A modification of a problem of Diophantus
-
Joshua Harrington
and
Lenny Jones
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 ft,d: 𝔽t → 𝔽 as
For any nonempty subset S of 𝔽, we say
For any integer n, with t≤ n≤ |𝔽|, let 𝒰(n,t,d) be the union of all ft,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).
(Communicated by Filippo Nuccio)
References
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© 2018 Mathematical Institute Slovak Academy of Sciences
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Articles in the same Issue
- Idempotents, group membership and their applications
- Residuation in non-associative MV-algebras
- An extension of F. Šik’s theorem on modular lattices
- Weak pseudo-BCK algebras
- Witt functor of a quadratic order
- A modification of a problem of Diophantus
- Cauchy problems involving a Hadamard-type fractional derivative
- Caratheodory’s solution of the Cauchy problem and a question of Z. Grande
- Sturm-Picone comparison theorems for nonlinear impulsive differential equations
- On oscillatory fourth order nonlinear neutral differential equations – III
- Local-periodic solutions for functional dynamic equations with infinite delay on changing-periodic time scales
- On the representation of involutive Jamesian functions
- Refinements of the heinz inequalities for operators and matrices
- Generalizations of Reid inequality
- Probabilistic convergence transformation groups
- Cluster sets and topology
- Some characterizations for Markov processes as mixed renewal processes
- Equivalent conditions of complete convergence for weighted sums of ANA random variables
