A059771 Second solution of x^2 = 2 mod p for primes p such that a solution exists.

0, 4, 11, 18, 23, 24, 40, 59, 41, 70, 64, 83, 65, 62, 111, 106, 105, 154, 134, 141, 179, 208, 148, 140, 219, 197, 153, 175, 149, 245, 193, 311, 186, 340, 288, 246, 348, 312, 243, 227, 418, 419, 377, 260, 292, 396, 346, 272, 368, 543, 451, 433, 379, 413, 321

Offset

1,2

Comments

Solutions mod p are represented by integers from 0 to p-1. For p > 2: If x^2 = 2 has a solution mod p, then it has exactly two solutions and their sum is p; i is a solution mod p of x^2 = 2 iff p-i is a solution mod p of x^2 = 2. No integer occurs more than once in this sequence. Moreover, no integer (except 0) occurs both in this sequence and in sequence A059770 of the first solutions (Cf. A059772).

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

a(n) = second (larger) solution of x^2 = 2 mod p, where p is the n-th prime such that x^2 = 2 mod p has a solution, i.e. p is the n-th term of A038873. a(n) = 0 if x^2 = 2 mod p has one solution (only for p = 2).

Example

a(6) = 24 since 41 is the sixth term of A038873, 17 and 24 are the solutions mod 41 of x^2 = 2 and 24 is the larger one.

Maple

R:= 0: p:= 2: count:= 1:
while count < 100 do
  p:= nextprime(p);
  if NumberTheory:-QuadraticResidue(2, p)=1 then
    v:= NumberTheory:-ModularSquareRoot(2, p);
    R:= R, max(v, p-v);
    count:= count+1
  fi
od:
R; # Robert Israel, Sep 07 2023

Crossrefs

Cf. A038873, A059770, A059772.

Sequence in context: A133725 A050395 A368424 * A325618 A361131 A043409

Adjacent sequences: A059768 A059769 A059770 * A059772 A059773 A059774

Keyword

nonn,look

Author

Klaus Brockhaus, Feb 21 2001

Status

approved