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A modification of a problem of Diophantus

In: Mathematica Slovaca, vol. 68, no. 6
Joshua Harrington - Lenny Jones
Detaily:
Rok, strany: 2018, 1343 - 1352
Kľúčové slová:
Diophantine $k$-tuple, fields, characteristic, closure
O článku:
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 $t\ge 2$ be an integer, and let $\mathbb{F}$ be a field. For $d\in \F$, define $f_{t,d}\:\F^{t} \longrightarrow \F$ as \[f_{t,d}(x_1,x_2,\ldots,x_{t}):=x_1x_2\cdots x_{t}+d.\] For any nonempty subset $S$ of $\F$, we say \[S \mbox{is \emph{$f_{t,d}$-closed} if} \left\{f_{t,d}(x_1,x_2,\ldots,x_{t}):x_i\in S \text{and distinct}\right\}\subseteq S.\] For any integer $n$, with $t\le n\le \abs{\F}$, let $\U\left(n,t,d\right)$ be the union of all $f_{t,d}$-closed subsets $S$ of $\F$ with $\abs{S}=n$. In this article, we investigate values of $n,t,d$ for which $\U\left(n,t,d\right)=\F$, with particular focus on $t=n-1$, where $n\in \{3,4\}$. Moreover, if $\U\left(n,t,d\right)\ne \F$, we determine in many cases the exact elements of the set $\F\smallsetminus \mathcal{U}\left(n,t,d\right)$.
Ako citovať:
ISO 690:
Harrington, J., Jones, L. 2018. A modification of a problem of Diophantus. In Mathematica Slovaca, vol. 68, no.6, pp. 1343-1352. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0185

APA:
Harrington, J., Jones, L. (2018). A modification of a problem of Diophantus. Mathematica Slovaca, 68(6), 1343-1352. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0185
O vydaní:
Publikované: 3. 12. 2018