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A001220 Wieferich primes: primes p such that p^2 divides 2^(p-1) - 1. 63
1093, 3511 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Sequence is believed to be infinite.

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) and Catalan pseudoprimes (A163209). - 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, Sep 29 2007

A002326((a(n)^2 - 1)/2) = A002326((a(n)-1)/2). - Vladimir Shevelev, Jul 09 2008, Aug 24 2008

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

There are no other terms below 1.2*10^17 as established by PrimeGrid (see link below). - Max Alekseyev, Oct 06 2013

REFERENCES

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.

Pace Nielsen, Wieferich primes, heuristics, computations, Abstracts Amer. Math. Soc., 33 (#1, 20912), #1077-11-48.

P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 263.

D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 163.

LINKS

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 (sic)

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.

J. Knauer and J. Richstein, The continuing search for Wieferich primes, Math. Comp., 75 (2005), 1559-1563.

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

Christian Perfect, Integer sequence reviews on Numberphile (or vice versa), 2013.

J. Silverman, Wieferich's Criterion and the abc Conjecture, J. Number Th. 30 (1988) 226-237.

J. Sondow, Lerch quotients, Lerch primes, Fermat-Wilson quotients, and the Wieferich-non-Wilson primes 2, 3, 14771, arXiv 2011.

PrimeGrid, Wieferich Prime Search statistics

V. Shevelev, Overpseudoprimes, Mersenne Numbers and Wieferich Primes, arxiv:0806.3412

Michel Waldschmidt, Lecture on the abc conjecture and some of its consequences, Abdus Salam School of Mathematical Sciences (ASSMS), Lahore, 6th World Conference on 21st Century Mathematics 2013.

Michel Waldschmidt, Lecture on the abc conjecture and some of its consequences, Abdus Salam School of Mathematical Sciences (ASSMS), Lahore, 6th World Conference on 21st Century Mathematics 2013.

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

FORMULA

A178815(A000720(p))^(p-1) - 1 mod p^2 = A178900(n), where p = a(n). - Jonathan Sondow, Jun 29 2010

Odd primes p such that A002326((p^2-1)/2) = A002326((p-1)/2). See A182297. - Thomas Ordowski, Feb 04 2014

MAPLE

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

MATHEMATICA

Select[Prime[Range[50000]], Divisible[2^(#-1)-1, #^2]&]  (* Harvey P. Dale, Apr 23 2011 *)

PROG

(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

CROSSREFS

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

KEYWORD

nonn,hard,bref,nice,more

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified April 21 11:44 EDT 2014. Contains 240824 sequences.