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Infinite set of solutions [ILL 1977]

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January 11, 2011
number theory unsolvednumber theory

Problem Statement

Let A1,A2,…,An+1A_1,A_2,\ldots ,A_{n+1} be positive integers such that (Ai,An+1)=1(A_i,A_{n+1})=1 for every i=1,2,…,ni=1,2,\ldots ,n. Show that the equation x1A1+x2A2+…+xnAn=xn+1An+1x_1^{A_1}+x_2^{A_2}+\ldots + x_n^{A_n}=x_{n+1}^{A_{n+1} } has an infinite set of solutions (x1,x2,…,xn+1)(x_1,x_2,\ldots , x_{n+1}) in positive integers.
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