A067319 Numbers n such that phi(n)^phi(n)+1 is prime.

1, 2, 3, 4, 5, 6, 8, 10, 12

Offset: 1

Author

Labos Elemer, Jan 15 2002

Keywords

• nonn

Comments

It is conjectured that the sequence of Fermat primes (A019434) is complete; if so then this sequence is complete:

Suppose that x is a positive integer for which x^x+1 is prime. If x has an odd prime factor p, then x^x + 1 = (x^(x/p))^p + 1 is divisible by x^(x/p) + 1, so it is not prime. So x must be a power of 2. Hence x^x is also a power of 2, so x^x+1 is a Fermat prime.

If there are no Fermat primes beyond the known ones (as is widely believed), then x must be 1, 2, or 4. Letting x=phi(n), it is easy to see that n must be one of the numbers listed. - Dean Hickerson, Feb 11 2002

Examples

			Cases n=1-12 are based on the primes 2, 5, 257.
		

Crossrefs

Cf. A063439, A000010.

Programs

Mathematica

ephiQ[n_]:=Module[{eph=EulerPhi[n]},PrimeQ[eph^eph+1]]; Select[ Range[ 20],ephiQ] (* Harvey P. Dale, Feb 23 2021 *)

PARI

isok(n) = isprime(eulerphi(n)^eulerphi(n) + 1); \\ Michel Marcus, Oct 07 2013