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A152156
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Minimal residues of Pepin's Test for Fermat Numbers using either 5 or 10 for the base.
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4
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-1, 0, -1, -1, -1, -810129131, -1220845804166146754, 6964187975677595099156927503004398881, 14553806122642016769237504145596730952769427034161327480375008633175279343120
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,6
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COMMENTS
| For n=0 or n>=2 the Fermat Number F(n) is prime if and only if 5^((F(n) - 1)/2) is congruent to -1 (mod F(n)).
5 was the base originally used by Pepin. The base 10 gives the same results.
Any positive integer k for which the Jacobi symbol (k|F(n)) is -1 can be used as the base instead.
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REFERENCES
| M. Krizek, F. Luca & L. Somer, 17 Lectures on Fermat Numbers, Springer-Verlag NY 2001, pp. 42-43.
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LINKS
| Dennis Martin, Table of n, a(n) for n = 0..11
Chris Caldwell, The Prime Pages: Pepin's Test.
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FORMULA
| a(n) = 5^((F(n) - 1)/2) (mod F(n)), where F(n) is the n-th Fermat Number
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EXAMPLE
| a(4) = 5^(32768) (mod 65537) = 65536 = -1 (mod F(4)), therefore F(4) is prime.
a(5) = 5^(2147483648) (mod 4294967297) = -810129131 (mod F(5)), therefore F(5) is composite.
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CROSSREFS
| Cf. A000215, A019434, A152153, A152154, A152155.
Sequence in context: A198173 A204499 A166072 * A017540 A132216 A091340
Adjacent sequences: A152153 A152154 A152155 * A152157 A152158 A152159
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KEYWORD
| sign,changed
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AUTHOR
| Dennis Martin (dennis.martin(AT)dptechnology.com), Nov 27 2008
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