S-parts of terms of integer linear recurrence sequences
نویسندگان
چکیده
Let S = {q1, . . . , qs} be a finite, non-empty set of distinct prime numbers. For a non-zero integer m, write m = q1 1 . . . q rs s M , where r1, . . . , rs are non-negative integers and M is an integer relatively prime to q1 . . . qs. We define the S-part [m]S of m by [m]S := q r1 1 . . . q rs s . Let (un)n≥0 be a linear recurrence sequence of integers. Under certain necessary conditions, we establish that for every ε > 0, there exists an integer n0 such that [un]S ≤ |un| holds for n > n0. Our proof is ineffective in the sense that it does not give an explicit value for n0. Under various assumptions on (un)n≥0, we also give effective, but weaker, upper bounds for [un]S of the form |un|, where c is positive and depends only on (un)n≥0 and S.
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