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

1,1

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

Links

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

Example

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).

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));

Crossrefs

Contains A074924.

Sequence in context: A046363 A101086 A342648 * A074924 A064166 A373678

Adjacent sequences: A297617 A297618 A297619 * A297621 A297622 A297623

Keyword

nonn

Author

Robert Israel and Thomas Ordowski, Jan 01 2018

Status

approved