A Little-Known Pythagorean Curiosity // 61

that time, the Pentium was the central processing unit of most of the world's computers. See

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A Little-Known Pythagorean Curiosity It is well known that any two Pythagorean triples can be combined to yield another one. In fact, if

a2 þ b2 ¼ c 2

and

A2 þ B2 ¼ C2

then

ðaA À bBÞ2 þ ðaB þ bAÞ2 ¼ ðcCÞ2

However, there is a lesser-known feature of this method for combining Pythagorean triples. If you think of it as a kind of `multiplication' for triples, then we can define a triple to be prime if it is not the product of two smaller triples. Then every Pythagorean triple is a product of distinct prime Pythagorean triples; moreover, this `prime factorisation' of triples is essen- tially unique, except for some trivial distinctions which I won't go into here.

It turns out that the prime triples are those for which the hypotenuse is a prime number of the form 4k þ 1 and the other two sides are both non-zero, or the hypotenuse is 2 or a prime of the form 4k À 1 and one of the other sides is zero (a `degenerate' triple).

For instance, the 3­4­5 triple is prime, and so is the 5­12­13 triple, because their hypotenuses are both 4k þ 1 primes. The 0­7­7 triple is also prime. The 33­56­65 triple is not prime ­ it is the `product' of the 3­4­5 and 5­12­13 triples.

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Just thought you'd like to know.

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