login
A236574
Primes p with prime(p)^3 + 2*p^3 and p^3 + 2*prime(p)^3 both prime.
2
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
1,1
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.
LINKS
D. R. Heath-Brown, Primes represented by x^3 + 2y^3. Acta Mathematica 186 (2001), pp. 1-84.
EXAMPLE
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.
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Jan 29 2014
STATUS
approved