On a Problem of Diophantus with Polynomials
Andrej Dujella, Florian Luca · Rocky Mountain Journal of Mathematics · 2007
Let m ≥ 2 and k ≥ 2 be integers, and let R be a commutative ring with a unit element denoted by 1.A kth power diophantine m-tuple in R is an m-tuple (a 1 , a 2 , . . ., a m ) of nonzero elements of R such that a i a j + 1 is a kth power of an element of R for 1 ≤ i < j ≤ m.In this paper, we investigate the case when k ≥ 3 and R = K[X], the ring of polynomials with coefficients in a field K of characteristic zero.We prove the following upper bounds on m, the size of diophantine m-tuple: m ≤ 5 if k = 3; m ≤ 4 if k = 4; m ≤ 3 for k ≥ 5; m ≤ 2 for k even and k ≥ 8. 2000