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