A088790 Numbers k such that (k^k-1)/(k-1) is prime.

2, 3, 19, 31, 7547

Offset: 1

Author

T. D. Noe, Oct 16 2003

Keywords

• hard
• more
• nonn

Comments

Note that (k^k-1)/(k-1) is prime only if k is prime, in which case it equals cyclotomic(k,k), the k-th cyclotomic polynomial evaluated at x=k. This sequence is a subsequence of A070519. The number cyclotomic(7547,7547) is a probable prime found by H. Lifchitz. Are there only a finite number of these primes?

From T. D. Noe, Dec 16 2008: (Start)

The standard heuristic implies that there are an infinite number of these primes and that the next k should be between 10^10 and 10^11.

Let N = (7547^7547-1)/(7547-1) = A023037(7547). If N is prime, then the period of the Bell numbers modulo 7547 is N. See A054767. (End)

References

• R. K. Guy, Unsolved Problems in Theory of Numbers, 1994, A3.

Links

• Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avril-juin 2010, pages 30-38; included here with permission from the editors of Quadrature.
• Eric Weisstein's World of Mathematics, Cyclotomic Polynomial

Crossrefs

Cf. A070519 (cyclotomic(n, n) is prime).

Cf. A056826 ((n^n+1)/(n+1) is prime).

Programs

Mathematica

Do[p=Prime[n]; If[PrimeQ[(p^p-1)/(p-1)], Print[p]], {n, 100}]

PARI

is(n)=ispseudoprime((n^n-1)/(n-1)) \\ Charles R Greathouse IV, Feb 17 2017