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

1,1

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).

Links

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

Example

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.

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)]);

Crossrefs

Cf. A124865.

Sequence in context: A253297 A163385 A368277 * A288449 A178218 A314323

Adjacent sequences: A358310 A358311 A358312 * A358314 A358315 A358316

Keyword

nonn

Author

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

Status

approved