# Understand the problem

Find all pairs of positive integers so that .

##### Source of the problem

Singapore MO 2008

##### Topic

Number Theory

##### Difficulty Level

Medium

##### Suggested Book

An Excursion in Mathematics

# Start with hints

Do you really need a hint? Try it first!

Note that has to be a prime, because any proper prime divisor of would divide too.

Say . Note that, for large enough , both and are factors of . Thus, the factor occurs twice in the expansion of .

Prove that, if then .

Hint 3 implies that . Hence .
This is obviously false. Hence, has to be small enough to avoid this situation. It is avoided precisely when . This corresponds to .
The equations to be solved are and . Hence, the solutions are .

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