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Successive terms of a sequence which are divisible by m

Source: Polish Mathematical Olympiad 2004 Final Round Problem 6

February 7, 2010

number theory proposednumber theory

Problem Statement

An integer

m > 1

is given. The inﬁnite sequence

(x_n)_{n\ge 0}

is deﬁned by x_i\equal{}2^i for

i<m

and x_i\equal{}x_{i\minus{}1}\plus{}x_{i\minus{}2}\plus{}\cdots \plus{}x_{i\minus{}m} for

i\ge m

. Find the greatest natural number

k

such that there exist

k

successive terms of this sequence which are divisible by

m