On a problem of Diophantus (Q2717587)

A set of positive integers \(\{a_1, \dots , a_m\}\) is called a Diophantine \(m\)-tuple if \(a_ia_j + 1\) is a square for every \(1 \leq i < j \leq m\). The aim of the present paper is the examination of the generalizations of this classical concept. The first set of theorems deals with the size of sets \({\mathcal A, \mathcal B} \subseteq \{1, \dots , N\}\) such that \(ab + 1\) is a \(k\)-th power for every \(a \in {\mathcal A}, b \in {\mathcal B}\). For \(k=2\) it is proved that \(\min\{|{\mathcal A}|,|{\mathcal B}|\} \leq \log N / \log 2\). This result is nearly best possible, as the following example of the paper shows. Let \({\mathcal A}=\{1,2\}\) for the moment. Then there exists \({\mathcal B}\) such that \(ab + 1\) is a square for every \(a \in {\mathcal A}, b \in {\mathcal B}\) and \(|{\mathcal B}|\geq \log N / \log 36\). Remark that \textit{A. Dujella} [J. Number Theory 89, No. 1, 126-150 (2001; Zbl 1010.11019)] recently proved that there exists no Diophantine 9-tuple. NEWLINENEWLINENEWLINEThe second set of results concern the analogous problem for prime fields. In the sequel, \(p\) denotes a prime. In contrast to the classical case, there exist large Diophantine sets in \(\mathbb{Z}_p\). Assuming \(p\) is large enough, the author proves the existence of \({\mathcal A} \subseteq\mathbb{Z}_p\) of size at least \(\log p / 6\log 3\) and such that \(aa' + 1\) is a square for every \(a, a' \in {\mathcal A}\). On the other hand she shows that if \({\mathcal A, B} \subseteq \{1, \dots , p-1\}\) such that \(ab + 1\) is a quadratic residue or 0 modulo \(p\) for every \(a \in {\mathcal A}, b \in {\mathcal B}\) then \(|{\mathcal A}||{\mathcal B}|\leq (\sqrt{p} + 1)^2\). NEWLINENEWLINENEWLINEThe final results concern analogous problems if \(ab + 1\) as well as if \(a+b\) is a \(k\)-th power residue, \(\gcd(k,p-1)>1\), modulo \(p\) for every \(a \in {\mathcal A}, b \in {\mathcal B}\). NEWLINENEWLINENEWLINEChanging the point of view, the problem is considered as to the solvability of the congruence \(ab \equiv x^k - 1 \pmod{p}\) as well as \(a + b \equiv x^k - 1 \pmod{p}\). NEWLINENEWLINEThe last theorem deals with the generalization for suitable polynomials \(h(x)\in\mathbb{Z}_p[x]\) of degree \(n>p\). Let \({\mathcal A, \mathcal B} \subseteq \{1, \dots , p-1\}\) such that \(|{\mathcal A}||{\mathcal B}|\geq p((p-1)(n-1)/(p-n))^2\). Assume for any \(d\mid p-1, d>1\) that \(h(x)\) is not a constant multiple of a \(d\)-th power, then there exist \(a \in {\mathcal A}, b \in {\mathcal B}\) such that the congruences \(ab \equiv h(x) \pmod{p}\) as well as \(a + b \equiv h(x) \pmod{p}\) are solvable. Bounds are also given for the number of solutions.