OFFSET
1,1
COMMENTS
From Jon E. Schoenfield, Dec 02 2016, paraphrasing information from the Munafo link: (Start)
The decimal expansion of 10^10^10^10^2 - 1 would be 1 googolplex digits long, with each digit a 9. Many factors of this number can be identified using simple facts of modular arithmetic.
Since its digits are all 9's, it is divisible by 9=3*3. Since its digits are all 9's and the number of digits is even, it is divisible by 99 (as are 9999=99*101, 999999=99*10101, 99999999=99*1010101, etc.), and thus divisible by 11.
By the same principle, it is divisible by 9999, 99999, 99999999, and by any other number whose decimal expansion consists of k 9's where k is of the form 2^a * 5^b, where a and b are nonnegative integers up to 10^100 (see A003592) and all their divisors. Additional factors can be found using Fermat's Little Theorem.
Consequently, a large number of factors of 10^10^10^10^2 - 1 are known. (End)
LINKS
Robert G. Wilson v, Table of n, a(n) for n = 1..587
Dario Alejandro Alpern, Known prime factors of Googolduplex - 1
Dario Alejandro Alpern, Known 1-digit prime factors of Googolduplex - 1
Robert P. Munafo, Notable Properties of Specific Numbers
EXAMPLE
10^10^10^10^2 - 1 = 10^10^10^100 - 1 = 999...999 (a total of a googolplex of nines).
CROSSREFS
KEYWORD
nonn,fini
AUTHOR
Robert Munafo and Robert G. Wilson v, Nov 28 2016
STATUS
approved