A190797 For primes p and q=p+6 create primitive Pythagorean triangles with sides (q^2 - p^2)/2, (p^2 + q^2)/2, and p*q. If the two remainders of the middle and longest side modulo the shortest side are both prime, then p is in the sequence.

11, 23, 41, 83, 107, 167, 191, 263, 307, 347, 367, 461, 641, 653, 877, 881, 1103, 1187, 1367, 2081, 2393, 2677, 3607, 4283, 4357, 4967, 5081, 5231, 5387, 5471, 5651, 6037, 6197, 6311, 6353, 6857, 7823, 8117, 8693, 8747, 9221, 9743, 9851, 9923

Offset: 1

Author

J. M. Bergot, May 20 2011

Keywords

• nonn

Comments

The short side is 6p+18, the middle side p^2+6p, the long side 6p+18+p^2.

The first few values have more terms == 3 (mod 4) than 1 (mod 4), but this does not appear to be the case for later terms. - Franklin T. Adams-Watters, May 22 2011

Examples

			For p=41 and q=47, the sides are (47^2 - 41^20)/2=264, 41*47=1927 and (41^2 + 43^2)/2=1945; divide 1927 and 1945 through 264 to get remainders 79 and 97.  Since both are primes, p=41 is in the sequence.
		

Crossrefs

Cf. A023201.

Programs

PARI

forprime(p=5,10000,if(isprime(q=p+6),x=(q^2-p^2)/2;if(isprime(((q^2+p^2)/2)%x)&isprime(p*q%x),print1(p", "))))

Formula

If p=6k+5, then the remainders are 7 + 12*k and 25 + 12*k.

If p=6k+1, then the remainders are 7 + 24*k and 25 + 24*k.

Extensions

Corrected and extended by Franklin T. Adams-Watters, May 22 2011