
Glossary: Prime Pages: Top 5000: 
Almost everyone knows the following result credited to
the school of Pythagorus (though it was known to others much earlier):
Integer triples which satisfy this equation are Pythagorean triples. The most well known examples are (3,4,5) and (5,12,13). Notice we can multiple the entries in a triple by any integer and get another triple. For example (6,8,10), (9,12,15) and (15,20,25). The triples for which the entries are relatively prime are called primitive. So what have these to do with primes? Look at the two examples abovein each case two of the legs are prime numbers. Can all three be prime? (Try to answer this before reading on!) Hopefully you answered 'no.' In any primitive Pythagorean triple one of the three entries must be even, and it is easy to show that 2 can not be the side of a Pythagorean triple (look modulo 8). But two sides can be prime, and it is conjectured that they are infinitely often [Ribenboim95]. We will explore this further below. Most elementary number theory texts prove that all primitive triples (a,b,c) are given by the following: a = u^{2}  v^{2}, b = 2uv, c = u^{2} + v^{2}where u and v are relatively prime integers, not both odd. Notice that a is a difference of squares, so for it to be prime we need that u and v differ by 1. So a = 2v + 1, b = 2v^{2} + 2v, and c = 2v^{2} + 2v + 1.By Schinzel and Sierpinski's Hypothesis H we then expect to see infinitely many triples with two prime entries. Here are the first few: Dubner and Forbes [DF2000] not only found many such examles involving titanic primes, but te also looked for chains of triples (triangles) where the prime hypotenuse of one triple was a prime leg of the next. This requires finding a sequennce of primes p_{0}, p_{1}, p_{2}, ... satisfing p_{n+1} = (p_{n}^{2} + 1)/2. Here are some of their examples:
References:
Chris K. Caldwell © 19992018 (all rights reserved)
