common factors.
Suppose d |(c + b) and d |(c − b). Then, we have
d |((c + b) + (c − b))⇒ d |2cd |((c + b)− (c − b))⇒ d |2b
Implication?
Since (b, c) = 1, d = 1 or d = 2.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Proof of Euclid’s Formula
Proof.Consider
a2 + b2 = c2 ⇒ a2 = c2 − b2
= (c + b)(c − b)
Assertion 1: (c + b) and (c − b) share no common factors.
Suppose d |(c + b) and d |(c − b). Then, we have
d |((c + b) + (c − b))⇒ d |2cd |((c + b)− (c − b))⇒ d |2b
Implication? Since (b, c) = 1, d = 1 or d = 2.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Proof of Euclid’s Formula
Also,
d |(c + b)(c − b)⇒ d |(c2 − b2)
⇒ d |a2
How does this help? a2 is odd ... why? Because s and t areodd and a = st.
So, d = 1, implying that (a, b, c) = 1. That is, (c + b) and(c − b) share no common factors.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Proof of Euclid’s Formula
Also,
d |(c + b)(c − b)⇒ d |(c2 − b2)
⇒ d |a2
How does this help? a2 is odd ... why? Because s and t areodd and a = st.
So, d = 1, implying that (a, b, c) = 1. That is, (c + b) and(c − b) share no common factors.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Proof of Euclid’s Formula
Also,
d |(c + b)(c − b)⇒ d |(c2 − b2)
⇒ d |a2
How does this help?
a2 is odd ... why? Because s and t areodd and a = st.
So, d = 1, implying that (a, b, c) = 1. That is, (c + b) and(c − b) share no common factors.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Proof of Euclid’s Formula
Also,
d |(c + b)(c − b)⇒ d |(c2 − b2)
⇒ d |a2
How does this help? a2 is odd ... why?
Because s and t areodd and a = st.
So, d = 1, implying that (a, b, c) = 1. That is, (c + b) and(c − b) share no common factors.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Proof of Euclid’s Formula
Also,
d |(c + b)(c − b)⇒ d |(c2 − b2)
⇒ d |a2
How does this help? a2 is odd ... why? Because s and t areodd and a = st.
So, d = 1, implying that (a, b, c) = 1. That is, (c + b) and(c − b) share no common factors.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Proof of Euclid’s Formula
Also,
d |(c + b)(c − b)⇒ d |(c2 − b2)
⇒ d |a2
How does this help? a2 is odd ... why? Because s and t areodd and a = st.
So, d = 1, implying that (a, b, c) = 1. That is, (c + b) and(c − b) share no common factors.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Proof of Euclid’s Formula
Assertion 2: (c + b) and (c − b) are squares.
We need this to be true because our formula is built on s, t,s2 and t2.
(c + b)(c − b) = a2
What does this tell us? Either c + b = c − b or both aresquares. Why?
a ∈ N, so√
(c + b)(c − b) ∈ N, which is only possible ifboth are squares themselves.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Proof of Euclid’s Formula
Assertion 2: (c + b) and (c − b) are squares.
We need this to be true because our formula is built on s, t,s2 and t2.
(c + b)(c − b) = a2
What does this tell us? Either c + b = c − b or both aresquares. Why?
a ∈ N, so√
(c + b)(c − b) ∈ N, which is only possible ifboth are squares themselves.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Proof of Euclid’s Formula
Assertion 2: (c + b) and (c − b) are squares.
We need this to be true because our formula is built on s, t,s2 and t2.
(c + b)(c − b) = a2
What does this tell us? Either c + b = c − b or both aresquares. Why?
a ∈ N, so√
(c + b)(c − b) ∈ N, which is only possible ifboth are squares themselves.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Proof of Euclid’s Formula
Assertion 2: (c + b) and (c − b) are squares.
We need this to be true because our formula is built on s, t,s2 and t2.
(c + b)(c − b) = a2
What does this tell us?
Either c + b = c − b or both aresquares. Why?
a ∈ N, so√
(c + b)(c − b) ∈ N, which is only possible ifboth are squares themselves.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Proof of Euclid’s Formula
Assertion 2: (c + b) and (c − b) are squares.
We need this to be true because our formula is built on s, t,s2 and t2.
(c + b)(c − b) = a2
What does this tell us? Either c + b = c − b or both aresquares. Why?
a ∈ N, so√
(c + b)(c − b) ∈ N, which is only possible ifboth are squares themselves.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Proof of Euclid’s Formula
Assertion 2: (c + b) and (c − b) are squares.
We need this to be true because our formula is built on s, t,s2 and t2.
(c + b)(c − b) = a2
What does this tell us? Either c + b = c − b or both aresquares. Why?
a ∈ N, so√
(c + b)(c − b) ∈ N, which is only possible ifboth are squares themselves.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Proof of Euclid’s Formula
If c + b = c − b, then we have that b = −b, or that b = 0.Since this is a contradiction, we know c + b is a square, as isc − b.
Since they are squares, we have c + b = s2 and c − b = t2
for some s, t ∈ N, where s > t ≥ 1 and both are oddintegers such that (s, t) = 1.
Solving for a:
a2 = (c + b)(c − b)
= s2t2
⇒ a = st
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Proof of Euclid’s Formula
If c + b = c − b, then we have that b = −b, or that b = 0.Since this is a contradiction, we know c + b is a square, as isc − b.
Since they are squares, we have c + b = s2 and c − b = t2
for some s, t ∈ N, where s > t ≥ 1 and both are oddintegers such that (s, t) = 1.
Solving for a:
a2 = (c + b)(c − b)
= s2t2
⇒ a = st
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Proof of Euclid’s Formula
If c + b = c − b, then we have that b = −b, or that b = 0.Since this is a contradiction, we know c + b is a square, as isc − b.
Since they are squares, we have c + b = s2 and c − b = t2
for some s, t ∈ N, where s > t ≥ 1 and both are oddintegers such that (s, t) = 1.
Solving for a:
a2 = (c + b)(c − b)
= s2t2
⇒ a = st
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Proof of Euclid’s Formula
If c + b = c − b, then we have that b = −b, or that b = 0.Since this is a contradiction, we know c + b is a square, as isc − b.
Since they are squares, we have c + b = s2 and c − b = t2
for some s, t ∈ N, where s > t ≥ 1 and both are oddintegers such that (s, t) = 1.
Solving for a:
a2 = (c + b)(c − b)
= s2t2
⇒ a = st
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Proof of Euclid’s Formula
If c + b = c − b, then we have that b = −b, or that b = 0.Since this is a contradiction, we know c + b is a square, as isc − b.
Since they are squares, we have c + b = s2 and c − b = t2
for some s, t ∈ N, where s > t ≥ 1 and both are oddintegers such that (s, t) = 1.
Solving for a:
a2 = (c + b)(c − b)
= s2t2
⇒ a = st
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Proof of Euclid’s Formula
Solving for c:
c + b = s2
c − b = t2
2c = s2 + t2
c =s2 + t2
2
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Proof of Euclid’s Formula
Solving for c:
c + b = s2
c − b = t2
2c = s2 + t2
c =s2 + t2
2
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Proof of Euclid’s Formula
Solving for b:
c + b = s2
c − b = t2
2b = s2 − t2
b =s2 − t2
2
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Proof of Euclid’s Formula
Solving for b:
c + b = s2
c − b = t2
2b = s2 − t2
b =s2 − t2
2
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
The Babylonian Way
The Babylonians had their own way to get triples, thoughnot necessarily primitive.
Here is a translation from Babylonian cuneiform:
I 4 is the length and 5 is the diagonal. What is thebreadth?
I Its size is not known.
I 4 times 4 is 165 times 5 is 25
I You take 16 from 25 and there remains 9. What timeswhat shall take in order to get 9?
I 3 times 3 is 9
I 3 is the breadth.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
The Babylonian Way
The Babylonians had their own way to get triples, thoughnot necessarily primitive.
Here is a translation from Babylonian cuneiform:
I 4 is the length and 5 is the diagonal. What is thebreadth?
I Its size is not known.
I 4 times 4 is 165 times 5 is 25
I You take 16 from 25 and there remains 9. What timeswhat shall take in order to get 9?
I 3 times 3 is 9
I 3 is the breadth.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
The Babylonian Way
The Babylonians had their own way to get triples, thoughnot necessarily primitive.
Here is a translation from Babylonian cuneiform:
I 4 is the length and 5 is the diagonal. What is thebreadth?
I Its size is not known.
I 4 times 4 is 165 times 5 is 25
I You take 16 from 25 and there remains 9. What timeswhat shall take in order to get 9?
I 3 times 3 is 9
I 3 is the breadth.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
The Babylonian Way
The Babylonians had their own way to get triples, thoughnot necessarily primitive.
Here is a translation from Babylonian cuneiform:
I 4 is the length and 5 is the diagonal. What is thebreadth?
I Its size is not known.
I 4 times 4 is 165 times 5 is 25
I You take 16 from 25 and there remains 9. What timeswhat shall take in order to get 9?
I 3 times 3 is 9
I 3 is the breadth.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
The Babylonian Way
The Babylonians had their own way to get triples, thoughnot necessarily primitive.
Here is a translation from Babylonian cuneiform:
I 4 is the length and 5 is the diagonal. What is thebreadth?
I Its size is not known.
I 4 times 4 is 165 times 5 is 25
I You take 16 from 25 and there remains 9. What timeswhat shall take in order to get 9?
I 3 times 3 is 9
I 3 is the breadth.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
The Babylonian Way
The Babylonians had their own way to get triples, thoughnot necessarily primitive.
Here is a translation from Babylonian cuneiform:
I 4 is the length and 5 is the diagonal. What is thebreadth?
I Its size is not known.
I 4 times 4 is 165 times 5 is 25
I You take 16 from 25 and there remains 9. What timeswhat shall take in order to get 9?
I 3 times 3 is 9
I 3 is the breadth.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
The Babylonian Way
The Babylonians had their own way to get triples, thoughnot necessarily primitive.
Here is a translation from Babylonian cuneiform:
I 4 is the length and 5 is the diagonal. What is thebreadth?
I Its size is not known.
I 4 times 4 is 165 times 5 is 25
I You take 16 from 25 and there remains 9. What timeswhat shall take in order to get 9?
I 3 times 3 is 9
I 3 is the breadth.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
More Babylonians
I The Babylonians used π, but as 318 .
I They knew of theorems dealing with ratios of the sidesof similar triangles but lacking the concept of anglemeasure, they studied sides of triangles instead.
I They also used Fourier analysis (using simpletrigonometric functions) to compute astrologicalposition tables
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
More Babylonians
I The Babylonians used π, but as 318 .
I They knew of theorems dealing with ratios of the sidesof similar triangles but lacking the concept of anglemeasure, they studied sides of triangles instead.
I They also used Fourier analysis (using simpletrigonometric functions) to compute astrologicalposition tables
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
More Babylonians
I The Babylonians used π, but as 318 .
I They knew of theorems dealing with ratios of the sidesof similar triangles but lacking the concept of anglemeasure, they studied sides of triangles instead.
I They also used Fourier analysis (using simpletrigonometric functions) to compute astrologicalposition tables
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Other Formulas
For m > 1, we can get a Pythagorean triple from
(2m,m2 − 1,m2 + 1)
The early Greeks used, for (u, v) = 1 and v > u with oneeven and one odd,
(v2 − u2, 2uv , v2 + u2)
We could even use Fibonacci numbers.
(FnFn+3, 2Fn+1Fn+2,F2n+1F
2n+2)
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Other Formulas
For m > 1, we can get a Pythagorean triple from
(2m,m2 − 1,m2 + 1)
The early Greeks used, for (u, v) = 1 and v > u with oneeven and one odd,
(v2 − u2, 2uv , v2 + u2)
We could even use Fibonacci numbers.
(FnFn+3, 2Fn+1Fn+2,F2n+1F
2n+2)
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Other Formulas
For m > 1, we can get a Pythagorean triple from
(2m,m2 − 1,m2 + 1)
The early Greeks used, for (u, v) = 1 and v > u with oneeven and one odd,
(v2 − u2, 2uv , v2 + u2)
We could even use Fibonacci numbers.
(FnFn+3, 2Fn+1Fn+2,F2n+1F
2n+2)
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Middle-School Friendly Formulas
These formulas work, but are not necessarily accessible tomiddle school students. Some others, however, are. Onemethod uses unit fractions based on either consecutive evenor odd numbers.
Consider 3 and 5. The associated unit fractions would be 13
and 15 . Now, add them.
1
3+
1
5=
5
15+
3
15=
8
15
The two numbers in the sum are always the lengths of thelegs of a primitive Pythagorean triangle (here, (8, 15, 17)).
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Middle-School Friendly Formulas
These formulas work, but are not necessarily accessible tomiddle school students. Some others, however, are. Onemethod uses unit fractions based on either consecutive evenor odd numbers.
Consider 3 and 5. The associated unit fractions would be 13
and 15 . Now, add them.
1
3+
1
5=
5
15+
3
15=
8
15
The two numbers in the sum are always the lengths of thelegs of a primitive Pythagorean triangle (here, (8, 15, 17)).
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Middle-School Friendly Formulas
These formulas work, but are not necessarily accessible tomiddle school students. Some others, however, are. Onemethod uses unit fractions based on either consecutive evenor odd numbers.
Consider 3 and 5. The associated unit fractions would be 13
and 15 . Now, add them.
1
3+
1
5=
5
15+
3
15=
8
15
The two numbers in the sum are always the lengths of thelegs of a primitive Pythagorean triangle (here, (8, 15, 17)).
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Middle-School Friendly Formulas
These formulas work, but are not necessarily accessible tomiddle school students. Some others, however, are. Onemethod uses unit fractions based on either consecutive evenor odd numbers.
Consider 3 and 5. The associated unit fractions would be 13
and 15 . Now, add them.
1
3+
1
5=
5
15+
3
15=
8
15
The two numbers in the sum are always the lengths of thelegs of a primitive Pythagorean triangle (here, (8, 15, 17)).
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Middle-School Friendly Formulas
These formulas work, but are not necessarily accessible tomiddle school students. Some others, however, are. Onemethod uses unit fractions based on either consecutive evenor odd numbers.
Consider 3 and 5. The associated unit fractions would be 13
and 15 . Now, add them.
1
3+
1
5=
5
15+
3
15=
8
15
The two numbers in the sum are always the lengths of thelegs of a primitive Pythagorean triangle (here, (8, 15, 17)).
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Middle-School Friendly Formulas
Now, consider 4 and 6. The associated unit fractions are 14
and 16 . Again, add them.
1
4+
1
6=
3
12+
2
12=
5
12
The two numbers here in the (reduced) sum are always thelengths of the legs of a primitive Pythagorean triangle (here,(5, 12, 13)).
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Middle-School Friendly Formulas
Now, consider 4 and 6. The associated unit fractions are 14
and 16 . Again, add them.
1
4+
1
6=
3
12+
2
12=
5
12
The two numbers here in the (reduced) sum are always thelengths of the legs of a primitive Pythagorean triangle (here,(5, 12, 13)).
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Middle-School Friendly Formulas
Now, consider 4 and 6. The associated unit fractions are 14
and 16 . Again, add them.
1
4+
1
6=
3
12+
2
12=
5
12
The two numbers here in the (reduced) sum are always thelengths of the legs of a primitive Pythagorean triangle (here,(5, 12, 13)).
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Middle-School Friendly Formulas
Now, consider 4 and 6. The associated unit fractions are 14
and 16 . Again, add them.
1
4+
1
6=
3
12+
2
12=
5
12
The two numbers here in the (reduced) sum are always thelengths of the legs of a primitive Pythagorean triangle (here,(5, 12, 13)).
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Middle-School Friendly FormulasHere is another method that does not use unit fractions.
I Take any two fractions or whole numbers whoseproduct is 2.Here let’s use 1
5 and 10.Note: The fractions do not need to be in lowest form.
I Add 2 to each.We now have 11
5 and 12
I Now, cross-multiply (here we only have one fraction butwe could have had two)Now, we have 11 and 60
I These are the lengths of the two legs of a Pythagoreantriangle and we can use the Pythagorean theorem tofind the third length.
112 + 602 = 3721⇒ c = 61
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Middle-School Friendly FormulasHere is another method that does not use unit fractions.
I Take any two fractions or whole numbers whoseproduct is 2.Here let’s use 1
5 and 10.
Note: The fractions do not need to be in lowest form.
I Add 2 to each.We now have 11
5 and 12
I Now, cross-multiply (here we only have one fraction butwe could have had two)Now, we have 11 and 60
I These are the lengths of the two legs of a Pythagoreantriangle and we can use the Pythagorean theorem tofind the third length.
112 + 602 = 3721⇒ c = 61
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Middle-School Friendly FormulasHere is another method that does not use unit fractions.
I Take any two fractions or whole numbers whoseproduct is 2.Here let’s use 1
5 and 10.Note: The fractions do not need to be in lowest form.
I Add 2 to each.We now have 11
5 and 12
I Now, cross-multiply (here we only have one fraction butwe could have had two)Now, we have 11 and 60
I These are the lengths of the two legs of a Pythagoreantriangle and we can use the Pythagorean theorem tofind the third length.
112 + 602 = 3721⇒ c = 61
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Middle-School Friendly FormulasHere is another method that does not use unit fractions.
I Take any two fractions or whole numbers whoseproduct is 2.Here let’s use 1
5 and 10.Note: The fractions do not need to be in lowest form.
I Add 2 to each.We now have 11
5 and 12
I Now, cross-multiply (here we only have one fraction butwe could have had two)Now, we have 11 and 60
I These are the lengths of the two legs of a Pythagoreantriangle and we can use the Pythagorean theorem tofind the third length.
112 + 602 = 3721⇒ c = 61
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Middle-School Friendly FormulasHere is another method that does not use unit fractions.
I Take any two fractions or whole numbers whoseproduct is 2.Here let’s use 1
5 and 10.Note: The fractions do not need to be in lowest form.
I Add 2 to each.We now have 11
5 and 12
I Now, cross-multiply (here we only have one fraction butwe could have had two)Now, we have 11 and 60
I These are the lengths of the two legs of a Pythagoreantriangle and we can use the Pythagorean theorem tofind the third length.
112 + 602 = 3721⇒ c = 61
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Middle-School Friendly FormulasHere is another method that does not use unit fractions.
I Take any two fractions or whole numbers whoseproduct is 2.Here let’s use 1
5 and 10.Note: The fractions do not need to be in lowest form.
I Add 2 to each.We now have 11
5 and 12
I Now, cross-multiply (here we only have one fraction butwe could have had two)Now, we have 11 and 60
I These are the lengths of the two legs of a Pythagoreantriangle and we can use the Pythagorean theorem tofind the third length.
112 + 602 = 3721⇒ c = 61
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
How Many Proofs Are There?
Anyone want to guess how many distinct proofs there are forthe Pythagorean theorem?
There is one book alone, by early 20th century professorElisha Scott Loomis, that contains 367 proofs. And, JamesGarfield is credited with a proof from before his presidency.
Some are not all that interesting, but you can check outsome of them here:
Proofs of the Pythagorean Theorem
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
How Many Proofs Are There?
Anyone want to guess how many distinct proofs there are forthe Pythagorean theorem?
There is one book alone, by early 20th century professorElisha Scott Loomis, that contains 367 proofs. And, JamesGarfield is credited with a proof from before his presidency.
Some are not all that interesting, but you can check outsome of them here:
Proofs of the Pythagorean Theorem
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
How Many Proofs Are There?
Anyone want to guess how many distinct proofs there are forthe Pythagorean theorem?
There is one book alone, by early 20th century professorElisha Scott Loomis, that contains 367 proofs. And, JamesGarfield is credited with a proof from before his presidency.
Some are not all that interesting, but you can check outsome of them here:
Proofs of the Pythagorean Theorem
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Euclid’s Proof
1. Let ABC be a right triangle with right angle CAB.
B
A
C
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Euclid’s Proof
2. On each of the sides BC , AC and AB, draw squaresCBDE , ACIH and ABFG .
B
A
C
F
G
H
I
D E
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Euclid’s Proof
2. On each of the sides BC , AC and AB, draw squaresCBDE , ACIH and ABFG .
B
A
C
F
G
H
I
D E
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Euclid’s Proof
3. From A, draw a line parallel to BD. It will beperpendicular to BC at K and DE at L.
B
A
C
F
G
H
I
D E
K
L
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Euclid’s Proof
3. From A, draw a line parallel to BD. It will beperpendicular to BC at K and DE at L.
B
A
C
F
G
H
I
D E
K
L
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Euclid’s Proof
4. Join CF and AD to form triangles BCF and BDA.
B
A
C
F
G
H
I
D E
K
L
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Euclid’s Proof
4. Join CF and AD to form triangles BCF and BDA.
B
A
C
F
G
H
I
D E
K
L
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Euclid’s Proof
5. Angles CABand BAG are both right angles, so C , A andG are collinear. Same for B, A and H.
B
A
C
F
G
H
I
D E
K
L
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Euclid’s Proof6. Angles CBD and FBA are both right angles, so themeasure of angle ABD equals the measure of angle FBCsince both are the sum of a right angle and angle ABC .
B
A
C
F
G
H
I
D E
K
L
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Euclid’s Proof
7. Since mAB = mFB and mBD = mBC , triangles ABDand FBC are congruent.
B
A
C
F
G
H
I
D E
K
L
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Euclid’s Proof8. Since AKL is a straight line that is parallel to BD, thenparallelogram BDLK has twice the area of the triangle ABDbecause they share BD and have the same altitude BK .
B
A
C
F
G
H
I
D E
K
L
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Euclid’s Proof
9. Since C is collinear with A and G , square BAGF must betwice the area to triangle FBC .
B
A
C
F
G
H
I
D E
K
L
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Euclid’s Proof
10. Therefore, rectangle BDLK must have the same area assquare BAGF , which is AB2.
B
A
C
F
G
H
I
D E
K
L
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Euclid’s Proof11. Similarly, rectangle CKLE must have the same area assquare ACIH, which is AC 2. Note: similarly here means toconstruct segments AE and HL and proceed with a similarargument.
B
A
C
F
G
H
I
D E
K
L
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Euclid’s Proof
12. Adding gives AB2 + AC 2 = BD × BK + KL× KC .
B
A
C
F
G
H
I
D E
K
L
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Euclid’s Proof
13. Since BD = KL,BD × BK + KL× KC = BD(BK + KC ) = BD × BC .
B
A
C
F
G
H
I
D E
K
L
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Euclid’s Proof
14. Therefore, AB2 + AC 2 = BC 2 since CBDE is a square.
B
A
C
F
G
H
I
D E
K
L
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Now For Middle School
Of course, a proof like this would not be a happy time foryour students. But there are several ways we can prove thePythagorean theorem using means they can follow. Twosuch ways we will use here involve Sketchpad.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Exactly one of a and b is divisible by 4
Let a = st, b = s2−t2
2 and c = s2+t2
2 .
If 4|s or 4|t, we are done. So suppose neither is true.
What possible remainders could we get if we divide s and tby 4? We could get 1, 2 or 3. But s and t were necessarilychosen to be odd, so we would have to get 1 or 3 as ourreminder. We will think of 3 as −1.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Exactly one of a and b is divisible by 4
Let a = st, b = s2−t2
2 and c = s2+t2
2 .
If 4|s or 4|t, we are done. So suppose neither is true.
What possible remainders could we get if we divide s and tby 4? We could get 1, 2 or 3. But s and t were necessarilychosen to be odd, so we would have to get 1 or 3 as ourreminder. We will think of 3 as −1.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Exactly one of a and b is divisible by 4
Let a = st, b = s2−t2
2 and c = s2+t2
2 .
If 4|s or 4|t, we are done. So suppose neither is true.
What possible remainders could we get if we divide s and tby 4?
We could get 1, 2 or 3. But s and t were necessarilychosen to be odd, so we would have to get 1 or 3 as ourreminder. We will think of 3 as −1.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Exactly one of a and b is divisible by 4
Let a = st, b = s2−t2
2 and c = s2+t2
2 .
If 4|s or 4|t, we are done. So suppose neither is true.
What possible remainders could we get if we divide s and tby 4? We could get 1, 2 or 3. But s and t were necessarilychosen to be odd, so we would have to get 1 or 3 as ourreminder. We will think of 3 as −1.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Exactly one of a and b is divisible by 4
If these are the remainders for s and t, we will get aremainder of 1 for s2 and t2 when dividing by 4.
Whether s and t had the same remainder or different ones,s2 and t2 have the same remainder when dividing by 4, sos2 − t2 will be divisible by 4.
So, if a = st is not divisible by 4, then s2 − t2 must be.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Exactly one of a and b is divisible by 4
If these are the remainders for s and t, we will get aremainder of 1 for s2 and t2 when dividing by 4.
Whether s and t had the same remainder or different ones,s2 and t2 have the same remainder when dividing by 4, sos2 − t2 will be divisible by 4.
So, if a = st is not divisible by 4, then s2 − t2 must be.
1 History ofPythagoreanTriples
2 PrimitivePythagoreanTriples
3 Other Properties
4 Finding PrimitivePythagoreanTriples
5 The Babylonians
6 Other Ways toFind Triples
7 Proving thePythagoreanTheorem
8 Middle SchoolFriendly Proofs
9 That Proof IOwe You
Exactly one of a and b is divisible by 4
If these are the remainders for s and t, we will get aremainder of 1 for s2 and t2 when dividing by 4.
Whether s and t had the same remainder or different ones,s2 and t2 have the same remainder when dividing by 4, sos2 − t2 will be divisible by 4.
So, if a = st is not divisible by 4, then s2 − t2 must be.