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Two-variable exponential diophantine equation with 2001

Source: ItaMO 2001 Problem 3

February 17, 2013

number theory proposednumber theory

Problem Statement

Consider the equation

x^{2001}=y^x .

[*] Find all pairs

(x,y)

of solutions where

x

is a prime number and

y

is a positive integer.
[*] Find all pairs

(x,y)

of solutions where

x

and

y

are positive integers.
(Remember that

2001=3 \cdot 23 \cdot 29