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A358313 Primes p such that 24*p is the difference of two squares of primes in three different ways.

%I A358313 #10 Nov 10 2022 07:43:04
%S A358313 5,7,13,17,23,103,6863,7523,11807,11833,22447,91807,100517,144167,
%T A358313 204013,221077,478937,531983,571867,752293,1440253,1647383,1715717,
%U A358313 1727527,1768667,2193707,2381963,2539393,2957237,3215783,3290647,3873713,4243997,4512223,4880963,4895777,5226107,5345317,5540063
%N A358313 Primes p such that 24*p is the difference of two squares of primes in three different ways.
%C A358313 The positive integer solutions of 24*p = x^2 - y^2 are (x = p+6, y = p-6), (x = 2*p+3, y = 2*p - 3), (x = 3*p+2, y = 3*p-2) and (x = 6*p+1, y=6*p-1). Since at least one of these is always divisible by 7, it is impossible for 24*p to be the difference of two squares of primes in 4 different ways.
%C A358313 Primes p such that three of the pairs (p +- 6), (2*p +- 3), (3*p +- 2), (6*p +- 1) are pairs of primes.
%C A358313 Except for 5, all terms == 3 or 7 (mod 10).
%H A358313 Robert Israel, <a href="/A358313/b358313.txt">Table of n, a(n) for n = 1..1000</a>
%e A358313 a(3) = 13 is a term because 13 is prime, 13 +- 6 = 19 and 7 are primes, 2*13 +- 3 = 29 and 23 are primes, and 3*13 +- 2 = 37 and 41 are primes.
%p A358313 filter:= proc(p) local t;
%p A358313   if not isprime(p) then return false fi;
%p A358313   t:= 0;
%p A358313   if isprime(p+6) and isprime(p-6) then t:= t+1 fi;
%p A358313   if isprime(2*p+3) and isprime(2*p-3) then t:= t+1 fi;
%p A358313   if isprime(3*p+2) and isprime(3*p-2) then t:= t+1 fi;
%p A358313   if isprime(6*p+1) and isprime(6*p-1) then t:= t+1 fi;
%p A358313   t = 3
%p A358313 end proc:
%p A358313 select(filter, [seq(i,i=3..10^7,2)]);
%Y A358313 Cf. A124865.
%K A358313 nonn
%O A358313 1,1
%A A358313 _J. M. Bergot_ and _Robert Israel_, Nov 08 2022