HMMT 二月 2004 · 冲刺赛 · 第 38 题

HMMT February 2004 — Guts Round — Problem 38

[15] Let S = { p p · · · p | p , p , . . . , p are distinct primes and p , . . . , p < 30 } . As- 1 2 n 1 2 n 1 n sume 1 is in S . Let a be an element of S . We define, for all positive integers n : 1 a = a / ( n + 1) if a is divisible by n + 1; n +1 n n a = ( n + 2) a if a is not divisible by n + 1 . n +1 n n How many distinct possible values of a are there such that a = a for infinitely many 1 j 1 j ’s?

解析

Let S = { p p · · · p | p , p , . . . , p are distinct primes and p , . . . , p < 30 } . Assume 1 2 n 1 2 n 1 n 1 is in S . Let a be an element of S . We define, for all positive integers n : 1 a = a / ( n + 1) if a is divisible by n + 1; n +1 n n a = ( n + 2) a if a is not divisible by n + 1 . n +1 n n How many distinct possible values of a are there such that a = a for infinitely many 1 j 1 j ’s? Solution: 512 If a is odd, then we can see by induction that a = ( j +1) a when j is even and a = a 1 j 1 j 1 when j is odd (using the fact that no even j can divide a ). So we have infinitely many 1 j ’s for which a = a . j 1 If a > 2 is even, then a is odd, since a = a / 2, and a may have only one factor of 1 2 2 1 1