Naslov
On the Extensibility of Diophanitne D(−1)-Pairs to Quadruples

Autori
Jukić Bokun, Mirela

Vrsta, podvrsta i kategorija rada
Sažeci sa skupova, neobjavljeni rad, znanstveni

Skup
22nd International Mathematics Conference 2021

Mjesto i datum
Dhaka, Bangladeš, 10.12.2021. - 11.12.2021

Vrsta sudjelovanja
Predavanje

Vrsta recenzije
Recenziran

Ključne riječi
Pellian equation ; Diophantine m-tuple ; Commutative ring

Sažetak
Let R be a commutative ring with unity. A set of m distinct elements in R such that the product of any two distinct elements increased by -1 is a perfect square is called a D(−1) − m-tuple in R. The existence of D(−1)-quadruples in a certain ring is closely related with the existence of positive integer solutions of some Pellian equations. We consider the solubility of the Pellian equation x2 − (n2k + 1)y2 = −n2l−1, (1) where n is a positive integer. We derived a generalized result concerning the equation of type (1) with the right-hand side equal to −m with some positive integer m. That result generate the results about the solvability of our initial equation (1). Moreover, we consider the equation x2 − (p2q2 + 1)y2 = −pq2, where p, q are primes and we completely solve the problem of its solvability in integers. By combining that results with other known results on the existence of Diophantine quadruples, we proved results on the ex- tensibility of some parametric families of D(−1)- pairs to quadruples in the ring Z[√−t], t > 0.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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