* Warut Roonguthai <warut822@gmail.com> [Apr 15. 2011 15:11]:
There are primes of the form 4n+3 that are not hypotenuses; the smallest ones are:
67, 127, 151, 227, 283, 307, 367, 439, 487, 547, 571, 587, 607, 643, 683, 727, 739, 751, 787, 811, 823, 907, 947, 967, 991, 1051.
This really want to be a sequence in the OEIS! (how did you determine these?)
If p is a prime of the form 4n+3 and (p+1)/4 is also prime but (p+3)/2 is composite (e.g., 67, 283, 547, 787, 907, 1051), then p cannot be a hypothenuse. So, assuming the first Hardy–Littlewood conjecture, there should be infinitely many non-hypotenuse primes of the form 4n+3.
This as well ? forprime(p=3,10^4,if( (p%4==3)&&isprime((p+1)/4)&&(!isprime((p+3)/2)),print1(p,", "))); 67, 283, 547, 787, 907, 1051, 1531, 1867, 2011, 2083, 2251, 2347, 2467, 2707, 2803, 3187, 3307, 3547, 3907, 3931, 4243, 4363, 4603, 4651, 4723, 5107, 5227, 5443, 6091, 6211, 6427, 6451, 6547, 6883, 7507, 8443, 8971, 9067, 9187, 9643, 9787, 9907,
Also by assuming the same conjecture, for every prime q there should be infinitely many primes r such that (q-1)(r-1)-1 is prime. In particular, every prime should be a leg.
Warut
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