login
A305184
Multiplicative order of 2 (mod p^2), where p is the n-th Wieferich prime (A001220).
0
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
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.
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
KEYWORD
nonn,hard,bref,more
AUTHOR
Felix Fröhlich, May 30 2018
STATUS
approved