

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



0, 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
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OFFSET

1,4


COMMENTS

For prime 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.):13, 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


EXAMPLE

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


MATHEMATICA

Table[p=Prime[n]; x=PrimitiveRoot[p]; prims=Select[Range[p1], GCD[ #1, p1]==1&]; s=0; Do[g=PowerMod[x, prims[[i]], p]; Do[If[PowerMod[g, h, p]==h, s++ ], {h, p1}], {i, Length[prims]}]; s, {n, 3, 100}]


CROSSREFS

Sequence in context: A265356 A083164 A094962 * A033820 A095259 A260596
Adjacent sequences: A084790 A084791 A084792 * A084794 A084795 A084796


KEYWORD

nonn


AUTHOR

T. D. Noe, Jun 03 2003


STATUS

approved



