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max v_p are equal in consecutive values

Source: Malaysian SST 2024 P7

September 5, 2024
number theory

Problem Statement

Let PP be the set of all primes. Given any positive integer nn, define f(n)=max⁡p∈Pvp(n)\displaystyle f(n) = \max_{p \in P}v_p(n) Prove that for any positive integer k≥2k\ge 2, there exists infinitely many positive integers mm such that f(m+1)=f(m+2)=⋯=f(m+k) f(m+1) = f(m+2) = \cdots = f(m+k)
Proposed by Ivan Chan Guan Yu
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