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A039678
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Smallest a>1 such that a^(p-1)-1 is divisible by p^2, p = n-th prime.
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5
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5, 8, 7, 18, 3, 19, 38, 28, 28, 14, 115, 18, 51, 19, 53, 338, 53, 264, 143, 11, 306, 31, 99, 184, 53, 181, 43, 164, 96, 68, 38, 58, 19, 328, 313, 78, 226, 65, 253, 259, 532, 78, 176, 276, 143, 174, 165, 69, 330, 44, 33, 332, 94, 263, 48, 79, 171, 747, 731, 20, 147, 91, 40
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| Using Fermat's little theorem twice, it is easy to see that a=p^2-1 solves this problem for all odd primes p. In fact, there appear to be exactly p-1 values of a with 1 <= a <= p^2 for which a^(p-1)=1 (mod p^2). See A096082 for the related open problem. - T. D. Noe (noe(AT)sspectra.com), Aug 24 2008
That there are exactly p-1 values of 1 <= a <= p^2 for which a^(p-1)==1 (mod p^2) follows immediately from Hensel's lifting lemma and Fermat's little theorem - every solution mod p corresponds to a unique solution mod p^2. [Phil Carmody, Jan 10 2011]
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REFERENCES
| P. Ribenboim, The New Book of Prime Number Records, Springer, 1996, 345-349.
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LINKS
| T. D. Noe, Table of n, a(n) for n = 1..10000
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EXAMPLE
| For n=3, p=5 is 3rd prime and 5^2 = 25 divides 7^4 - 1 = 2401.
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CROSSREFS
| Sequence in context: A143618 A177056 A053787 * A131040 A007450 A200297
Adjacent sequences: A039675 A039676 A039677 * A039679 A039680 A039681
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KEYWORD
| nonn,nice
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AUTHOR
| Felice Russo (frusso(AT)micron.com)
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EXTENSIONS
| More terms from David W. Wilson (davidwwilson(AT)comcast.net)
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