A236574 Primes p with prime(p)^3 + 2*p^3 and p^3 + 2*prime(p)^3 both prime.

3, 79, 997, 2657, 3697, 4513, 6947, 8887, 9547, 16187, 22697, 26479, 31319, 37463, 39139, 39887, 43573, 43987, 45667, 47387, 47743, 47819, 48221, 54217, 56923, 57373, 74017, 74149, 74707, 75533, 93251, 100043

Offset: 1

Author

Zhi-Wei Sun, Jan 29 2014

Keywords

• nonn

Comments

Conjecture: This sequence has infinitely many terms.

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

Examples

			a(1) = 3 since prime(3)^3 + 2*3^3 = 125 + 54 = 179 and 3^3 + 2*prime(3)^3 = 27 + 2*125 = 277 are both prime, but 2^3 + 2*prime(2)^3 = 62 is composite.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• D. R. Heath-Brown, Primes represented by x^3 + 2y^3. Acta Mathematica 186 (2001), pp. 1-84.

Crossrefs

Cf. A000040, A000578, A173587, A220413, A236193.

Programs

Mathematica

p[n_]:=PrimeQ[Prime[n]^3+2*n^3]&&PrimeQ[n^3+2*Prime[n]^3]
n=0;Do[If[p[Prime[k]],n=n+1;Print[n," ",Prime[k]]],{k,1,10000}]
Select[Prime[Range[10000]],AllTrue[{Prime[#]^3+2*#^3,#^3+2*Prime[ #]^3}, PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Aug 20 2017 *)