Higher power rational Diophantine tuples

Abstract Let $$k \ge 2$$ k ≥ 2 be a positive integer. A k th power rational Diophantine n -tuple is a set of n pairwise distinct nonzero rational numbers $$\{ a_{1}, a_{2}, \dots , a_{n} \}$$ { a 1 , a 2 , ⋯ , a n } such that $$a_{i} a_{j} + 1 = r_{i, j}^{k}$$ a i a j + 1 = r i , j k holds for each $$1 \le i < j \le n$$ 1 ≤ i < j ≤ n with some rational $$r_{i, j}$$ 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 $$k = 3$$ 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 .