Discussiones
Mathematicae General Algebra and Applications 20(2) (2000) 199-206
DOI: https://doi.org/10.7151/dmgaa.1017
DIOPHANTINE EQUATIONS AND CLASS NUMBERS OF IMAGINARY QUADRATIC FIELDS
Zhenfu Cao and Xiaolei Dong
Department of Mathematics, Harbin Institute of
Technology
Harbin 150001, P. R. China
e-mail:zfcao@hope.hit.edu.cn
Abstract
Let A, D, K, k ∈ N with D square free and 2 | /k,B = 1,2 or 4 and μi ∈ {-1,1}(i = 1,2), and let h(-21-eD)(e = 0 or 1) denote the class number of the imaginary quadratic field Q(√{-21-eD}). In this paper, we give the all-positive integer solutions of the Diophantine equation Ax2+μ1B = K((Ay2+μ2B)/K)n, 2 | / n, n > 1 and we prove that if D > 1, then h(-21-eD) ≡ 0 (mod n), where D, and n satisfy kn-2e+1 = Dx2,x ∈ N,2 | / n, n > 1. The results are valuable for the realization of quadratic field cryptosystem.
Keywords: Diophantine equation, imaginary quadratic field, class number, cryptographic problem.
1991 Mathematics Subject Classification: 11D41, 11R11, 11R29, 94A60.
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Received 20 July 1998
Revised 30 October 2000
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