On the diophantine equation x_1x_2···x_n = h(n)(x_1 + x_2 +...+ x_n)
Abstract
We are concerned with the equation of the title, where n is a fixed positive integer, h(n) is a given integer-valued arithmetic function and the unknowns take positive integral values 1 ≤ x1 ≤ x2 ≤ ... ≤ xn. We estimate the number m for which xi = 1 (i = 1, 2,...,m) in every solution. Next we give an upper bound for the number of coordinates of a solution which can be greater than 1. Further we estimate the number of all solution of the equation, and the paper concludes with a list of open problems.
References
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4. C. Viola, On the diophantine equations Π_0^k x_i - Σ_0^k x_i = n and Σ_0^k 1/x_i = a/n, Acta Arithmetica, XXII (1973), 339-352.
Department of Mathematics, Faculty of Natural Sciences, Constantine the Philosopher University, Slovakia Slovakia
Nové Zámky, Slovakia Slovakia
Department of Mathematics, Faculty of Natural Sciences, Constantine the Philosopher University, Slovakia Slovakia
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