The exponential diophantine equation AX2 + BY2 = λkz and its applications

Zhenfu Cao*, Chuan I. Chu, Wai Chee SHIU

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)

Abstract

Let A, B ∈ N with A > 1, B > 1 and gcd(A, B) = 1, k ≥ 2 be an integer coprime with AB, and let λ ∈ {1, 2, 4} be such that if λ = 4, then A ≠ 4 and B ≠ 4; and if k is even, then λ = 4. In this paper, we shall describe all solutions of the equation AX2 + BY2 = λkZ, X, Y, Z ∈ Z, gcd(X, Y) = 1, Z > 0 with X|*A or Y |*B, where the symbol X|*A means that every prime divisor of X divides A. Then, using this result, we give some more general results on the number of solutions of the equation lax + mby = λcz, x > 1, y > 1, z >1. In addition, using Cao's resulton Pell equation, we obtain some improvement of Terai's results on the equations ax + 2 = cz, ax + 4 = cz and ax + 2y = cz.

Original languageEnglish
Pages (from-to)1015-1034
Number of pages20
JournalTaiwanese Journal of Mathematics
Volume12
Issue number5
DOIs
Publication statusPublished - Aug 2008

Scopus Subject Areas

  • Mathematics(all)

User-Defined Keywords

  • Exponential diophantine equation
  • Quadratic field

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