A084793 For p = prime(n), the number of solutions (g,h) to the equation g^h == h (mod p), where 0 < h < p and g is a primitive root of p; fixed points of the discrete logarithm with base g.

1, 0, 1, 3, 2, 4, 10, 3, 13, 15, 7, 7, 16, 16, 27, 25, 20, 13, 18, 30, 29, 30, 32, 51, 33, 34, 37, 44, 21, 53, 27, 39, 62, 35, 69, 28, 43, 43, 93, 89, 74, 42, 94, 62, 81, 54, 35, 73, 98, 74, 110, 101, 67, 86, 120, 143, 121, 109, 96, 89, 84, 135, 102, 139, 108, 159, 99, 108

Offset

1,4

Comments

For primes p > 3, there is always a solution to the equation.

References

R. K. Guy, Unsolved Problems in Number Theory, Second Edition, Springer, 1994, Section F9.

W. P. Zhang, On a problem of Brizolis, Pure Appl. Math., 11(suppl.):1-3, 1995.

Links

T. D. Noe, Table of n, a(n) for n = 1..1000

J. Holden and P. Moree, New conjectures and results for small cycles of the discrete logarithm, arXiv:math/0305305 [math.NT], 2003, published in: High Primes and Misdemeanours: lectures in honour of the 60th birthday of Hugh Cowie Williams, AMS, 2004, pp. 245-254.

Example

a(1) = 1 because p = 2, g = 1, and 1^1 == 1 (mod 2).
a(3) = 1 because p = 5 and 2^3 == 3 (mod 5) is the only solution.

Mathematica

Table[p=Prime[n]; x=PrimitiveRoot[p]; prims=Select[Range[p-1], GCD[ #1, p-1]==1&]; s=0; Do[g=PowerMod[x, prims[[i]], p]; Do[If[PowerMod[g, h, p]==h, s++ ], {h, p-1}], {i, Length[prims]}]; s, {n, 3, 100}]

Crossrefs

Sequence in context: A338246 A368225 A371267 * A033820 A095259 A260596

Adjacent sequences: A084790 A084791 A084792 * A084794 A084795 A084796

Keyword

nonn

Author

T. D. Noe, Jun 03 2003

Extensions

a(1) corrected by N. J. A. Sloane, Apr 14 2024 at the suggestion of José Hdz. Stgo.

Status

approved