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Diophantine equations and class number of imaginary quadratic fields

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Let A, D, K, k ∈ ℕ with D square free and 2 ∤ k,B = 1,2 or 4 and $μ_{i} ∈ {-1,1}(i = 1,2)$, and let $h(-2^{1-e}D)(e = 0 or 1)$ denote the class number of the imaginary quadratic field $ℚ(√(-2^{1-e}D))$. 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 $h(-2^{1-e}D) ≡ 0 (mod n)$, where D, and n satisfy $kⁿ - 2^{e+1} = Dx²$, x ∈ ℕ, 2 ∤ n, n > 1. The results are valuable for the realization of quadratic field cryptosystem.
Twórcy
autor
  • Department of Mathematics, Harbin Institute of Technology, Harbin 150001, P. R. China
autor
  • Department of Mathematics, Harbin Institute of Technology, Harbin 150001, P. R. China
Bibliografia
  • [1] J. Buchmann and H.C. Williams, Quadratic fields and cryptography, 'Number Theory and Cryptography', University Press, Cambridge 1990, 9-25.
  • [2] Z. Cao, An Erdös conjecture, Pell sequences and Diophantine equations(Chinese), J. Harbin Inst. Tech. 2 (1987), 122-124.
  • [3] Z. Cao, On the equation $Dx² ± 1 = y^{p}$, xy ≠ 0 (Chinese), J. Math. Res. & Exposition 7 (1987), no. 3, 414.
  • [4] Z. Cao, On the equation $ax^{m}-byⁿ = 2$ (Chinese), Chinese Sci. Bull. 35 (1990), 558-559.
  • [5] Z. Cao, On the Diophantine equation $(ax^{m}-4c)/(abx-4c) = by²$ (Chinese), J. Harbin Inst. Tech. 23 (1991), Special Issue, 110-112.
  • [6] Z. Cao, The Diophantine equation $cx⁴+dy⁴ = z^{p}$, C.R. Math. Rep. Acad. Sci. Canada 14 (1992), 231-234.
  • [7] Z. Cao and A. Grytczuk, Some classes of Diophantine equations connected with McFarland's and Ma's conjectures, Discuss. Math. - Algebra and Applications 2 (2000), 193-198.
  • [8] G. Degert, Über die Bestimung der Grundeinheit gewisser reell-quadratischer Zahlkorper, Abh. Math. Sem. Univ. Hamburg 22 (1958), 92-97.
  • [9] K. Inkeri, On the diophantine equations $2y² = 7^{k}+1$ and x² + 11 = 3ⁿ, Elem. Math. 34 (1979), 119-121.
  • [10] V.A. Lebesgue, Sur l'impossibilitéon nombres entiers de l'équation $x^{m} = y²+1$, Nouv. Ann. Math. 9 (1850), no. 1, p. 178-181.
  • [11] W. Ljunggren, Über die Gleichungen 1 + Dx² = 2yⁿ und 1 + Dx² = 4yⁿ, Norske Vid. Selsk. Forhandl. 15 (30) (1942), 115-118.
  • [12] R.A. Mollin, Solutions of Diophantine equations and divisibility of class numbers of complex quadratic fields, Glasgow Math. J. 38 (1996), 195-197.
  • [13] T. Nagell, Sur l'impossibilité de quelques équations a deux indéterminées, Norsk Matem. Forenings Skr. Serie I 13 (1923), 65-82.
  • [14] C. Richaud, Sur la résolution des équations x² - Ay² = ±1, Atti Acad. Pontif. Nuovi Lincei (1866), 177-182.
  • [15] C. Størmer, Solution compléte en nombres entiers m, n,x, y, k de l'équation marctg 1/x + narctg1/y = kπ/4, Christiania Vid. Selsk. Skr. I, 11 (1895).
  • [16] D.T. Walker, On the Diophantina equation mx² - ny² = ±1, Amer. Math. Monthly 74 (1967), 504-513.
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Bibliografia
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bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1017
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