A087037 Smallest integer x > 0 such that x^x + n is prime, or 0 if no such x exists.

1, 1, 2, 1, 444, 1, 2

Offset: 1

Author

Hugo Pfoertner, Jul 31 2003

Keywords

• nonn
• more
• hard

Comments

The sequence with the unknown terms a(8), a(11), a(17), a(24), a(41), a(53), a(59), a(65) indicated by ? (each of which exceeds 6000) begins: 1, 1, 2, 1, 444, 1, 2, ?, 2, 1, ?, 1, 2, 3, 2, 1, ?, 1, 2, 3, 4, 1, 6, ?, 2, 3, 2, 1, 30, 1, 6, 3, 2, 3, 6, 1, 2, 5, 2, 1, ?, 1, 2, 3, 58, 1, 6, 7, 2, 3017, 4, 1, ?, 35, 2, 3, 2, 1, ?, 1, 4, 3, 2, 19, ?, 1, 2, 27, 2, 1, 6, 1, 8, 3, 2, ..., where the value a(50)=3017 corresponds to a probable prime. [extended by Jon E. Schoenfield, Mar 17 2018, Mar 19 2018]

It is conjectured that such x always exists. - Dean Hickerson

From Farideh Firoozbakht and M. F. Hasler, Nov 27 2009: (Start)

We can show that for all n=(6k-1)^3, k > 0, there is no such x, which disproves the conjecture:

Since n=(6k-1)^3 is odd, x must be even, else x^x+n is even and composite.

If x == +-1 (mod 3), then x^x + n == (+-1)^2 + (-1)^3 == 0 (mod 3), i.e., divisible by 3 and therefore composite.

Finally, if x == 0 (mod 3), then x^x + n = (x^(x/3) + 6k-1)*(x^(2x/3) - x^(x/3)*(6k-1) + (6k-1)^2) is again composite. (End)

a(8) >= 36869. - Max Alekseyev, Sep 16 2013

Examples

			a(7)=2 because 2^2 + 7 = 11 is prime.
		

Links

• OpenPFGW Project, Primality Tester

Crossrefs

Cf. A000312 (n^n), A087038 (x^x+n is prime, x>1).

Cf. A166853 (x^x-n is prime). - Farideh Firoozbakht and M. F. Hasler, Nov 27 2009

Programs

PARI

a(n) = {my(x = 1); while (!isprime(x^x+n), x++); x;} \\ Michel Marcus, Mar 20 2018

Extensions

Name edited by Altug Alkan, Apr 01 2018