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OFFSET
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1,1
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COMMENTS
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Joseph Silverman showed that the abc-conjecture implies that there are infinitely many primes which are not in the sequence. - Benoit Cloitre, Jan 09 2003
Graves and Murty (2013) improved Silverman's result by showing that for any fixed k > 1, the abc-conjecture implies that there are infinitely many primes == 1 (mod k) which are not in the sequence. - Jonathan Sondow, Jan 21 2013
The squares of these numbers are Fermat pseudoprimes to base 2 (A001567). - T. D. Noe, May 22 2003
Primes p that divide the numerator of the harmonic number H((p-1)/2); that is, p divides A001008((p-1)/2). - T. D. Noe, Mar 31 2004
In a 1977 paper, Wells Johnson, citing a suggestion from Lawrence Washington, pointed out the repetitions in the binary representations of the numbers which are one less than the two known Wieferich primes; i.e. 1092 = 10001000100 (base 2); 3510 = 110110110110 (base 2). It is perhaps worth remarking that 1092 = 444 (base 16) and 3510 = 6666 (base 8), so that these numbers are small multiples of repunits in the respective bases. Whether this is mathematically significant does not appear to be known. - John Blythe Dobson (j.dobson(AT)uwinnipeg.ca), Sep 29 2007
A002326((a(n)^2 - 1)/2) = A002326((a(n)-1)/2). - Vladimir Shevelev, Jul 09 2008, Aug 24 2008
Dorais and Klyve (see link) reported on Nov 27 2008, that there are no other Wieferich primes up to 6.7*10^15. [Peter Luschny, Feb 10 2009]
From a posting to the Math Fun mailing list by R. W. Gosper, Dec 03 2009:(Start)
Subject: I just reordered checks.
Bank Lady: Where would you like the numbering to start?
rwg: What was my last one?
Bank Lady: 1093
rwg: Well then obviously 3511.
Bank Lady: [Opens mouth. Decides not to ask. Resumes typing.]
(End)
It is believed that p^2 does not divide 3^(p-1) - 1 if p = a(n). This is true for n = 1 and 2. See A178815, A178844, A178900, and Ostafe-Shparlinski (2010) Section 1.1. [Jonathan Sondow, Jun 29 2010]
These primes also divide the numerator of the harmonic number H(floor((p-1)/4)); [H. Eskandari (hamid.r.eskandari(AT)gmail.com), Sep 28 2010]
1093 and 3511 are prime numbers p satisfying congruence 429327^(p-1) == 1 (mod p^2). Why? [Arkadiusz Wesolowski, Apr 07 2011]
A196202(a049084(a(1)) = A196202(a049084(a(2)) = 1. [Reinhard Zumkeller, Sep 29 2011]
If q is prime and q^2 divides a prime-exponent Mersenne number, then q must be a Wieferich prime. Neither of the two known Wieferich primes divide Mersenne numbers. See Will Edgington's Mersenne page in the links below. [Daran Gill, Apr 04 2013].
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REFERENCES
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R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see p. 28.
R. K. Guy, Unsolved Problems in Number Theory, A3.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 91.
Y. Hellegouarch, "Invitation aux mathematiques de Fermat Wiles", Dunod, 2eme Edition, pp. 340-341.
J. Knauer and J. Richstein, The continuing search for Wieferich primes, Math. Comp., 75 (2005), 1559-1563.
P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 263.
V. Shevelev, Overpseudoprimes, Mersenne Numbers and Wieferich Primes, arxiv.org/abs/0806.3412
J. Silverman, "Wieferich's Criterion and the abc Conjecture", J. Number Th. 30 (1988) 226-237.
D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 163.
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LINKS
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Table of n, a(n) for n=1..2.
Joerg Arndt, Fxtbook, p.780
C. K. Caldwell, The Prime Glossary, Wieferich prime
C. K. Caldwell, Prime-square Mersenne divisors are Wieferich
D. X. Charles, On Wieferich Primes
R. Crandall, K. Dilcher and C. Pomerance, A search for Wieferich and Wilson primes, Mathematics of Computation, Volume 66, 1997.
J. K. Crump, Joe's Number Theory Web, Weiferich Primes
John Blythe Dobson, A note on the two known Wieferich Primes
F. G. Dorais and D. W. Klyve, A Wieferich Prime Search up to 6.7*10^15, Journal of Integer Sequences, Vol. 14, 2011.
Will Edgington, Mersenne Page.
A. Granville, K. Soundararajan, A binary additive problem of Erdos and the order of 2 mod p^2, Raman. J. 2 (1998) 283-298
Hester Graves and M. Ram Murty, The abc conjecture and non-Wieferich primes in arithmetic progressions, Journal of Number Theory, 133 (2013), 1809-1813.
Lorenz Halbeisen and Norbert Hungerbuehler, Number theoretic aspects of a combinatorial function, Notes on Number Theory and Discrete Mathematics 5 (1999) 138-150. (ps, pdf)
W. Johnson, On the nonvanishing of Fermat quotients (mod p), Journal f. die Reine und Angewandte Mathematik 292 (1977): 196-200.
C. McLeman, PlanetMath.org, Wieferich prime
Sihem Mesnager and Jean-Pierre Flori, A note on hyper-bent functions via Dillon-like exponents
A. Ostafe and I. Shparlinski (2010), Pseudorandomness and Dynamics of Fermat Quotients
J. Sondow, Lerch quotients, Lerch primes, Fermat-Wilson quotients, and the Wieferich-non-Wilson primes 2, 3, 14771, arXiv 2011.
Eric Weisstein's World of Mathematics, Wieferich Prime
Eric Weisstein's World of Mathematics, abc Conjecture
Eric Weisstein's World of Mathematics, Integer Sequence Primes
Wieferich Home Page, Search for Wieferich primes
Wikipedia, Wieferich prime
P. Zimmermann, Records for Prime Numbers
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FORMULA
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A178815(A000720(p))^(p-1) - 1 mod p^2 = A178900(n), where p = a(n). [Jonathan Sondow, Jun 29 2010]
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MAPLE
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wieferich := proc (n) local nsq, remain, bin, char: if (not isprime(n)) then RETURN("not prime") fi: nsq := n^2: remain := 2: bin := convert(convert(n-1, binary), string): remain := (remain * 2) mod nsq: bin := substring(bin, 2..length(bin)): while (length(bin) > 1) do: char := substring(bin, 1..1): if char = "1"
then remain := (remain * 2) mod nsq fi: remain := (remain^2) mod nsq: bin := substring(bin, 2..length(bin)): od: if (bin = "1") then remain := (remain * 2) mod nsq fi: if remain = 1 then RETURN ("Wieferich prime") fi: RETURN ("non-Wieferich prime"): end: # UlrSchimke(AT)aol.com, Nov 01, 2001
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MATHEMATICA
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Select[Prime[Range[10^3*5]], Round[(2^(#-1)-1)/#^2]==((2^(#-1)-1)/#^2) &] (* Vladimir Joseph Stephan Orlovsky, May 01 2008 *)
Select[Prime[Range[50000]], Divisible[2^(#-1)-1, #^2]&] (* Harvey P. Dale, Apr 23 2011 *)
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PROG
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(Haskell)
import Data.List (elemIndices)
a001220 n = a001220_list !! (n-1)
a001220_list = map (a000040 . (+ 1)) $ elemIndices 1 a196202_list
-- Reinhard Zumkeller, Sep 29 2011
(PARI)
N=10^9; default(primelimit, N);
forprime(n=2, N, if(Mod(2, n^2)^(n-1)==1, print1(n, ", ")));
\\ Joerg Arndt, May 01 2013
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CROSSREFS
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See A007540 for a similar problem.
Sequences "primes p such that p^2 divides X^(p-1)-1": A014127 (X=3), A123692 (X=5), A212583 (X=6), A123693 (X=7), A045616 (X=10).
Cf. A001567, A077816, A001008, A039951, A049094, A126196, A126197, A178815, A178844, A178871, A178900.
Sequence in context: A023698 A038469 A077816 * A203858 A115192 A091674
Adjacent sequences: A001217 A001218 A001219 * A001221 A001222 A001223
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KEYWORD
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nonn,hard,bref,nice,more
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Sequence is believed to be infinite.
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STATUS
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approved
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