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Infinite Sequences of Integers

Source: USAJMO 2022/1

March 24, 2022
USAUSAJMOnumber theoryarithmetic sequencegeometric sequenceSequenceUSA(J)MO

Problem Statement

For which positive integers mm does there exist an infinite arithmetic sequence of integers a1,a2,...a_1, a_2, . . . and an infinite geometric sequence of integers g1,g2,...g_1, g_2, . . . satisfying the following properties?
[*] an−gna_n - g_n is divisible by mm for all integers n≥1n \ge 1; [*] a2−a1a_2 - a_1 is not divisible by mm.
Holden Mui
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