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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

%I

%S 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,

%T 37,44,21,53,27,39,62,35,69,28,43,43,93,89,74,42,94,62,81,54,35,73,98,

%U 74,110,101,67,86,120,143,121,109,96,89,84,135,102,139,108,159,99,108

%N 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.

%C For prime p > 3, there is always a solution to the equation.

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

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

%H T. D. Noe, <a href="/A084793/b084793.txt">Table of n, a(n) for n = 1..1000</a>

%H J. Holden and P. Moree, <a href="http://www.arXiv.org/abs/math/0305305">New conjectures and results for small cycles of the discrete logarithm</a>

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

%t 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}]

%K nonn

%O 1,4

%A _T. D. Noe_, Jun 03 2003

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Last modified June 1 09:53 EDT 2020. Contains 334762 sequences. (Running on oeis4.)