Diophantine Equations Relating Sums and Products of Positive Integers: Computation-Aided Study of Parametric Solutions, Bounds, and Distinct-Term Solutions

Abstract

Diophantine equations ∏i=1nxi=F∑i=1nxi with xi,F∈ℤ+ associate the products and sums of n natural numbers. Only special cases have been studied so far. Here, we provide new parametric solutions depending on F and the divisors of F or F2. One of these solutions shows that the equation of any degree with any F is solvable. For n = 2, exactly two solutions exist if and only if F is a prime. These solutions are (2F, 2F) and (F + 1, F(F + 1)). We generalize an upper bound for the sum of solution terms from n = 3 established by Crilly and Fletcher in 2015 to any n to be (F+1)(F+n−1) and determine a lower bound to be nnFn−1. Confining the solutions to n-tuples consisting of distinct terms, equations of the 4th degree with any F are solvable but equations of the 5th to 9th degree are not. An upper bound for the sum of terms of distinct-term solutions is conjectured to be (F+1)[F+(n−2)(n−1)!/2+1]/(n−2)!. The conjecture is supported by computation, which also indicates that the upper bound equals the largest sum of solution terms if and only if F=(n+k−2)(n−2)!−1, k∈ℤ+. Computation provides further insights into the relationships between F and the sum of terms of distinct-term solutions.

Keywords

• Diophantine equation
• parametric solution
• Lagrange multiplier
• sums and products
• upper bound
• lower bound