Glasnik Matematicki, Vol. 54, No. 1 (2019), 21-52.

A POLYNOMIAL VARIANT OF A PROBLEM OF DIOPHANTUS AND ITS CONSEQUENCES

Alan Filipin and Ana Jurasić

Faculty of Civil Engineering, University of Zagreb, Fra Andrije Kačića-Miošića 26, 10 000 Zagreb, Croatia
e-mail: filipin@grad.hr

Department of Mathematics, University of Rijeka, Radmile Matejčić 2, 51 000 Rijeka, Croatia
e-mail: ajurasic@math.uniri.hr

Abstract.   In this paper we prove that every Diophantine quadruple in ℝ [X] is regular. In other words, we prove that if {a, b, c, d} is a set of four non-zero elements of ℝ[X], not all constant, such that the product of any two of its distinct elements increased by 1 is a square of an element of ℝ[X], then

(a+b-c-d)2=4(ab+1)(cd+1).
Some consequences of the above result are that for an arbitrary nℕ there does not exist a set of five non-zero elements from ℤ[X], which are not all constant, such that the product of any two of its distinct elements increased by n is a square of an element of ℤ[X]. Furthermore, there can exist such a set of four non-zero elements of ℤ[X] if and only if n is a square.

2010 Mathematics Subject Classification.   11D09,11D45

Key words and phrases.   Diophantine m-tuples, polynomials

Full text (PDF) (free access)

https://doi.org/10.3336/gm.54.1.03

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