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a^ab^{a + b} = c^cd^{c + d}, gcd (a, b) = gcd $(c, d) = 1=>(a,b)=(c,d)

Source: INAMO Shortlist 2015 N4

May 14, 2019
number theoryExponential equationexponentialgreatest common divisorDiophantine equation

Problem Statement

Suppose that the natural number a,b,c,da, b, c, d satisfy the equation aaba+b=ccdc+da^ab^{a + b} = c^cd^{c + d}. (a) If gcd (a,b)=(a, b) = gcd (c,d)=1(c, d) = 1, prove that a=ca = c and b=db = d. (b) Does the conclusion a=ca = c and b=db = d apply, without the condition gcd (a,b)=(a, b) = gcd (c,d)=1(c, d) = 1?
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