On Diophantine tuples involving star numbers

In this paper, we discuss two related Diophantine tuple problems involving star numbers $S_n=6n(n-1)+1$. First, we investigate classical $D(k)$-tuples whose elements are star numbers. We construct infinitely many $D(k)$-pairs for several positive and negative admissible values of $k$, and we prove several nonexistence results for $D(k)$-triples and quadruples. Second, we introduce a modified notion of a star number $D(k)$-tuple, requiring $a_ia_j+k$ itself to be a star number. We construct several infinite families of $D(1)$-triples, and study their extensions to quadruples. In particular, we prove that $\{4,108,165\}$ has the unique extension $\{4,108,165,360\}$ to a star number $D(1)$-quadruple. The main tools used in this paper are generalized Pell equations, Nagell's theorem, lower bounds for linear forms in logarithms due to Matveev and Dujella-Pethő version of the Baker-Davenport reduction method in diophantine approximation.