A polynomial variant of a problem of Diophantus and its consequences
In this paper we prove that every Diophantine quadruple in [X] is regular. In other words, we prove that if {a, b, c, d} is a set of four non-zero elements of [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 [X], then (a+b-c...
| Permalink: | http://skupnikatalog.nsk.hr/Record/nsk.NSK01001075563/Details |
|---|---|
| Matična publikacija: |
Glasnik matematički (Online) 54 (2019), 1 ; str. 21-52 |
| Glavni autori: | Filipin, Alan (Author), Jurasić, Ana |
| Vrsta građe: | e-članak |
| Jezik: | eng |
| Predmet: | |
| Online pristup: |
https://doi.org/10.3336/gm.54.1.03 Glasnik matematički (Online) Hrčak |
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| 024 | 7 | |2 doi |a 10.3336/gm.54.1.03 | |
| 035 | |a (HR-ZaNSK)001075563 | ||
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| 080 | 1 | |a 51 |2 2011 | |
| 100 | 1 | |a Filipin, Alan |4 aut | |
| 245 | 1 | 2 | |a A polynomial variant of a problem of Diophantus and its consequences |h [Elektronička građa] / |c Alan Filipin, Ana Jurasić. |
| 500 | |a Bilješke uz tekst. | ||
| 504 | |a Bibliografija: 30 jed. | ||
| 504 | |a Abstract. | ||
| 520 | |a In this paper we prove that every Diophantine quadruple in [X] is regular. In other words, we prove that if {a, b, c, d} is a set of four non-zero elements of [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 [X], then (a+b-c-d)2=4(ab+1)(cd+1). Some consequences of the above result are that for an arbitrary n there does not exist a set of five non-zero elements from [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 [X]. Furthermore, there can exist such a set of four non-zero elements of [X] if and only if n is a square. | ||
| 653 | 0 | |a Polinomi |a Diofantove m-torke | |
| 700 | 1 | |a Jurasić, Ana |4 aut | |
| 773 | 0 | |t Glasnik matematički (Online) |x 1846-7989 |g 54 (2019), 1 ; str. 21-52 |w nsk.(HR-ZaNSK)000659858 | |
| 981 | |b Be2019 |b B05/19 | ||
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| 856 | 4 | 0 | |u https://hrcak.srce.hr/220841 |y Hrčak |
| 856 | 4 | 1 | |y Digitalna.nsk.hr |