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p(m-1) < p(m) < p(m + 1), p(n) is the largest prime divisor of n

Source: New Zealand NZMOC Camp Selection Problems 2013 p12

September 19, 2021
prime divisorsdivisorprimenumber theoryinequalities

Problem Statement

For a positive integer nn, let p(n)p(n) denote the largest prime divisor of nn. Show that there exist infinitely many positive integers m such that p(māˆ’1)<p(m)<p(m+1)p(m-1) < p(m) < p(m + 1).
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