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Odd numbers k such that x^2 == 2 (mod k) has no solution.
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%I #13 Aug 22 2020 17:43:20

%S 3,5,9,11,13,15,19,21,25,27,29,33,35,37,39,43,45,51,53,55,57,59,61,63,

%T 65,67,69,75,77,81,83,85,87,91,93,95,99,101,105,107,109,111,115,117,

%U 121,123,125,129,131,133,135,139,141,143,145,147,149,153,155,157,159,163

%N Odd numbers k such that x^2 == 2 (mod k) has no solution.

%C Complement of A058529 over the odd numbers: odd numbers k such that x^2 == 2 (mod k) has solutions.

%C Odd numbers k such that at least one prime factor of k is congruent to 3 or 5 modulo 8 (at least one prime factor is in A003629).

%C Also odd terms in A025020.

%H Robert Israel, <a href="/A329095/b329095.txt">Table of n, a(n) for n = 1..10000</a>

%e x^2 == 2 (mod 45) has no solution, so 45 is a term.

%p filter:= proc(t) (numtheory:-factorset(t) mod 8) intersect {3,5} <> {} end proc:

%p select(filter, [seq(i,i=1..1000,2)]); # _Robert Israel_, Nov 05 2019

%t Reap[Do[If[AnyTrue[FactorInteger[k][[All, 1]], MatchQ[Mod[#, 8], 3|5]&], Sow[k]], {k, 1, 999, 2}]][[2, 1]] (* _Jean-François Alcover_, Aug 22 2020 *)

%o (PARI) isA329095(k) = (k%2) && !issquare(Mod(2,k))

%Y Cf. A003629. A047621 is a subsequence.

%Y Cf. A058529, A057126, A025020 (numbers k such that x^2 == 2 (mod k) has no solution).

%K nonn,easy

%O 1,1

%A _Jianing Song_, Nov 04 2019