On a problem of Diophantus with polynomials
نویسندگان
چکیده
Let m ≥ 2 and k ≥ 2 be integers and let R be a commutative ring with a unit element denoted by 1. A k-th power diophantine m-tuple in R is an m-tuple (a1, a2, . . . , am) of non-zero elements of R such that aiaj +1 is a k-th 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.
منابع مشابه
When Newton met Diophantus : A study of rational-derived polynomials and their extension to quadratic fields
We consider the problem of classifying all univariate polynomials, defined over a domain k, with the property that they and all their derivatives have all their roots in k. This leads to a number of interesting sub-problems such as finding k-rational points on a curve of genus 1 and rational points on a curve of genus 2.
متن کاملOn the Size of Sets in a Polynomial Variant of a Problem of Diophantus
In this paper, we prove that there does not exist a set of 8 polynomials (not all constant) with coefficients in an algebraically closed field of characteristic 0 with the property that the product of any two of its distinct elements plus 1 is a perfect square.
متن کاملComplete Solution of the Polynomial Version of a Problem of Diophantus
In this paper, we prove that if {a, b, c, d} is a set of four non-zero polynomials with integer coefficients, not all constant, such that the product of any two of its distinct elements plus 1 is a square of a polynomial with integer coefficients, then (a + b − c − d) = 4(ab + 1)(cd + 1). This settles the “strong” Diophantine quintuple conjecture for polynomials with integer coefficients.
متن کاملA Polynomial Variant of a Problem of Diophantus and Euler
In this paper, we prove that there does not exist a set of four polynomials with integer coefficients, which are not all constant, such that the product of any two of them is one greater than a square of a polynomial with integer coefficients.
متن کاملOn the Exceptional Set in the Problem of Diophantus and Davenport
The Greek mathematician Diophantus of Alexandria noted that the numbers x, x + 2, 4x + 4 and 9x + 6, where x = 1 16 , have the following property: the product of any two of them increased by 1 is a square of a rational number (see [4]). Fermat first found a set of four positive integers with the above property, and it was {1, 3, 8, 120}. Later, Davenport and Baker [3] showed that if d is a posi...
متن کامل