A057683 Numbers k such that k^2 + k + 1, k^3 + k + 1 and k^4 + k + 1 are all prime.

1, 2, 5, 6, 12, 69, 77, 131, 162, 426, 701, 792, 1221, 1494, 1644, 1665, 2129, 2429, 2696, 3459, 3557, 3771, 4350, 4367, 5250, 5670, 6627, 7059, 7514, 7929, 8064, 9177, 9689, 10307, 10431, 11424, 13296, 13299, 13545, 14154, 14286, 14306, 15137

Offset: 1

Author

Harvey P. Dale, Oct 20 2000

Keywords

• easy
• nice
• nonn

Comments

After a(0) = 1, k^5 + k + 1 is never prime. Proof: k^5 + k + 1 = (k^2 + k + 1)*(k^3 - k^2 + 1). - Jonathan Vos Post, Oct 17 2007, edited by Robert Israel, Aug 01 2016

For n > 1, no terms == 1 (mod 3) or == 3 (mod 5). - Robert Israel, Jul 31 2016

Examples

			5 is included because 5^2 + 5 + 1 = 31, 5^3 + 5 + 1 = 131 and 5^4 + 5 + 1 = 631 are all prime.
		

Links

• Reinhard Zumkeller and Robert Israel, Table of n, a(n) for n = 1..10000 (n = 1..100 from Reinhard Zumkeller).

Crossrefs

Cf. A049407.

Cf. Subsequence of A219117; A010051.

Programs

Haskell

a057683 n = a057683_list !! (n-1)
a057683_list = filter (all (== 1) . p) [1..] where
   p x = map (a010051 . (+ (x + 1)) . (x ^)) [2..4]
-- Reinhard Zumkeller, Nov 12 2012

Magma

[n: n in [0..20000]|IsPrime(n^2+n+1) and IsPrime(n^3+n+1) and IsPrime(n^4+n+1)] // Vincenzo Librandi, Dec 20 2010

Maple

select(n -> isprime(n^4+n+1) and isprime(n^3+n+1) and isprime(n^2+n+1), [$1..50000]); # Robert Israel, Jul 31 2016

Mathematica

Select[Range[16000],And@@PrimeQ/@(Table[n^i+n+1,{i,2,4}]/.n->#)&]  (* Harvey P. Dale, Mar 28 2011 *)

Python

from sympy import isprime
A057683_list = [n for n in range(10**5) if isprime(n**2+n+1) and isprime(n**3+n+1) and isprime(n**4+n+1)] # Chai Wah Wu, Apr 02 2021