A077816 Wieferich numbers (1): n > 1 such that 2^A000010(n) == 1 (mod n^2).

1093, 3279, 3511, 7651, 10533, 14209, 17555, 22953, 31599, 42627, 45643, 52665, 68859, 94797, 99463, 127881, 136929, 157995, 228215, 298389, 410787, 473985, 684645, 895167, 1232361, 2053935, 2685501, 3697083, 3837523, 6161805, 11512569

Offset: 1

Author

Reinhard Zumkeller, Nov 17 2002

Keywords

• nonn

Comments

A077815(a(n)) = 1.

The only known primes are a(1)=A001220(1)=1093 and a(3)=A001220(2)=3511, the Wieferich primes.

If there are finitely many Wieferich primes (A001220), this sequence is finite. In particular, unless there are other Wieferich primes besides 1093 and 3511, this sequence consists of 104 terms with the largest being 16547533489305 (Agoh et al., 1997).

a(105)=A001220(3) in the sense that either both numbers are well-defined and equal, or else neither number exists. - Jeppe Stig Nielsen, Oct 16 2016

Examples

			A077815(3279) = 2^A000010(3279) mod 3279^2 = 2^2184 mod 10751841 = 1, therefore 3279 is a term.
		

Links

• Max Alekseyev, Table of n, a(n) for n = 1..104 (all currently known terms)
• T. Agoh, K. Dilcher, and L. Skula, Fermat Quotients for Composite Moduli, Journal of Number Theory 66(1), 1997, 29-50. doi: 10.1006/jnth.1997.2162
• William D. Banks, Florian Luca, and Igor E. Shparlinski, Estimates for Wieferich Numbers, The Ramanujan Journal, December 2007, Volume 14, Issue 3, pp 361-378.
• R. Crandall, K. Dilcher, and C. Pomerance, A search for Wieferich and Wilson primes, Mathematics of Computation, Volume 66, 1997.
• R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001, p. 28.
• Jiří Klaška, A Simple Proof of Skula's Theorem on Prime Power Divisors of Mersenne Numbers, J. Int. Seq., Vol. 25 (2022), Article 22.4.3.
• Jiří Klaška, Jakóbczyk's Hypothesis on Mersenne Numbers and Generalizations of Skula's Theorem, J. Int. Seq., Vol. 26 (2023), Article 23.3.8.

Crossrefs

For another definition of Wieferich numbers, see A182297.

Cf. A001220.

Programs

Magma

[n: n in [1..8*10^5] | 2^EulerPhi(n) mod n^2 eq 1]; // Vincenzo Librandi, Dec 05 2015

Mathematica

Reap[For[k = 1, k <= 10^8, k++, If[PowerMod[2, EulerPhi[k], k^2] == 1, Print[k]; Sow[k]]]][[2, 1]] (* Jean-François Alcover, Nov 17 2021 *)

PARI

for(n=2, 10^9, if(Mod(2, n^2)^(eulerphi(n))==1, print1(n, ", "))); \\ Felix Fröhlich, May 27 2014

Extensions

More terms from Emeric Deutsch, Mar 05 2005

More terms from Sam Handler (sam_5_5_5_0(AT)yahoo.com), Jun 18 2005