On the extendibility of certain D(-1)-pairs in imaginary quadratic rings

被引:0
作者
Fujita, Yasutsugu [1 ]
Soldo, Ivan [2 ]
机构
[1] Nihon Univ, Dept Math, Coll Ind Technol, 2-11-1 Shin Ei, Narashino, Chiba, Japan
[2] Univ Osijek, Dept Math, Trg Ljudevita Gaja 6, Osijek 31000, Croatia
关键词
Diophantine m-tuple; Pell equation; D(-1)-TRIPLES; DIOPHANTUS; EQUATION;
D O I
10.1007/s13226-023-00465-4
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
Let R be a commutative ring with unity 1. A set of m different non-zero elements in R such that the product of any two distinct elements decreased by 1 is a perfect square in R is called a D(-1)-m-tuple in R. In the ring Z[root-k], with an integer k >= 2, we consider the D(-1)-pairs {a, 2(i)p(j)}, where i is an element of{0,1}, a, j are positive integers, p is an odd prime, gcd(a, 2(i)p(j)) = 1 and a < 2(i)p(j). We prove that there does not exist a D(-1)-quadruple of the form {a, 2(i)p(j), c, d} in Z[root-k] in the following cases: k does not divide 2(i)p(j) - a; k divides 2(i)p(j) - a and (2(i)p(j) - a)/k is a prime; k = 2(i)p(j) - a and a > 1.
引用
收藏
页码:156 / 162
页数:7
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