Pythagorean triples
Almost everyone knows the following result credited to the school of Pythagoras (though it was known to others much earlier):
- Pythagorean theorem
- The square of the length of the hypotenuse of a right triangle is the sum of the squares of the lengths of the two sides. This is usually expressed as a2 + b2 = c2.
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 above--in 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 = u2 - v2, b = 2uv, c = u2 + v2
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 = 2v2 + 2v, and c = 2v2 + 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:
| prime leg | even leg | hypotenuse |
|---|---|---|
| 3 | 4 | 5 |
| 5 | 12 | 13 |
| 11 | 60 | 61 |
| 19 | 180 | 181 |
| 29 | 420 | 421 |
| 59 | 1740 | 1741 |
| 61 | 1860 | 1861 |
| 71 | 2520 | 2521 |
| 79 | 3120 | 3121 |
Dubner and Forbes [DF2000] not only found many such examples 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 sequence of primes p0, p1, p2, ... satisfying pn+1 = (pn2 + 1)/2.
Here are some of their examples:
| number of triangles | first starting prime |
|---|---|
| 2 | 3 |
| 3 | 271 |
| 4 | 169219 |
| 5 | 356498179 |
| 6 | 2500282512131 |
References:
- DF2000
- H. Dubner and T. Forbes, "Prime Pythagorean triangles," (March 2000) Complete text: PDF. (Abstract available)
- Ribenboim95
- P. Ribenboim, The new book of prime number records, 3rd edition, Springer-Verlag, New York, NY, 1995. pp. xxiv+541, ISBN 0-387-94457-5. MR 96k:11112 [An excellent resource for those with some college mathematics. Basically a Guinness Book of World Records for primes with much of the relevant mathematics. The extensive bibliography is seventy-five pages.]