A197633 -id:A197633 - cp's OEIS frontend

A197634 Fermat-Wilson remainders: Fermat-Wilson quotients A197633 of non-Wilson primes p modulo p.

0, 0, 6, 7, 9, 7, 1, 6, 18, 17, 30, 11, 25, 30, 24, 46, 64, 16, 18, 4, 29, 66, 95, 11, 10, 9, 8, 64, 118, 77, 136, 15, 139, 64, 105, 9, 153, 167, 93, 4, 144, 180, 67, 179, 133, 51, 145, 130, 168, 41, 25, 163, 51, 42, 43, 100, 162, 212, 235, 2, 98, 232, 22

Offset: 1

Jonathan Sondow, Oct 16 2011

nonn

a(n) = 0 iff the n-th non-Wilson prime is a Wieferich-non-Wilson prime A197635. For example a(1) = a(2) = 0, so the 1st and 2nd non-Wilson primes 2 and 3 are Wieferich-non-Wilson primes. The next one is 14771, which is the 1728th non-Wilson prime, so the next zero in the sequence occurs at a(1728) = 0.

			The 3rd non-Wilson prime is 7, and A197633(3) = 170578899504 == 6 (mod 7), so a(3) = 6.

Jonathan Sondow, Lerch Quotients, Lerch Primes, Fermat-Wilson Quotients, and the Wieferich-non-Wilson Primes 2, 3, 14771, in Proceedings of CANT 2011, arXiv:1110.3113
Jonathan 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.

Cf. A197633, A197635, A197636.

nmax = 63; nonWilsonQ[p_] := Mod[((p-1)! + 1)/p, p] != 0; nwp = Select[ Prime[ Range[nmax + 2]], nonWilsonQ]; A197633[n_] := With[{p = nwp[[n]]}, ((((p-1)! + 1)/p)^(p-1) - 1)/p]; a[n_] := Mod[A197633[n], nwp[[n]]]; Table[a[n], {n, 1, nmax}] (* Jean-François Alcover, Oct 10 2012 *)

a(n) = A197633(n) mod A197636(n).

a(n) = ((((p-1)!+1)/p)^(p-1)-1)/p mod p, where p is the n-th non-Wilson prime.

A197635 Wieferich-non-Wilson primes: non-Wilson primes that divide their Fermat-Wilson quotient A197633.

Original entry on oeis.org

2, 3, 14771

Offset: 1

Jonathan Sondow, Oct 16 2011

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.

			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.

Carlos Rivera, Problem 59. Wieferich-non-Wilson primes, The Prime Puzzles and Problems Connection.
Jonathan Sondow, Lerch Quotients, Lerch Primes, Fermat-Wilson Quotients, and the Wieferich-non-Wilson Primes 2, 3, 14771, in Proceedings of CANT 2011, arXiv:1110.3113
Jonathan 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.

Cf. A007540, A197633, A197634, A197636, A197637.

A197634(A197637(a(n))) = 0.

(((p-1)!+1)/p)^(p-1) == 1 (mod p^2), where p = a(n).

A197636 Non-Wilson primes: primes p such that (p-1)! =/= -1 mod p^2.

Original entry on oeis.org

2, 3, 7, 11, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 569

Offset: 1

Jonathan Sondow, Oct 19 2011

nonn

All primes except 5, 13, 563, and any other Wilson prime A007540 that may exist.
Same as primes p that do not divide their Wilson quotient ((p-1)!+1)/p.
Wilson's theorem says that (p-1)! == -1 (mod p) if and only if p is prime.
p = prime(i) is a term iff A250406(i) != 0. - Felix Fröhlich, Jan 24 2016
Complement of A007540 in A000040. - Felix Fröhlich, Jan 24 2016

			2 is a non-Wilson prime since (2-1)! = 1 ==/== -1 (mod 2^2).

E. Costa, R. Gerbicz and D. Harvey, A search for Wilson primes, Mathematics of Computation, 83 (2014), 3071-3091 (arXiv:1209.3436 [math.NT], 2012).
Jonathan Sondow, Lerch Quotients, Lerch Primes, Fermat-Wilson Quotients, and the Wieferich-non-Wilson Primes 2, 3, 14771, in Proceedings of CANT 2011, arXiv:1110.3113 [math.NT], 2011-2012.
Jonathan 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.

Cf. A007540, A007619, A197633, A197634, A197635, A250406.

Select[Prime@ Range@ 104, Mod[Factorial[# - 1], #^2] != #^2 - 1 &] (* Michael De Vlieger, Jan 24 2016 *)

forprime(p=1, 500, if(Mod((p-1)!, p^2)!=-1, print1(p, ", "))) \\ Felix Fröhlich, Jan 24 2016

((p-1)!+1)/p =/= 0 (mod p), where p is prime.

A222207 Morley quotients: (2^(2p-2) - (-1)^((p-1)/2)binomial(p-1,(p-1)/2)) / p^3, where p = prime(n) and n >= 3.

Original entry on oeis.org

2, 12, 788, 7636, 874202, 10018884, 1445893544, 2954512034024, 38700329118256, 93229749133527532, 17540746936557672236, 243284404062970619608, 47694250379410432495952, 136236017676683906365850456, 404504597532158799519693872144, 5856120097210409121404621878992, 18102352585707069737371994385420772, 3894254646848417473467131712404310728

Offset: 3

Jonathan Sondow, Feb 22 2013

nonn

Morley (1894/95) proved 2^(2*p-2) == (-1)^((p-1)/2)*binomial(p-1,(p-1)/2) mod p^3 for all primes p > 3.

Morley quotients are even, since 2^(2*p-2) and binomial(p-1,(p-1)/2) are even and p^3 is odd.

			prime(3) = 5, so a(3) = (2^(2*5-2) - (-1)^((5-1)/2)*binomial(5-1,(5-1)/2))/5^3 = (2^8 - binomial(4,2))/5^3 = (256-6)/125 = 2.

Vincenzo Librandi, Table of n, a(n) for n = 3..200
C. Aebi, G. Cairns, Morley’s other miracle, Math. Mag., 85 (2012), 205-211.
F. Morley, Note on the Congruence 2^4n == (-1)^n*(2n)!/(n!)^2 where 2n+1 is a prime, Annals of Mathematics, Vol. 9 (1894 - 1895), pp. 168-170.

Cf. A007619, A007663, A034602, A197630, A197633.

m[p_] := (2^(2*p-2) - (-1)^((p-1)/2)*Binomial[p-1, (p-1)/2])/p^3; Table[ m[ Prime[n]], {n, 3, 20}]

A216867 Fermat-Wilson quotients divided by their GCD.

Original entry on oeis.org

0, 0, 7107454146, 57823016899195665754121432041221505493862544313363725

Offset: 1

Jonathan Sondow, Sep 18 2012

nonn

The GCD of all Fermat-Wilson quotients A197633 is 24.

Jonathan Sondow, Lerch Quotients, Lerch Primes, Fermat-Wilson Quotients, and the Wieferich-non-Wilson Primes 2, 3, 14771, in Proceedings of CANT 2011, arXiv:1110.3113
Jonathan 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.

Cf. A197633.

a(n) = A197633(n)/24.

cp's OEIS Frontend

A197634 Fermat-Wilson remainders: Fermat-Wilson quotients A197633 of non-Wilson primes p modulo p.

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A197635 Wieferich-non-Wilson primes: non-Wilson primes that divide their Fermat-Wilson quotient A197633.

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A197636 Non-Wilson primes: primes p such that (p-1)! =/= -1 mod p^2.

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A222207 Morley quotients: (2^(2*p-2) - (-1)^((p-1)/2)*binomial(p-1,(p-1)/2)) / p^3, where p = prime(n) and n >= 3.

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A216867 Fermat-Wilson quotients divided by their GCD.

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A222207 Morley quotients: (2^(2p-2) - (-1)^((p-1)/2)binomial(p-1,(p-1)/2)) / p^3, where p = prime(n) and n >= 3.