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A306501 Primes p such that 2, 3, 5, 7, ..., 37 are all quadratic nonresidues modulo p.

%I A306501 #17 Apr 02 2019 06:07:39
%S A306501 163,74093,92333,170957,222643,225077,253507,268637,292157,328037,
%T A306501 360293,517613,524453,530837,613637,641093,679733,781997,847997,
%U A306501 852893,979373,991027,1096493,1110413,1333963,1398053,1730357,1821893,2004917,2055307,2056147,2079173
%N A306501 Primes p such that 2, 3, 5, 7, ..., 37 are all quadratic nonresidues modulo p.
%C A306501 The prime number 163 is famous for having 2, 3, 5, 7, ..., 37 as quadratic nonresidues, because the smallest prime having 2, 3, 5, 7, ..., 41 as quadratic nonresidues, namely 74093, is 453.5 times larger. This is related to the fact that the quadratic field Q[sqrt(-163)] is a unique factorization domain.
%C A306501 If p is in the sequence then so are all primes q with q == p (mod 29682952539240), where 29682952539240 = 2^3*3*5*7*11*13*17*19*23*29*31*37. In particular, the sequence is infinite. - _Robert Israel_, Mar 31 2019
%H A306501 Robert Israel, <a href="/A306501/b306501.txt">Table of n, a(n) for n = 1..2694</a>
%p A306501 N:= 3*10^6: # to get all terms <= N
%p A306501 S:= {seq(8*i+3, i=1..(N-3)/8)} union {seq(8*i+5,i=1..(N-5)/8)}:
%p A306501 for p in select(isprime, [$3..37]) do
%p A306501   R:= select(t -> numtheory:-legendre(t,p) = 1, {$1..p-1});
%p A306501   if p mod 4 = 1 then S:= remove(t -> member(t mod p, R), S)
%p A306501   else S:= select(t -> member(t mod p, R) = evalb(t mod 4 = 3), S)
%p A306501   fi;
%p A306501 od:
%p A306501 sort(convert(select(isprime,S),list)); # _Robert Israel_, Mar 31 2019
%o A306501 (PARI) forprime(p=2, 1e6, if(sum(k=1, 37, isprime(k)*kronecker(k, p))==-12, print1(p, ", ")))
%Y A306501 Cf. A191089.
%K A306501 nonn
%O A306501 1,1
%A A306501 _Jianing Song_, Feb 19 2019