A283792 - OEIS

%I #29 Mar 18 2017 04:59:34

%S 109,157,229

%N Primes of the form (p^2 + q^2) / 2 such that (p^2 - q^2) / 24 is prime, where primes p > q > 3.

%C Union of primes of the form:

%C t^2 + 6^2 such that t and p = t+6 and q = t-6 are primes,

%C (2t)^2 + 3^2 such that t and p = 2t+3 and q = 2t-3 are primes,

%C (3t)^2 + 2^2 such that t and p = 3t+2 and q = 3t-2 are primes,

%C (6t)^2 + 1^2 such that t and p = 6t+1 and q = 6t-1 are primes.

%C Note: this last subset is empty.

%C We have p*q*(p^2-q^2)*(p^2+q^2) = p^5*q - p*q^5 == 0 (mod 5), so at least one of p, q, p^2-q^2, or p^2+q^2 must be divisible by 5. Thus, this sequence is finite and 229 is the last term. - _Robert Israel_, Mar 16 2017

%e Prime 109 = (13^2 + 7^2)/2 is a term since (13^2 - 7^2)/24 = 5 is prime.

%e Note: 109 = (2*5)^2 + 3^2, 157 = 11^2 + 6^2, and 229 = (3*5)^2 + 2^2.

%o (PARI) list(lim)=my(v=List(), p2, q2, t); lim\=1; lim=min(max(lim,9),229); forprime(p=3, sqrtint(2*lim-9), p2=p^2; forprime(q=3, min(sqrtint(2*lim-p2), p-2), q2=q^2; if((p2-q2)%24==0 && isprime(t=(p2+q2)/2) && isprime((p2-q2)/24), listput(v, t)))); Set(v) \\ _Charles R Greathouse IV_, Mar 17 2017

%Y Cf. A103739, A283562.

%K nonn,fini,full

%O 1,1

%A _Thomas Ordowski_ and _Altug Alkan_, Mar 16 2017