# 2018-15 Diophantine equation

Let $$n$$ be a positive integer. Suppose that $$a_1, a_2, \dots, a_n$$ are non-zero integers and $$b_1, b_2, \dots, b_n$$ are positive integers such that $$(b_i, b_n) = 1$$ for $$i = 1, 2, \dots, n-1$$. Prove that the Diophantine equation
$a_1 x_1^{b_1} + a_2 x_2^{b_2} + \dots + a_n x_n^{b_n} = 0$
has infinitely many integer solutions $$(x_1, x_2, \dots, x_n)$$.

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