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A197635
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Wieferich-non-Wilson primes: non-Wilson primes that divide their Fermat-Wilson quotient A197633.
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3
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OFFSET
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1,1
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COMMENTS
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A Wieferich prime base a is a prime p satisfying a^(p-1) == 1 (mod p^2). A non-Wilson prime p is called a Wieferich-non-Wilson prime if p is a Wieferich prime base w_p, where w_p = ((p-1)!+1)/p is the Wilson quotient of p.
Michael Mossinghoff has computed that if a 4th Wieferich-non-Wilson prime exists, it is > 10^7.
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LINKS
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J. Sondow, Lerch Quotients, Lerch Primes, Fermat-Wilson Quotients, and the Wieferich-non-Wilson Primes 2, 3, 14771, Combinatorial and Additive Number Theory, CANT 2011 and 2012, Springer Proc. in Math. & Stat., vol. 101 (2014), pp. 243-255.
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FORMULA
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(((p-1)!+1)/p)^(p-1) == 1 (mod p^2), where p = a(n).
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EXAMPLE
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The first two non-Wilson primes are 2 and 3, whose Fermat-Wilson quotients are 0. Since 2 and 3 divide 0, they are members.
The 1728th non-Wilson prime is prime(1731) = 14771, and A197634(1728) = 0, so 14771 is a member.
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CROSSREFS
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KEYWORD
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nonn,hard,more,bref
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AUTHOR
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STATUS
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approved
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