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

5, 7, 13, 17, 23, 103, 6863, 7523, 11807, 11833, 22447, 91807, 100517, 144167, 204013, 221077, 478937, 531983, 571867, 752293, 1440253, 1647383, 1715717, 1727527, 1768667, 2193707, 2381963, 2539393, 2957237, 3215783, 3290647, 3873713, 4243997, 4512223, 4880963, 4895777, 5226107, 5345317, 5540063

Offset: 1

Author

J. M. Bergot and Robert Israel, Nov 08 2022

Keywords

• nonn

Comments

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.

Primes p such that three of the pairs (p +- 6), (2*p +- 3), (3*p +- 2), (6*p +- 1) are pairs of primes.

Except for 5, all terms == 3 or 7 (mod 10).

Examples

			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.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A124865.

Programs

Maple

filter:= proc(p) local t;
  if not isprime(p) then return false fi;
  t:= 0;
  if isprime(p+6) and isprime(p-6) then t:= t+1 fi;
  if isprime(2*p+3) and isprime(2*p-3) then t:= t+1 fi;
  if isprime(3*p+2) and isprime(3*p-2) then t:= t+1 fi;
  if isprime(6*p+1) and isprime(6*p-1) then t:= t+1 fi;
  t = 3
end proc:
select(filter, [seq(i,i=3..10^7,2)]);