“Pythagorean triples” are integer solutions to the **Pythagorean Theorem, a ^{2} + b^{2} = c^{2}**. I like “triplets,” but “triples” seems to be the favored term. For a right triangle, the

**c**side is the hypotenuse, the side opposite the right angle. The

**a**side is the shorter of the two sides adjacent to the right angle. The first rules that I became aware of for determining a subset of Pythagorean triplets are as follows:

- Every odd number is the
**a**side of a Pythagorean triplet. - The
**b**side of a Pythagorean triplet is simply**(a**.^{2}– 1) / 2 - The
**c**side is**b + 1**.

Here, **a** and **c** are always odd; **b** is always even. These relationships hold because the difference between successive square numbers is successive odd numbers. Every odd number that is itself a square (and the square of every odd number is an odd number) thus makes for a Pythagorean triplet. Thus, the square of **7**, **49**, is the difference between**576**, the square of **24**, and **625**, the square of **25**, giving us the triplet **7,24,25**. Similarly, the square of **23**, **529**, is the difference between **69696**, the square of **264**, and **70225**, the square of **265**, giving us the triplet **23,264,265**.

The simplest triplet in the table, **1,0,1**, is not a triangle, and **a>b**, but it is a solution to the Pythagorean Theorem — a very trival one since **n,0,n**, where **n** is *any number*, works just as well (using **0** would even allow us to break Fermat’s Last Theorem).

Credit: http://www.friesian.com/pythag.htm

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