# Solution May 25, 2007

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### Problem

Given natural numbers , and that are pairwise distinct and satisfy ,

prove that at least one of the numbers , , is not prime.

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### Solution

Assume that *a*,*b*,*c* are distinct primes. Further assume without loss of generality that *a* < *b* < *c*.

Note that , so and similar for and . Also, are pairwise coprime, so we have

Let

- .

is an integer. Obviously, . The right side is a strictly decreasing function in all three variables. Let's check a few cases:

- if and , the right side becomes which is never an integer for any
- if , , the right side is not an integer for and less than for , so it's not an integer then either.
- if , then and again, .
- if , then and and, (surprise!) .

This shows that can never be an integer, in contradiction with our assumptions, completing the proof.