In that post I pointed out that:
- 8675309 is a prime.
- 8675309 is a twin prime (8675311 is also prime).
- 8675309 is the hypotenuse of a (primitive) Pythagorean triple: 86753092 = 24602602+83191412.
What is the smallest number that would meet all three conditions above, that is it is a twin prime and it is also the hypotenuse of a primitive Pythagorean triple…. Ok, way to easy … what is the NEXT smallest?
Well, after some failed attempts to get WolframAlpha to answer the question for me, I decided to go at it the old fashioned (i.e., pre-2009) way. Google the terms, get lists of twin primes and primitive Pythagorean triples, and start checking by hand.
What I found was that there are many twin primes that are also hypotenuses of Pythagorean triples. In fact, all of the first 35 twin prime pairs had one twin which was such a hypotenuse. For example:
(3,5) ~ (3,4,5)
(5,7) ~ (3,4,5)
(11,13) ~ (5,12,13)
(17,19) ~ (8,15,17)
(29,31) ~ (20,21,29)
(41,43) ~ (9,40,41)
(59, 61) ~ (11,60,61)
(71,73) ~ (48,55,73)
(101, 103) ~ (20,99,101)
(107, 109) ~ (60,91,109)
(137, 139) ~ (88,105,137)
(149, 151) ~ (51,140,149)
~ skip some pairs ~
(881, 883) ~ (369,800,881)
I am not a number theorist, so I don’t know if this has any significance or not. I do know that if the hypotenuse in a Pythagorean triple is prime, then it must have the form 4n+1. So that explains why we don’t see both twins of a twin prime pair in a Pythagorean triple.
So let me ask my own question: what is the smallest twin prime pair (p,q) such that neither p nor q is the hypotenuse of a Pythagorean triple?