A059771 - OEIS

%I #9 Sep 07 2023 16:00:10

%S 0,4,11,18,23,24,40,59,41,70,64,83,65,62,111,106,105,154,134,141,179,

%T 208,148,140,219,197,153,175,149,245,193,311,186,340,288,246,348,312,

%U 243,227,418,419,377,260,292,396,346,272,368,543,451,433,379,413,321

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

%C 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).

%H Robert Israel, <a href="/A059771/b059771.txt">Table of n, a(n) for n = 1..10000</a>

%F 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).

%e 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.

%p R:= 0: p:= 2: count:= 1:

%p while count < 100 do

%p   p:= nextprime(p);

%p   if NumberTheory:-QuadraticResidue(2,p)=1 then

%p     v:= NumberTheory:-ModularSquareRoot(2,p);

%p     R:= R, max(v,p-v);

%p     count:= count+1

%p   fi

%p od:

%p R; # _Robert Israel_, Sep 07 2023

%Y Cf. A038873, A059770, A059772.

%K nonn,look

%O 1,2

%A _Klaus Brockhaus_, Feb 21 2001