A232269 Number of ways to write 2*n + 1 = x + y (x, y > 0) with x^3 + y^2 and x^2 + y^2 both prime.

1, 3, 1, 2, 3, 2, 1, 6, 4, 1, 4, 6, 3, 8, 1, 1, 6, 1, 1, 9, 2, 4, 5, 3, 1, 2, 7, 4, 5, 8, 1, 12, 4, 4, 12, 3, 4, 9, 10, 1, 5, 9, 5, 11, 7, 4, 9, 2, 4, 19, 1, 1, 14, 4, 6, 16, 8, 5, 8, 7, 2, 11, 8, 1, 16, 3, 5, 9, 4, 3, 8, 8, 6, 16, 4, 3, 12, 13, 5, 11, 5, 3, 10, 10, 7, 12, 7, 4, 17, 20, 1, 17, 5, 6, 15, 4, 5, 18, 5, 7

Offset: 1

Author

Zhi-Wei Sun, Nov 22 2013

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for all n > 0. Also, any odd integer greater than one can be written as x + y (0 < x < y) with x^3 + y^2 prime.

The conjecture implies that there are infinitely many primes of the form x^3 + y^2 (x, y > 0) with x^2 + y^2 also prime.

Note that Ming-Zhi Zhang ever asked (cf. A036468) whether any odd integer greater than one can be written as x + y (x, y > 0) with x^2 + y^2 prime.

Examples

			a(10) = 1 since 2*10 + 1 = 1 + 20 with 1^2 + 20^2 = 1^3 + 20^2 = 401 prime.
a(15) = 1 since 2*15 + 1 = 25 + 6 with 25^2 + 6^2 = 661 and 25^3 + 6^2 = 15661 both prime.
a(40) = 1 since 2*40 + 1 = 55 + 26 with 55^2 + 26^2 = 3701 and 55^3 + 26^2 = 167051 both prime.
a(91) =1 since 2*91 + 1 = 85 + 98 with 85^2 + 98^2 = 16829 and 85^3 + 98^2 = 623729 both prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588.

Crossrefs

Cf. A000040, A036468, A066649, A220413, A232174, A232186.

Programs

Mathematica

a[n_]:=Sum[If[PrimeQ[x^3+(2n+1-x)^2]&&PrimeQ[x^2+(2n+1-x)^2],1,0],{x,1,2n}]
Table[a[n],{n,1,100}]