Diophantine m-tuples for linear polynomials II. Equal degrees

Periodica Mathematica Hungarica, 2002

In this paper, we prove that there does not exist a set with more than 26 polynomials with integer coefficients, such that the product of any two of them plus a linear polynomial is a square of a polynomial with integer coefficients.

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 k-th power diophantine m-tuple in R is an m-tuple (a 1 , a 2 , . . . , a m ) of non-zero elements of R such that a i a j + 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.

The problem of the construction of Diophantine m-tuples, i.e. sets with the property that the product of any two of its distinct elements is one less then a square, has a very long history. In this survey, we describe several conjectures and recent results concerning Diophantine m-tuples and their generalizations.

Mathematical proceedings of the Cambridge Philosophical Society, 2002

Let n be a nonzero integer and assume that a set S of positive integers has the property that xy + n is a perfect square whenever x and y are distinct elements of S. In this paper we find some upper bounds for the size of the set S. We prove that if |n| ≤ 400 then |S| ≤ 32, and if |n| > 400 then |S| < 267.81 log |n| (log log |n|) 2 . The question whether there exists an absolute bound (independent on n) for |S| still remains open.

Journal of Number Theory, 2001

A set of m positive integers is called a Diophantine m-tuple if the product of its any two distinct elements increased by 1 is a perfect square. We prove that if {a, b, c} is a Diophantine triple such that b > 4a and c > max{b 13 , 10 20 } or c > max{b 5 , 10 1029 }, then there is unique positive integer d such that d > c and {a, b, c, d} is a Diophantine quadruple. Furthermore, we prove that there does not exist a Diophantine 9-tuple and that there are only finitely many Diophantine 8-tuples.

The Ramanujan Journal, 2008

A set of m positive integers is called a Diophantine m-tuple if the product of any two of them is one less than a perfect square. It is known that there does not exist a Diophantine sextuple and that there are only finitely many Diophantine quintuples. On the other hand, there are infinitely many Diophantine m-tuples for m = 2, 3 and 4. In this paper, we derive asymptotic estimates for the number of Diophantine pairs, triples and quadruples with elements less than given positive integer N .

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.

2008

The problem of the construction of Diophantine m-tuples, i.e. sets with the property that the product of any two of its distinct elements is one less then a square, has a very long history. In this survey, we describe several conjectures and recent results concerning Diophantine m-tuples and their generalizations.

Mathematical Proceedings of the Cambridge …, 2002

2012

We study the following problem raised by von zur Gathen and Roche [GR97]: