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Search: id:A084793
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| A084793 |
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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. |
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+0
1
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| 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
(list; graph; listen)
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OFFSET
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1,4
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COMMENT
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For prime p > 3, there is always a solution to the equation.
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REFERENCES
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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.
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LINKS
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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
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EXAMPLE
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a(3) = 1 because 2^3 = 3 (mod 5) is the only solution.
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MATHEMATICA
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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}]
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CROSSREFS
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Adjacent sequences: A084790 A084791 A084792 this_sequence A084794 A084795 A084796
Sequence in context: A083762 A083164 A094962 this_sequence A033820 A095259 A137824
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KEYWORD
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nonn
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AUTHOR
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T. D. Noe (noe(AT)sspectra.com), Jun 03 2003
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