MathDB
M 26

Source:

May 25, 2007
modular arithmeticfunctioninequalitiesfloor functionRecursive Sequences

Problem Statement

Let pp be an odd prime pp such that 2h≠1  (modp)2h \neq 1 \; \pmod{p} for all h∈Nh \in \mathbb{N} with h<p−1h< p-1, and let aa be an even integer with a∈]p2,p[a \in] \tfrac{p}{2}, p [. The sequence {an}n≥0\{a_n\}_{n \ge 0} is defined by a0=aa_{0}=a, an+1=p−bna_{n+1}=p -b_{n} \; (n≥0)(n \ge 0), where bnb_{n} is the greatest odd divisor of ana_n. Show that the sequence {an}n≥0\{a_n\}_{n \ge 0} is periodic and find its minimal (positive) period.
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