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4

Part of 2019 Finnish National High School Mathematics Comp

Problems(1)

a_n = n^n + (n - 1)^{n+1}, eventually periodic modulo m

Source: Finland 2019, p4

9/8/2019
Define a sequence an=nn+(n−1)n+1 a_n = n^n + (n - 1)^{n+1} when nn is a positive integer. Define all those positive integer mm , for which this sequence of numbers is eventually periodic modulo mm, e.g. there are such positive integers KK and ss such that ak≡ak+sa_k \equiv a_{k+s} (mod mmod \,m), where kk is an integer with k≥Kk \ge K.
Periodic sequenceSequenceperiodicbir tinga qimmat masala