A305184 Multiplicative order of 2 (mod p^2), where p is the n-th Wieferich prime (A001220).

364, 1755

Offset

1,1

Comments

Meissner discovered the congruence 2^364 == 1 (mod 1093^2) and thus proved that 1093 is a Wieferich prime, i.e., a term of A001220 (cf. Meissner, 1913).

Later, Beeger discovered the congruence 2^1755 == 1 (mod 3511^2) and proved that 3511 is also a Wieferich prime (cf. Beeger, 1922).

Let b(n) = (A001220(n)-1)/a(n). Then b(1) = 3 and b(2) = 2.

From the fact that a(1) and a(2) are composite it follows that A001220(1) = 1093 and A001220(2) = 3511 do not divide any terms of A001348 (cf. Dobson).

Curiously, both 364 and 1755 are repdigits in some base. 364 = 444 in base 9 and 1755 = 3333 in base 8. Compare this with Dobson's observation that 1092 and 3510 are 444 in base 16 and 6666 in base 8, respectively (cf. Dobson).

Links

Table of n, a(n) for n=1..2.

N. G. W. H. Beeger, On a new case of the congruence 2^p-1 == 1 (mod p^2), Messenger of Mathematics 51 (1922), 149-150.

J. B. Dobson, A note on the two known Wieferich primes

W. Meissner, Über die Teilbarkeit von 2^p-2 durch das Quadrat der Primzahl p = 1093, Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften Berlin, 35 (1913), 663-667. [Annotated scanned copy]

Formula

a(n) = A014664(A000720(A001220(n))) = A243905(A000720(A001220(n))). [Corrected by Jianing Song, Sep 20 2019]

Prog

(PARI) forprime(p=1, , if(Mod(2, p^2)^(p-1)==1, print1(znorder(Mod(2, p^2)), ", ")))

Crossrefs

Cf. A001220, A001348, A014664, A243905, A282552, A282902.

Sequence in context: A303123 A043471 A203857 * A105920 A241617 A027799

Adjacent sequences: A305181 A305182 A305183 * A305185 A305186 A305187

Keyword

nonn,hard,bref,more

Author

Felix Fröhlich, May 30 2018

Status

approved