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Number of positive divisors

Source: European Mathematical Cup 2014, Senior Division, P1

December 14, 2014

modular arithmeticnumber theoryprime factorizationnumber theory unsolved

Problem Statement

Prove that there exist infinitely many positive integers which cannot be written in form

a^{d(a)}+b^{d(b)}

for some positive integers

a

and

b

For positive integer

d(a)

denotes number of positive divisors of

a

Proposed by Borna Vukorepa