Diophantine equations and class number of imaginary quadratic fields
Zhenfu Cao ; Xiaolei Dong
Discussiones Mathematicae - General Algebra and Applications, Tome 20 (2000), p. 199-206 / Harvested from The Polish Digital Mathematics Library
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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=0or1) 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(modn), 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.

Publié le : 2000-01-01
EUDML-ID : urn:eudml:doc:287661 
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     journal = {Discussiones Mathematicae - General Algebra and Applications},
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     year = {2000},
     pages = {199-206},
     zbl = {0979.11020},
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Zhenfu Cao; Xiaolei Dong. Diophantine equations and class number of imaginary quadratic fields. Discussiones Mathematicae - General Algebra and Applications, Tome 20 (2000) pp. 199-206. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1017/

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