A344173 Number of primes p < prime(n) of the form x^3 + 2*y^3 with x nonnegative and y + 1 prime such that p is a quadratic nonresidue modulo prime(n).

0, 1, 2, 1, 1, 1, 1, 2, 1, 3, 3, 3, 3, 3, 2, 2, 2, 4, 3, 1, 3, 4, 2, 3, 2, 3, 2, 3, 2, 4, 3, 4, 4, 4, 3, 1, 3, 5, 2, 4, 2, 3, 2, 3, 3, 3, 5, 1, 3, 3, 4, 1, 3, 4, 3, 2, 4, 4, 4, 2, 4, 4, 4, 3, 3, 5, 3, 3, 2, 3, 1, 2, 6, 4, 6, 2, 4, 3, 4, 3, 4, 5, 4, 2, 4, 5, 4, 1, 5, 3, 3, 6, 4, 4, 3, 4, 3, 3, 5, 4

Offset

1,3

Comments

In 2001 Heath-Brown proved that there are infinitely many primes of the form x^3 + 2*y^3 with x and y nonnegative integers.

Conjecture: (i) a(n) > 0 for all n > 1. In other words, for each odd prime p, there is a prime q < p of the form x^3 + 2*y^3 with x nonnegative and y + 1 prime such that q is a quadratic nonresidue modulo p.

(ii) For any prime p > 5 not equal to 29, there is a prime q < p of the form x^3 + 2*y^3 with x nonnegative and y + 1 prime such that q is a quadratic residue modulo p.

Part (i) of the conjecture verified for all odd primes p < 2*10^9.

We even conjecture further that for any prime p > 5 there is a prime q < p of the form x^3 + 2*y^3 with x nonnegative and y + 1 prime such that q is a primitive root modulo p.

See also A344174 for a similar conjecture.

Links

Table of n, a(n) for n=1..100.

D. R. Heath-Brown, Primes represented by x^3 + 2y^3, Acta Mathematica 186 (2001), 1-84.

Example

a(2) = 1, and the prime 0^3 + 2*(2-1)^3 = 2 is a quadratic nonresidue modulo prime(2) = 3.
a(6) = 1, and the prime 0^3 + 2*(2-1)^3 = 2 is a quadratic nonresidue modulo prime(6) = 13.
a(20) = 1, and the prime 1^3 + 2*(2-1)^3 = 17 is a quadratic nonresidue modulo prime(20) = 71.
a(48) = 1, and the prime 1^3 + 2*(2-1)^3 = 3 is a quadratic nonresidue modulo prime(48) = 223.
a(88) = 1, and the prime 3^3 + 2*(3-1)^3 = 43 is a quadratic nonresidue modulo prime(88) = 457.

Mathematica

tab={0}; Do[p:=p=Prime[n]; tt={}; Do[If[PrimeQ[b+1]&&PrimeQ[a^3+2b^3]&&JacobiSymbol[a^3+2b^3, p]==-1, tt=Append[tt, a^3+2b^3]], {a, 0, (p-1)^(1/3)}, {b, 1, ((p-1-a^3)/2)^(1/3)}]; tab=Append[tab, Length[Union[tt]]], {n, 2, 100}]; Print[tab]

Crossrefs

Cf. A000040, A000578, A173587, A220413, A344174.

Sequence in context: A358192 A156281 A002217 * A157047 A059342 A357481

Adjacent sequences: A344170 A344171 A344172 * A344174 A344175 A344176

Keyword

nonn

Author

Zhi-Wei Sun, May 10 2021

Status

approved