Pat B. wrote a response to my last post on the number 867-5309.

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: 8675309
^{2}= 2460260^{2}+8319141^{2}.

Pat asked:

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?

All primes of the form 4n+1 can br expressed as a sum of 2 squares. This result goes back to Fermat. (That’s the fellow whose famous conjecture was proved a few years back.)

I looked up a proof of the 4n+1 prime theorem.

But, it’s just a little too long to include here.

By:

Michael Welfordon August 27, 2009at 12:28 am

I am not sure that is possible. In any pair of twin primes, one of them must be congruent to 1 mod 4, and any prime congruent to one mod 4 can be written as the sum of two squares, r^2 + s^2, say, (I think you can use Gaussian integers to prove this) which means that the numbers r^2+s^2, 2rs and r^2-s^2 (assuming that r>s) form a Pythagorean triple, with r^2+s^2 the hypotenuse.

By:

Cathyon August 27, 2009at 6:37 am

Very nice! Thank you both for putting that to rest! Number theory never ceases to amaze me.

By:

Dave Richesonon August 27, 2009at 8:49 am

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Carnival of Mathematics #56 « Reasonable Deviationson August 28, 2009at 3:44 am

IS there such a thing as a twin primes in which neither prime is a Pythagorean hypotenuse? If the smaller of the pair is 4k+1, the larger is 4k+3. If the smaller is 4k+3, then the larger is 4k+5 or 4k+4+1 or 4(k+1)+1 or 4v+1. Since any prime 4v+1 is a Pythagorean hypotenuse, one twin prime is always a Pythagorean hypotenuse. Do you agree? Or have I overlooked something?

By:

dccon April 4, 2012at 9:02 am

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