On a conjecture of Franu\v sić and Jadrijevi' c: Counter-examples

2022/11/09 by Chakraborty, Kalyan, Gupta, Shubham, Hoque, Azizul
#11D09 #11R11 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2211.05010

Abstract

Let d≡ 2\pmod 4 be a square-free integer such that x² - dy² =- 1 and x² - dy² = 6 are solvable in integers. We prove the existence of infinitely many quadruples in ℤ[√(d)] with the property D(n) when n ∈ \(4m + 1) + 4k√(d), (4m + 1) + (4k + 2)√(d), (4m + 3) + 4k√(d), (4m + 3) + (4k + 2)√(d), (4m + 2) + (4k + 2)√(d)\ for m, k ∈ ℤ. As a consequence, we provide few counter examples to a conjecture of Franu\v sić and Jadrijevi' c (see Conjecture 1.1).

Related

• Infinitely Many Counter Examples of a Conjecture of Franušić and Jadrijević
• Diophantine D(n)-quadruples in ℤ[√(4k + 2)]
• On restricted sums of four squares and Zhi-Wei Sun's x+24y conjecture
• Sums of two squares and a power
• On a conjecture of Iizuka