The Pythagorean Theorem is usually stated as

*a*² +

*b*² =

*c*². If all three numbers are whole numbers, it is called a Pythagorean triple. Here are some examples with small numbers as the entries.

3² + 4² = 5²

12² + 5² = 13²

15² + 8² = 17²

24² + 7² = 25²

21² + 20² = 29²

35² + 12² = 37²

In these examples, no number bigger than 1 divides all three numbers in the set, so these are the relatively prime Pythagorean triples. It is true that 6² + 8² = 10² (36+64=100), but this is just taking the 3-4-5 triangle and multiply each number by two.

Here are a few rules about the relatively prime Pythagorean triples.

**They are always of the form**² +

*odd***² =**

*even***². We can have**

*odd**even*² +

*even*² =

*even*², like 6-8-10, but all the numbers are divisible by two, so they aren't relatively prime. It turns out that

*odd*² +

*odd*² will be even, of course, but it can never be the square of an even number. All the even squares are divisible by 4 and the sum of two odd perfect squares will always have remainder 2 when divided by 4.

**There is a generating function**. Take any two distinct whole numbers

*j*and

*k*and let

*j*be the larger. Here are the formulas for

*a, b*and

*c*.

*a*=

*j*² -

*k*²

*b*= 2

*jk*

*c*=

*j*² +

*k*²

Tomorrow, we will look at what choices for

*j*and

*k*will produce the relatively prime Pythagorean triples.

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