Diophantine quadruples in $$\mathbb {Z}[i][X]$$ Z [ i ] [ X ]
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Diophantine quadruples in Z[i][X] Alan Filipin1 · Ana Jurasi´c2
© Akadémiai Kiadó, Budapest, Hungary 2020
Abstract In this paper, we prove that every Diophantine quadruple in Z[i][X ] is regular. More precisely, we prove that if {a, b, c, d} is a set of four non-zero polynomials from Z[i][X ], not all constant, such that the product of any two of its distinct elements increased by 1 is a square of a polynomial from Z[i][X ], then (a + b − c − d)2 = 4(ab + 1)(cd + 1). Keywords Diophantine m-tuples · Polynomials · Regular quadruples Mathematics Subject Classification 11D09 · 11D45
1 Introduction A set consisting of m positive integers such that the product of any two of its distinct elements increased by 1 is a perfect square is called a Diophantine m-tuple. There is long history of finding such sets. One of the questions of interest, which various mathematicians try to solve, is how large those sets can be. Very recently, He, Togbé and Ziegler [22] proved the folklore conjecture that there cannot be 5 elements in Diophantine m-tuple, i.e. m < 5. However, there is also a stronger version of that conjecture that is still open, which states that every Diophantine triple can be extended to a quadruple with a larger element in a unique way (see [10]): Conjecture 1.1 If {a, b, c, d} is a Diophantine quadruple of integers and d > max{a,b,c}, √ then d = d+ = a + b + c + 2(abc + (ab + 1)(ac + 1)(bc + 1)). There are a lot of results supporting that conjecture. The history of the problem with recent results and up-to-date references can be found on the webpage [6].
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Ana Jurasi´c [email protected] Alan Filipin [email protected]
1
Faculty of Civil Engineering, University of Zagreb, Fra Andrije Kaˇci´ca-Mioši´ca 26, 10000 Zagreb, Croatia
2
Department of Mathematics, University of Rijeka, Radmile Matejˇci´c 2, 51000 Rijeka, Croatia
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A. Filipin, A. Jurasi´c
There are also many generalizations of the original problem, for example we can add a fixed integer n instead of 1 or consider the problem over domains other than Z or Q. We have the following more general definition: Definition 1.2 Let m ≥ 2 and let R be a commutative ring with unity. Let n ∈ R be a nonzero element and let {a1 , . . . , am } be a set of m distinct non-zero elements from R such that ai a j + n is a square of an element of R for 1 ≤ i < j ≤ m. The set {a1 , . . . , am } is called a Diophantine m-tuple with the property D(n) or simply a D(n)-m-tuple in R. Like in the original problem, we are interested in finding upper bounds for the number of elements of such sets. Dujella [8,9] found such bounds for the integer case. For other similar results, we refer the reader to [4,7,11,12,16,21]. In this paper, we consider a polynomial variant of the problem, which was firstly studied by Jones [23,24] for the case R = Z[X ] and n = 1. Also, a lot of other variants of such a polynomial problem were considered (see for example [12,14–19]). In the case where R is a polynomial ring and n is a constant, it is usually assumed that not all polynomials in such a
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