Diophantine equations and class number of imaginary quadratic fields

Zhenfu Cao; Xiaolei Dong

Discussiones Mathematicae - General Algebra and Applications (2000)

  • Volume: 20, Issue: 2, page 199-206
  • ISSN: 1509-9415

Abstract

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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 o r 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 ( m o d n ) , where D, and n satisfy k ⁿ - 2 e + 1 = D x ² , x ∈ ℕ, 2 ∤ n, n > 1. The results are valuable for the realization of quadratic field cryptosystem.

How to cite

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Zhenfu Cao, and Xiaolei Dong. "Diophantine equations and class number of imaginary quadratic fields." Discussiones Mathematicae - General Algebra and Applications 20.2 (2000): 199-206. <http://eudml.org/doc/287661>.

@article{ZhenfuCao2000,
abstract = {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.},
author = {Zhenfu Cao, Xiaolei Dong},
journal = {Discussiones Mathematicae - General Algebra and Applications},
keywords = {Diophantine equation; imaginary quadratic field; class number; cryptographic problem; Diophantine equations; cryptography},
language = {eng},
number = {2},
pages = {199-206},
title = {Diophantine equations and class number of imaginary quadratic fields},
url = {http://eudml.org/doc/287661},
volume = {20},
year = {2000},
}

TY - JOUR
AU - Zhenfu Cao
AU - Xiaolei Dong
TI - Diophantine equations and class number of imaginary quadratic fields
JO - Discussiones Mathematicae - General Algebra and Applications
PY - 2000
VL - 20
IS - 2
SP - 199
EP - 206
AB - 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.
LA - eng
KW - Diophantine equation; imaginary quadratic field; class number; cryptographic problem; Diophantine equations; cryptography
UR - http://eudml.org/doc/287661
ER -

References

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  12. [12] R.A. Mollin, Solutions of Diophantine equations and divisibility of class numbers of complex quadratic fields, Glasgow Math. J. 38 (1996), 195-197. Zbl0859.11058
  13. [13] T. Nagell, Sur l'impossibilité de quelques équations a deux indéterminées, Norsk Matem. Forenings Skr. Serie I 13 (1923), 65-82. 
  14. [14] C. Richaud, Sur la résolution des équations x² - Ay² = ±1, Atti Acad. Pontif. Nuovi Lincei (1866), 177-182. 
  15. [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). 
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