A306501 - OEIS

%I #17 Apr 02 2019 06:07:39

%S 163,74093,92333,170957,222643,225077,253507,268637,292157,328037,

%T 360293,517613,524453,530837,613637,641093,679733,781997,847997,

%U 852893,979373,991027,1096493,1110413,1333963,1398053,1730357,1821893,2004917,2055307,2056147,2079173

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

%C 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 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 Robert Israel, <a href="/A306501/b306501.txt">Table of n, a(n) for n = 1..2694</a>

%p N:= 3*10^6: # to get all terms <= N

%p S:= {seq(8*i+3, i=1..(N-3)/8)} union {seq(8*i+5,i=1..(N-5)/8)}:

%p for p in select(isprime, [$3..37]) do

%p   R:= select(t -> numtheory:-legendre(t,p) = 1, {$1..p-1});

%p   if p mod 4 = 1 then S:= remove(t -> member(t mod p, R), S)

%p   else S:= select(t -> member(t mod p, R) = evalb(t mod 4 = 3), S)

%p   fi;

%p od:

%p sort(convert(select(isprime,S),list)); # _Robert Israel_, Mar 31 2019

%o (PARI) forprime(p=2, 1e6, if(sum(k=1, 37, isprime(k)*kronecker(k, p))==-12, print1(p, ", ")))

%Y Cf. A191089.

%K nonn

%O 1,1

%A _Jianing Song_, Feb 19 2019