Recall that p+1 = (q-1)(r-1). So if p = 4n+3 is a non-hypotenuse, then for all divisors d of p+1, either q = d+1 or r = (p+1)/d+1 is composite. Here are the first 100 such primes: 67, 127, 151, 283, 307, 367, 439, 487, 547, 571, 587, 607, 643, 683, 727, 739, 751, 787, 811, 823, 907, 947, 967, 991, 1051, 1063, 1087, 1163, 1307, 1327, 1423, 1447, 1459, 1471, 1523, 1531, 1567, 1579, 1627, 1663, 1699, 1747, 1783, 1831, 1867, 1987, 1999, 2003, 2011, 2083, 2131, 2143, 2239, 2251, 2287, 2311, 2347, 2411, 2467, 2503, 2531, 2539, 2543, 2647, 2659, 2671, 2683, 2707, 2767, 2791, 2803, 2887, 3067, 3079, 3187, 3259, 3307, 3319, 3343, 3347, 3499, 3511, 3547, 3559, 3583, 3607, 3631, 3727, 3803, 3847, 3907, 3919, 3923, 3931, 3967, 4007, 4027, 4099, 4111, 4211 In my previous e-mail, the sequence there had one wrong term, and the condition I gave for a special class of non-hypotenuse primes was also wrong; it should read: If p is a prime of the form 4n+3 and (p+1)/4 is also prime but p+2 and (p+3)/2 are composite, then p cannot be a hypothenuse. If you think this stuff is interesting enough for OEIS, feel free to submit any related sequences. I'm not familiar with the new OEIS yet, and I don't have much time now. Thanks, Warut On Fri, Apr 15, 2011 at 9:20 PM, Joerg Arndt <arndt@jjj.de> wrote:
* 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
[...]
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun