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

A polynomial variant of a problem of Diophantus and its consequences

Alan Filipin ; Faculty of Civil Engineering, University of Zagreb, Fra Andrije Kačića-Miošića 26, 10 000 Zagreb, Croatia
Ana Jurasić ; Department of Mathematics, University of Rijeka, Radmile Matejčić 2, 51 000 Rijeka, Croatia


Puni tekst: engleski pdf 273 Kb

str. 21-52

preuzimanja: 536

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Sažetak

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.

Ključne riječi

Diophantine m-tuples; polynomials

Hrčak ID:

220841

URI

https://hrcak.srce.hr/220841

Datum izdavanja:

7.6.2019.

Posjeta: 1.350 *