Naslov
A Pellian equation with primes and its applications

Autori
Jukić Bokun, Mirela

Vrsta, podvrsta i kategorija rada
Sažeci sa skupova, sažetak, znanstveni

Izvornik
Fibonacci Conference / - , 2020, 8-8

Skup
The Nineteenth International Conference on Fibonacci Numbers and Their Applications

Mjesto i datum
Sarajevo, Bosna i Hercegovina, 21.07.2020. - 23.07.2020

Vrsta sudjelovanja
Predavanje

Vrsta recenzije
Recenziran

Ključne riječi
Diophantine equation, quadratic field, Diophantine $m$-tuple

Sažetak
The Pell numbers are numbers defined by the recurrence relation P_0 = 0, P_1 = 1, P_{; ; n+1}; ; = 2P_n +P_{; ; n−1}; ; , n ≥ 1. Those numbers are 2- Fibonacci numbers and they are closely related to solutions of the Diophantine equations x^2 −2y^2 = ±1: if (x, y) denote a non-negative integer solution of any of these equation, then y = Pi , for some non-negative integer i. The Diophantine equation x2 − dy^2 = n, where d is is a positive non-square integer and n is a non-zero integer is called Pellian equation. In this talk we will briefly describe main results concerning of a so- lution of such equations and present re- sults on the solubility of the Pellian equa- tion x2 − (p^{; ; 2k+2}; ; + 1)y2 = −p^{; ; 2l+1}; ; , l ∈ {; ; 0, 1, ... , k}; ; , k ≥ 0, (1) with a prime p. We will apply the obtained results on the Diophantine m-tuple problem. A set of m non-zero elements a_1, ... , a_m of of a commutative ring R is a Diophantine m-tuple with the property D(−1) or just a D(−1) − m- tuple if ai aj − 1 is a perfect square in R for all i and j with 1 = i < j = m. The existence of a positive integer solutions of the equation (1) is related to the existence of some D(−1)- quadruples in a certain ring. By combining results about the solubility of (1) with other known results on the existence of the Diophantine quadruples, we will present results on the extensibility of some parametric families of D(−1)- pairs (for example, pairs which contain Fermat prime) to quadruples in the ring Z[√−t], t > 0.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

POVEZANOST RADA