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Search: id:A152155
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| A152155 |
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Minimal residues of Pepin's Test for Fermat Numbers using the base 3. |
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4
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| 0, -1, -1, -1, -1, 10324303, -6586524273069171148, 110780954395540516579111562860048860420, 5864545399742183862578018016183410025465491904722516203269973267547486512819
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OFFSET
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0,6
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COMMENT
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For n>=1 the Fermat Number F(n) is prime if and only if 3^((F(n) - 1)/2) is congruent to -1 (mod F(n)).
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
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M. Krizek, F. Luca & L. Somer, 17 Lectures on Fermat Numbers, Springer-Verlag NY 2001, pp. 42-43.
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LINKS
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Dennis Martin, Table of n, a(n) for n = 0..11
Chris Caldwell, The Prime Pages: Pepin's Test.
M. Krizek, F. Luca & L. Somer, 17 Lectures on Fermat Numbers, pp. 42-43.
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FORMULA
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a(n) = 3^((F(n) - 1)/2) (mod F(n)), where F(n) is the n-th Fermat Number
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EXAMPLE
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a(4) = 3^(32768) (mod 65537) = 65536 = -1 (mod F(4)), therefore F(4) is prime.
a(5) = 3^(2147483648) (mod 4294967297) = 10324303 (mod F(5)), therefore F(5) is composite.
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CROSSREFS
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A000215, A019434, A152153, A152154, A152156
Sequence in context: A030093 A118213 A061407 this_sequence A159837 A066870 A157762
Adjacent sequences: A152152 A152153 A152154 this_sequence A152156 A152157 A152158
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KEYWORD
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sign
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AUTHOR
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Dennis Martin (dennis.martin(AT)dptechnology.com), Nov 27 2008
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