Glasnik matematički, Vol. 54 No. 1, 2019.
Izvorni znanstveni članak
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
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
Datum izdavanja:
7.6.2019.
Posjeta: 1.069 *