Search: id:A084793
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%I A084793
%S A084793 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 A084793 37,44,21,53,27,39,62,35,69,28,43,43,93,89,74,42,94,62,81,54,35,73,98,
%U A084793 74,110,101,67,86,120,143,121,109,96,89,84,135,102,139,108,159,99,108
%N 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.
%C A084793 For prime p > 3, there is always a solution to the equation.
%D A084793 R. K. Guy, Unsolved Problems in Number Theory, Second Edition, Springer, 
               1994, Section F9.
%D A084793 W. P. Zhang, On a problem of Brizolis, Pure Appl. Math., 11(suppl.):1-3, 
               1995.
%H A084793 T. D. Noe, <a href="b084793.txt">Table of n, a(n) for n = 1..1000</a>
%H A084793 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 A084793 a(3) = 1 because 2^3 = 3 (mod 5) is the only solution.
%t A084793 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}]
%Y A084793 Sequence in context: A083762 A083164 A094962 this_sequence A033820 A095259 
               A137824
%Y A084793 Adjacent sequences: A084790 A084791 A084792 this_sequence A084794 A084795 
               A084796
%K A084793 nonn
%O A084793 1,4
%A A084793 T. D. Noe (noe(AT)sspectra.com), Jun 03 2003

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