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The pairs of positive integers

Source: Chinese TST 2007 6th quiz P1

January 4, 2009
modular arithmeticnumber theory proposednumber theory

Problem Statement

Find all the pairs of positive integers (a,b) (a,b) such that a^2 \plus{} b \minus{} 1 is a power of prime number ; a^2 \plus{} b \plus{} 1 can divide b^2 \minus{} a^3 \minus{} 1, but it can't divide (a \plus{} b \minus{} 1)^2.