A297620 Positive numbers n such that n^2 == p (mod q) and n^2 == q (mod p) for some consecutive primes p,q.

6, 10, 12, 24, 42, 48, 62, 72, 76, 84, 90, 93, 108, 110, 120, 122, 128, 145, 146, 174, 187, 188, 194, 204, 208, 215, 220, 228, 232, 240, 241, 264, 297, 306, 310, 314, 317, 326, 329, 336, 349, 357, 366, 372, 386, 408, 410, 423, 426, 431, 444, 454, 456, 468, 470, 474, 518, 522, 535, 538, 546, 548

Offset: 1

Author

Robert Israel and Thomas Ordowski, Jan 01 2018

Keywords

• nonn

Comments

Positive numbers n such that n^2 == p+q mod (p*q) for some consecutive primes p, q.

Each pair of consecutive primes p,q such that p is a quadratic residue mod q and p and q are not both == 3 (mod 4) contributes infinitely many members to the sequence.

Odd terms of this sequence are 93, 145, 187, 215, 241, 297, 317, 329, 349, 357, 423, 431, 535, ... - Altug Alkan, Jan 01 2018

Examples

			a(3) = 12 is in the sequence because 71 and 73 are consecutive primes with 12^2 == 73 (mod 71) and 12^2 == 71 (mod 73).
		

Links

• Robert Israel, Table of n, a(n) for n = 1..2305

Crossrefs

Contains A074924.

Programs

Maple

N:= 1000: # to get all terms <= N
R:= {}:
q:= 3:
while q < N^2 do
  p:= q;
  q:= nextprime(q);
  if ((p mod 4 <> 3) or (q mod 4 <> 3)) and numtheory:-quadres(q,p) = 1 then
    xp:= numtheory:-msqrt(q,p); xq:= numtheory:-msqrt(p,q);
    for sp in [-1,1] do for sq in [-1,1] do
      v:= chrem([sp*xp,sq*xq],[p,q]);
      R:= R union {seq(v+k*p*q, k = 0..(N-v)/(p*q))}
    od od;
  fi;
od:
sort(convert(R,list));