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number theory

Source: iranian TST 2015 third exam day 1 P2

June 12, 2015
number theoryIranIranian TSTleast common multiple

Problem Statement

Assume that a1,a2,a3a_1, a_2, a_3 are three given positive integers consider the following sequence: an+1=lcm[an,an−1]−lcm[an−1,an−2]a_{n+1}=\text{lcm}[a_n, a_{n-1}]-\text{lcm}[a_{n-1}, a_{n-2}] for n≥3n\ge 3 Prove that there exist a positive integer kk such that k≤a3+4k\le a_3+4 and ak≤0a_k\le 0. ([a,b][a, b] means the least positive integer such thata∣[a,b],b∣[a,b] a\mid[a,b], b\mid[a, b] also because lcm[a,b]\text{lcm}[a, b] takes only nonzero integers this sequence is defined until we find a zero number in the sequence)
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