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%I A001220
%S A001220 1093,3511
%N A001220 Wieferich primes: primes p with the property that p^2 divides 2^(p-1) 
               - 1.
%C A001220 Joseph Silverman showed that the abc-conjecture implies that there are 
               infinitely many primes which are not in the sequence. - Benoit Cloitre 
               (benoit7848c(AT)orange.fr), Jan 09 2003
%C A001220 The squares of these numbers are Fermat pseudoprimes to base 2 (A001567). 
               - T. D. Noe (noe(AT)sspectra.com), May 22 2003
%C A001220 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 (noe(AT)sspectra.com), 
               Mar 31 2004
%C A001220 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
%C A001220 A002326((a(n)^2 - 1)/2) = A002326((a(n)-1)/2). - Vladimir Shevelev (shevelev(AT)bgu.ac.il), 
               Jul 09 2008, Aug 24 2008
%C A001220 Dorais and Klyve (see reference) reported on November 27, 2008, that 
               there are no other Wieferich primes up to 6.7*10^15. [From Peter 
               Luschny (peter(AT)luschny.de), Feb 10 2009]
%D A001220 R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, 
               Springer, NY, 2001; see p. 28.
%D A001220 R. K. Guy, Unsolved Problems in Number Theory, A3.
%D A001220 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 
               5th ed., Oxford Univ. Press, 1979, th. 91.
%D A001220 Y. Hellegouarch, "Invitation aux mathematiques de Fermat Wiles", Dunod, 
               2eme Edition, pp. 340-341.
%D A001220 J. Knauer and J. Richstein, The continuing search for Wieferich primes, 
               Math. Comp., 75 (2005), 1559-1563.
%D A001220 P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 
               2nd ed., 1989, p. 263.
%D A001220 J. Silverman, "Wieferich's Criterion and the abc Conjecture", J. Number 
               Th. 30 (1988) 226-237.
%D A001220 D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. 
               Penguin Books, NY, 1986, 163.
%D A001220 V. Shevelev, Overpseudoprimes, Mersenne Numbers and Wieferich Primes, 
               arxiv.org/abs/0806.3412
%H A001220 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Fxtbook</a>
%H A001220 C. K. Caldwell, The Prime Glossary, <a href="http://www.utm.edu/research/
               primes/glossary/WieferichPrime.html">Wieferich prime</a>
%H A001220 C. K. Caldwell, <a href="http://primes.utm.edu/notes/proofs/SquareMerDiv.html">
               Prime-square Mersenne divisors are Wieferich</a>
%H A001220 D. X. Charles, <a href="http://www.cs.wisc.edu/~cdx/Criterion.pdf">On 
               Wieferich Primes</a>
%H A001220 R. Crandall, K. Dilcher and C. Pomerance, <a href="http://www.math.dartmouth.edu/
               ~carlp/PDF/paper111.pdf">A search for Wieferich and Wilson primes</
               a>, Mathematics of Computation, Volume 66, 1997.
%H A001220 J. K. Crump, Joe's Number Theory Web, <a href="http://www.immortaltheory.com/
               NumberTheory/Wieferich.htm">Weiferich Primes</a>
%H A001220 John Blythe Dobson, <a href="http://cybrary.uwinnipeg.ca/people/Dobson/
               mathematics/Wieferich_primes.html">A note on the two known Wieferich 
               Primes</a>
%H A001220 Lorenz Halbeisen and Norbert Hungerbuehler, Number theoretic aspects 
               of a combinatorial function, Notes on Number Theory and Discrete 
               Mathematics 5 (1999) 138-150. (<a href="http://math.berkeley.edu/
               ~halbeis/publications/psf/seq.ps">ps</a>, <a href="http://math.berkeley.edu/
               ~halbeis/publications/pdf/seq.pdf">pdf</a>)
%H A001220 W. Johnson, On the nonvanishing of Fermat quotients (mod p), <a href="http:/
               /www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN243919689_0292">
               Journal f. die Reine und Angewandte Mathematik 292</a> (1977): 196-200.
%H A001220 C. McLeman, PlanetMath.org, <a href="http://planetmath.org/encyclopedia/
               WieferichPrime.html">Wieferich prime</a>
%H A001220 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               WieferichPrime.html">Link to a section of The World of Mathematics.</
               a>
%H A001220 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               abcConjecture.html">Link to a section of The World of Mathematics.</
               a>
%H A001220 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               IntegerSequencePrimes.html">Integer Sequence Primes</a>
%H A001220 Wieferich Home Page, <a href="http://www.elmath.org/">Search for Wieferich 
               primes</a>
%H A001220 Wikipedia, <a href="http://en.wikipedia.org/wiki/Wieferich_prime">Wieferich 
               prime</a>
%H A001220 P. Zimmermann, <a href="http://www.loria.fr/~zimmerma/records/primes.html">
               RECORDS FOR PRIME NUMBERS</a>
%H A001220 F.G. Dorais and D.W. Klyve, Near Wieferich Primes up to 6.7*10^15, November 
               27, 2008, <a href="http://www-personal.umich.edu/~dorais/docs/wieferich.pdf">
               PDF</a> [From Peter Luschny (peter(AT)luschny.de), Feb 10 2009]
%p A001220 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"
%p A001220 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: # from UlrSchimke(AT)aol.com, 
               Nov 01, 2001
%t A001220 Select[Prime[Range[10^3*5]], Round[(2^(#-1)-1)/#^2]==((2^(#-1)-1)/#^2) 
               &] (from Vladimir Orlovsky (4vladimir(AT)gmail.com), May 01 2008)
%Y A001220 See A007540 for a similar problem. Cf. A001567, A077816.
%Y A001220 Sequence in context: A023698 A038469 A077816 this_sequence A115192 A091674 
               A022197
%Y A001220 Adjacent sequences: A001217 A001218 A001219 this_sequence A001221 A001222 
               A001223
%K A001220 nonn,hard,bref,nice,more
%O A001220 1,1
%A A001220 N. J. A. Sloane (njas(AT)research.att.com).
%E A001220 Sequence is believed to be infinite, although there are no other terms 
               < 4*10^12.
%E A001220 Wieferich Home Page link from Filip Zaludek (filip.zaludek(AT)gtsnovera.cz), 
               Feb 05 2008

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