

A306501


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


1



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
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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



MAPLE

N:= 3*10^6: # to get all terms <= N
S:= {seq(8*i+3, i=1..(N3)/8)} union {seq(8*i+5, i=1..(N5)/8)}:
for p in select(isprime, [$3..37]) do
R:= select(t > numtheory:legendre(t, p) = 1, {$1..p1});
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



KEYWORD

nonn


AUTHOR



STATUS

approved



