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

1,1

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

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

Prog

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

Crossrefs

Cf. A191089.

Sequence in context: A217963 A138200 A146504 * A247269 A145741 A247273

Adjacent sequences: A306498 A306499 A306500 * A306502 A306503 A306504

Keyword

nonn

Author

Jianing Song, Feb 19 2019

Status

approved