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We prove an asymptotic formula for the average number of solutions to the Diophantine equation axyxy=n in which a is fixed and n varies.

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We prove that if k is a positive integer and d is a positive integer such that the product of any two distinct elements of the set {k + 1, 4k, 9k + 3, d} increased by 1 is a perfect square, then d = 144k 3 + 192k 2 + 76k + 8.

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