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number of solutions to the equation

Source: Iran 3rd round 2011-number theory exam-p3

September 5, 2011
modular arithmeticnumber theory proposednumber theory

Problem Statement

Let kk be a natural number such that k≥7k\ge 7. How many (x,y)(x,y) such that 0≤x,y<2k0\le x,y<2^k satisfy the equation 7373x≡99y(mod2k)73^{73^x}\equiv 9^{9^y} \pmod {2^k}?
Proposed by Mahyar Sefidgaran
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