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

163, 74093, 92333, 170957, 222643, 225077, 253507, 268637, 292157, 328037, 360293, 517613, 524453, 530837, 613637, 641093, 679733, 781997, 847997, 852893, 979373, 991027, 1096493, 1110413, 1333963, 1398053, 1730357, 1821893, 2004917, 2055307, 2056147, 2079173

Offset: 1

Author

Jianing Song, Feb 19 2019

Keywords

• nonn

Comments

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.

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

Links

• Robert Israel, Table of n, a(n) for n = 1..2694

Crossrefs

Cf. A191089.

Programs

Maple

N:= 3*10^6: # to get all terms <= N
S:= {seq(8*i+3, i=1..(N-3)/8)} union {seq(8*i+5,i=1..(N-5)/8)}:
for p in select(isprime, [$3..37]) do
  R:= select(t -> numtheory:-legendre(t,p) = 1, {$1..p-1});
  if p mod 4 = 1 then S:= remove(t -> member(t mod p, R), S)
  else S:= select(t -> member(t mod p, R) = evalb(t mod 4 = 3), S)
  fi;
od:
sort(convert(select(isprime,S),list)); # Robert Israel, Mar 31 2019

PARI

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