INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A23 ITERATED EXPONENTS IN NUMBER THEORY

We show that if a1, a2, a3,... is a sequence of positive integers and k is given, then the sequence a1, a a2 1, a a a3 2 1,... becomes constant when reduced (mod k). We also consider the sequence 11, 22, 33,... (mod k), showing that this sequence, and related ones like nn n (mod k), are eventually periodic. –Dedicated to the memory of Prof. Arnold Ross 1