A modification of a problem of Diophantus

Abstract

Abstract An old question, due to Diophantus, asks to find sets of rational numbers such that 1 added to the product of any two elements from the set is a square. We are concerned here with a modification of this question. Let t ≥ 2 be an integer, and let 𝔽 be a field. For d ∈ 𝔽, define f t,d : 𝔽 t → 𝔽 as f t , d ( x 1 , x 2 , … , x t ) := x 1 x 2 ⋯ x t + d . $$\begin{array}{} \displaystyle f_{t,d}(x_1,x_2,\ldots,x_{t}):=x_1x_2\cdots x_{t}+d. \end{array}$$ For any nonempty subset S of 𝔽, we say S     is     f t , d − c l o s e d     if     f t , d ( x 1 , x 2 , … , x t ) : x i ∈ S  and distinct ⊆ S . $$\begin{array}{} \displaystyle S ~~\text{is}~~ {f_{t,d}-closed} ~~\text{if}~~ \left\{f_{t,d}(x_1,x_2,\ldots,x_{t}):x_i\in S\text{ and distinct}\right\}\subseteq S. \end{array}$$ For any integer n, with t≤ n≤ |𝔽|, let 𝒰(n,t,d) be the union of all f t,d -closed subsets S of 𝔽 with |S|=n. In this article, we investigate values of n,t,d for which 𝒰(n,t,d) = 𝔽, with particular focus on t = n – 1, where n ∈ {3,4}. Moreover, if 𝒰(n,t,d)≠ 𝔽, we determine in many cases the exact elements of the set 𝔽∖ 𝔽(n,t,d).