Does the arithmetic sequence 1, 2188, 4375, 6562, ...... contain infinitely many powers of 10?
Give reasons for your answer.
Well, of course it does.
We are looking for values 10^N which equal 1 mod 2187.
Because 10^0 = 1, mod N, the sequence 10^N must eventually equal 1 for some higher N, because powers cycle with respect to any mod.
There is nothing magic about 2187 or 10 for that matter. For any k and s, the arithmetic series 1, 1+k, 1+2k, ... contains infinitely many powers of s for any positive integers k and s.
In this case, using excel, I see that 10^244 = 1 mod 2187, so the cycle has length 244. 10^n = 1 mod 2187 for n = 244k, where k is any integer.
Edited on July 14, 2015, 2:57 pm
Pwerful answer (spoiler)
