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- Title
The non-existence of D(−1)-quadruples extending certain pairs in imaginary quadratic rings.
- Authors
Fujita, Y.; Soldo, I.
- Abstract
A D(n)-m-tuple, where n is a non-zero integer, is a set of m distinct elements in a commutative ring R such that the product of any two distinct elements plus n is a perfect square in R. In this paper, we prove that there does not exist a D(−1)-quadruple { a , b , c , d } in the ring Z [ - k ] , k ≥ 2 with positive integers a < b < 16 a 2 - a - 2 + 2 k (8 a 2 + 3 a + 1) and integers c and d satisfying d < 0 < c . By combining that result with [14, Theorem 1.1] we were able to obtain a general result on the non-existence of a D(−1)-quadruple { a , b , c , d } in Z [ - k ] with integers a, b, c, d satisfying a < b ≤ 8 a - 3 . Furthermore, for a non-negative integer i and a positive integer j, we apply the obtained results in proving of the non-existence of D(−1)-quadruples containing powers of primes p i , q j with an arbitrary different primes p and q.
- Subjects
NONCOMMUTATIVE algebras; COMMUTATIVE rings; INTEGERS
- Publication
Acta Mathematica Hungarica, 2023, Vol 170, Issue 2, p455
- ISSN
0236-5294
- Publication type
Article
- DOI
10.1007/s10474-023-01356-0