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
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
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, May 10 2021
STATUS
approved