A283792 Primes of the form (p^2 + q^2) / 2 such that (p^2 - q^2) / 24 is prime, where primes p > q > 3.

109, 157, 229

Offset: 1

Author

Thomas Ordowski and Altug Alkan, Mar 16 2017

Keywords

• nonn
• fini
• full

Comments

Union of primes of the form:

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

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

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

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

Note: this last subset is empty.

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

Examples

			Prime 109 = (13^2 + 7^2)/2 is a term since (13^2 - 7^2)/24 = 5 is prime.
Note: 109 = (2*5)^2 + 3^2, 157 = 11^2 + 6^2, and 229 = (3*5)^2 + 2^2.
		

Crossrefs

Cf. A103739, A283562.

Programs

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