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

Diophantine equations ∏ i = 1 n x i = F ∑ i = 1 n x i with x i , 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 F 2 . 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 (2 F , 2 F ) 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 n n F n − 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.