With too quick a glance to tell me whether the increasing sequences are of very unusual length or not, I am not so surprised that the sequences should be mostly increasing. I would expect it to increase most of the time (not all of the time), based only on the fact that the modulus is increasing in each term. If you assume that a^n mod b^n is uniformly distributed within [0, b^n), and that a^(n+1) mod b^(n+1) is uniformly distributed within [0..b^(n+1)), then there is only roughly a 1 in 2b chance that successive terms will decrease. Perhaps there is some structure in the sequence that makes the likelihood of decreases even lower. If they are really (eventually) monotonically increasing then that does seem mysterious. Dan Asimov wrote:
For primes q > p, the sequence q^n mod p^n, n = 1,2,3,... seems mysterious.
E..g, 2^n mod 3^n is:
1, 1, 3, 1, 19, 25, 11, 161, [next 25 or so terms continue to increase],... [tip o' the hat to OEIS A002380: http://www.research.att.com/~njas/sequences/A002380 ]
5^n mod 2^n is:
1, 1, 5, 1, 21, 9, 45, 225, 225, 357, 761, 1757, 2641,...
5^n mod 3^n is:
2, 7, 17, 58, 209, 316, 5180, 3526, 4508, 22540, 112700, 209206,...
All of these (based on only a few terms) appear to be eventually monotonic; 5^n mod 3^n appears to be monotonic, period.
Is anything known about this phenomenon?
--Dan
_____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele
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