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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.
3
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
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: A368225 A371267 A374726 * A033820 A095259 A260596
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