Missouri Journal of Mathematical Sciences

A Note on the Diophantine Equation $lx^3 - kx^2 + kx - l = y^2$: The Cases $k=3l \pm 1$

Konstantine Zelator

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Abstract

In this work, we investigate the Diophantine equation $lx^3-kx^2+kx-l = y^2$ where $k$ and $l$ are positive integers. The two results are Theorems 1.1 and 1.2. The first theorem states that if $k=3l-1$ and $l = \rho^2$, the above equation has a unique integer solution, namely $(x,y) = (1,0)$. The second theorem says that if $k=3l+1$ and $l\equiv 0,1,4,5,7 \pmod 8$ the above equation also has a unique solution, the pair $(x,y) = (1,0)$.

Article information

Source
Missouri J. Math. Sci., Volume 21, Issue 2 (2009), 136-140.

Dates
First available in Project Euclid: 14 September 2011

Permanent link to this document
https://projecteuclid.org/euclid.mjms/1316027246

Digital Object Identifier
doi:10.35834/mjms/1316027246

Mathematical Reviews number (MathSciNet)
MR2529016

Zentralblatt MATH identifier
1187.11009

Subjects
Primary: 11Dxx: Diophantine equations [See also 11Gxx, 14Gxx]

Citation

Zelator, Konstantine. A Note on the Diophantine Equation $lx^3 - kx^2 + kx - l = y^2$: The Cases $k=3l \pm 1$. Missouri J. Math. Sci. 21 (2009), no. 2, 136--140. doi:10.35834/mjms/1316027246. https://projecteuclid.org/euclid.mjms/1316027246


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