In case you don't know, a Pythagorean triple is a set of three positive integers

There are many other examples of Pythagorean triples, but are there only so many of them or an infinite number of them? It seems likely that there is an infinite number of Pythagorean triples, but we mathematicians dare not rely on intuition. So here's a proof I came up with that demonstrates that Pythagorean triples are infinite in number:

I will be happy to answer any polite questions about this proof that you may have.

*a*,*b*, and*c*such that*a^*2 +*b*^2 =*c^*2. For example, if*a*= 3,*b*= 4, and*c*= 5, then*a^*2 +*b*^2 =*c^*2 = 3*^*2 + 4^2 = 5*^*2 which is true, of course, so*a*= 3,*b*= 4, and*c*= 5 is a Pythagorean triple.There are many other examples of Pythagorean triples, but are there only so many of them or an infinite number of them? It seems likely that there is an infinite number of Pythagorean triples, but we mathematicians dare not rely on intuition. So here's a proof I came up with that demonstrates that Pythagorean triples are infinite in number:

**Theorem**: There is an infinite number of Pythagorean triples.

*Proof*: Let

*n*be any nonnegative integer. Then 2^

*n*∙ 3, 2^

*n*∙ 4, and 2^

*n*∙ 5 are positive integers because the set of positive integers is closed under multiplication. So using these three values we have

(2^

*n*∙ 3)^2 + (2^*n*∙ 4)^2 = 9(2^(2*n*)) + 16(2^(2*n*)) = 25(2^(2*n*)) = (2^*n*∙ 5)^2.

So by the definition of a Pythagorean triple 2^n ∙ 3, 2^n ∙ 4, and 2^n ∙ 5 constitutes a Pythagorean triple. And since the set of nonnegative integers is infinite, then there is an infinite number of Pythagorean triples of the form 2^

*n*∙ 3, 2^*n*∙ 4, and 2^*n*∙ 5 each corresponding to some nonnegative integer*n*. Therefore, there is an infinite number of Pythagorean triples. □I will be happy to answer any polite questions about this proof that you may have.

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