Let \(k \ge 2\) be a positive integer. A kth power rational Diophantine n-tuple is a set of n pairwise distinct nonzero rational numbers \(\{ a_{1}, a_{2}, \dots , a_{n} \}\) such that \(a_{i} a_{j} + 1 = r_{i, j}^{k}\) holds for each \(1 \le i < j \le n\) with some rational \(r_{i, j}\) ’s. In this paper, we prove that there exist infinitely many rational Diophantine triples for an arbitrary exponent. When the exponent is \(k = 3\) , we characterize certain pairs that can be extended to triples; additionally, we also prove the existence of infinitely many quadruples. The primary tool in our arguments is what we call curves induced by rational Diophantine pairs.
