Quadruples (a, b, c, d) of positive integers a < b < c < d with the property that the product of any two of them is one more than a perfect square are studied. Improved lower and upper bounds for the entries b and c are established. As an application of these results, a bound for the number of such quadruples is obtained.
Further remarks on Diophantine quintuples by Mihai Cipu (Bucureşti) 1. Introduction. Let m and n be integers, with m positive. A set of m positive integers is called a D(n)-m-tuple if the product of any two distinct elements increased by n is a perfect square. In this paper we study exclusively D(1)-m-sets, which will be called Diophantine m-tuples. When m = 2 (3, 4, 5 or 6), we shall speak of Diophantine pairs (triples, quadruples, quintuples or sextuples, respectively).Since Dujella [6] proved that there are no Diophantine sextuples and only finitely many Diophantine quintuples, the major concern is to confirm the folklore conjecture according to which no Diophantine quintuple exists. An essential ingredient of any strategy seems to be a convenient stratification of the set of Diophantine tuples. Quite early the notion of regular Diophantine tuple appeared (see [1]): a Diophantine triple {a, b, c} with a < b < c is called regular if c = c + , where c ± = a + b ± 2 √ ab + 1, and a Diophantine quadruple {a, b, c, d} with a < b < c < d is called regular if d = d + , where d ± = a + b + c + 2abc ± 2 (ab + 1)(ac + 1)(bc + 1). A stronger conjecture put forward by [1] and independently by [13] claims that every Diophantine quadruple is regular. We shall also employ another useful classification and adopt Fujita's point of view, calling {a, b, c} a standard triple if it satisfies one of the following: • c > b 5 (standard of the first kind ); • b > 4a and c ≥ b 2 (standard of the second kind ); • b > 12a and b 5/3 < c < b 2 (standard of the third kind ).Each Diophantine quadruple contains at least one standard triple, not necessarily unique, since the properties required in the classification are not mutually exclusive. Large gaps of entries in a Diophantine set facilitate theoretical analysis (e.g., they are necessary in order to employ the
Abstract. We provide irreducibility conditions for polynomials of the form f (X)+p k g(X), with f and g relatively prime polynomials with integer coefficients, deg f < deg g, p a prime number and k a positive integer. In particular, we prove that if k is prime to deg g − deg f and p k exceeds a certain bound depending on the coefficients of f and g, then f (X)+p k g(X)is irreducible over Q.
A set of positive integers with the property that the product of any two of them is the successor of a perfect square is called Diophantine 𝐷(−1)-set. Such objects are usually studied via a system of generalized Pell equations naturally attached to the set under scrutiny. In this paper, an innovative technique is introduced in the study of Diophantine 𝐷(−1)-quadruples. The main novelty is the uncovering of a quadratic equation relating various parameters describing a hypothetical 𝐷(−1)-quadruple with integer entries. In combination with extensive computations, this idea leads to the confirmation of the conjecture according to which there is no Diophantine 𝐷(−1)-quadruples.
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