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- Title
On a Conjecture of Franušić and Jadrijević: Counter-Examples.
- Authors
Chakraborty, Kalyan; Gupta, Shubham; Hoque, Azizul
- Abstract
Let d ≡ 2 (mod 4) be a square-free integer such that x 2 - d y 2 = - 1 and x 2 - d y 2 = 6 are solvable in integers. We prove the existence of infinitely many quadruples in Z [ d ] with the property D(n) when n ∈ { (4 m + 1) + 4 k d , (4 m + 1) + (4 k + 2) d , (4 m + 3) + 4 k d , (4 m + 3) + (4 k + 2) d , (4 m + 2) + (4 k + 2) d } for m , k ∈ Z . As a consequence, we provide few counter examples to Conjecture 1 of Franušić and Jadrijević [Math. Slovaca 69, 1263–1278 (2019)].
- Subjects
PELL'S equation; DIOPHANTINE equations; QUADRATIC fields; ALGEBRAIC field theory; ALGEBRAIC fields
- Publication
Results in Mathematics / Resultate der Mathematik, 2023, Vol 78, Issue 1, p1
- ISSN
1422-6383
- Publication type
Article
- DOI
10.1007/s00025-022-01794-2