Geodesic
Diophantine equations and class number of imaginary quadratic fields
Discussiones Mathematicae. General Algebra and Applications, Tome 20 (2000) no. 2, pp. 199-206.

Voir la notice de l'article provenant de la source Library of Science

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-eD)(e = 0 or 1) denote the class number of the imaginary quadratic field ℚ(√(-2^1-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(-2^1-eD) ≡ 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.
Keywords: Diophantine equation, imaginary quadratic field, class number, cryptographic problem 
@article{DMGAA_2000_20_2_a4,
     author = {Cao, Zhenfu and Dong, Xiaolei},
     title = {Diophantine equations and class number of imaginary quadratic fields},
     journal = {Discussiones Mathematicae. General Algebra and Applications},
     pages = {199--206},
     publisher = {mathdoc},
     volume = {20},
     number = {2},
     year = {2000},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DMGAA_2000_20_2_a4/}
}
TY  - JOUR
AU  - Cao, Zhenfu
AU  - Dong, Xiaolei
TI  - Diophantine equations and class number of imaginary quadratic fields
JO  - Discussiones Mathematicae. General Algebra and Applications
PY  - 2000
SP  - 199
EP  - 206
VL  - 20
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DMGAA_2000_20_2_a4/
LA  - en
ID  - DMGAA_2000_20_2_a4
ER  - 
%0 Journal Article
%A Cao, Zhenfu
%A Dong, Xiaolei
%T Diophantine equations and class number of imaginary quadratic fields
%J Discussiones Mathematicae. General Algebra and Applications
%D 2000
%P 199-206
%V 20
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DMGAA_2000_20_2_a4/
%G en
%F DMGAA_2000_20_2_a4
Cao, Zhenfu; Dong, Xiaolei. Diophantine equations and class number of imaginary quadratic fields. Discussiones Mathematicae. General Algebra and Applications, Tome 20 (2000) no. 2, pp. 199-206. http://geodesic.mathdoc.fr/item/DMGAA_2000_20_2_a4/

[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.