|
|
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
|
|
|
FORMULA
|
|
|
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
|
|
|
STATUS
|
approved
|
|
|
|