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Search: All articles in the CMB digital archive with keyword Diophantine equations

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1. CMB Online first

Fujita, Yasutsugu; Miyazaki, Takafumi
Jeśmanowicz' Conjecture with Congruence Relations. II
Let $a,b$ and $c$ be primitive Pythagorean numbers such that $a^{2}+b^{2}=c^{2}$ with $b$ even. In this paper, we show that if $b_0 \equiv \epsilon \pmod{a}$ with $\epsilon \in \{\pm1\}$ for certain positive divisors $b_0$ of $b$, then the Diophantine equation $a^{x}+b^{y}=c^z$ has only the positive solution $(x,y,z)=(2,2,2)$.

Keywords:exponential Diophantine equations, Pythagorean triples, Pell equations
Categories:11D61, 11D09

2. CMB 2006 (vol 49 pp. 560)

Luijk, Ronald van
A K3 Surface Associated With Certain Integral Matrices Having Integral Eigenvalues
In this article we will show that there are infinitely many symmetric, integral $3 \times 3$ matrices, with zeros on the diagonal, whose eigenvalues are all integral. We will do this by proving that the rational points on a certain non-Kummer, singular K3 surface are dense. We will also compute the entire N\'eron--Severi group of this surface and find all low degree curves on it.

Keywords:symmetric matrices, eigenvalues, elliptic surfaces, K3 surfaces, Néron--Severi group, rational curves, Diophantine equations, arithmetic geometry, algebraic geometry, number theory
Categories:14G05, 14J28, 11D41

3. CMB 2005 (vol 48 pp. 121)

Mollin, R. A.
Necessary and Sufficient Conditions for the Central Norm to Equal $2^h$ in the Simple Continued Fraction Expansion of $\sqrt{2^hc}$ for Any Odd $c>1$
We look at the simple continued fraction expansion of $\sqrt{D}$ for any $D=2^hc $ where $c>1$ is odd with a goal of determining necessary and sufficient conditions for the central norm (as determined by the infrastructure of the underlying real quadratic order therein) to be $2^h$. At the end of the paper, we also address the case where $D=c$ is odd and the central norm of $\sqrt{D}$ is equal to $2$.

Keywords:quadratic Diophantine equations, simple continued fractions,, norms of ideals, infrastructure of real quadratic fields
Categories:11A55, 11D09, 11R11

4. CMB 2002 (vol 45 pp. 428)

Mollin, R. A.
Criteria for Simultaneous Solutions of $X^2 - DY^2 = c$ and $x^2 - Dy^2 = -c$
The purpose of this article is to provide criteria for the simultaneous solvability of the Diophantine equations $X^2 - DY^2 = c$ and $x^2 - Dy^2 = -c$ when $c \in \mathbb{Z}$, and $D \in \mathbb{N}$ is not a perfect square. This continues work in \cite{me}--\cite{alfnme}.

Keywords:continued fractions, Diophantine equations, fundamental units, simultaneous solutions
Categories:11A55, 11R11, 11D09

5. CMB 2002 (vol 45 pp. 247)

Kihel, O.; Levesque, C.
On a Few Diophantine Equations Related to Fermat's Last Theorem
We combine the deep methods of Frey, Ribet, Serre and Wiles with some results of Darmon, Merel and Poonen to solve certain explicit diophantine equations. In particular, we prove that the area of a primitive Pythagorean triangle is never a perfect power, and that each of the equations $X^4 - 4Y^4 = Z^p$, $X^4 + 4Y^p = Z^2$ has no non-trivial solution. Proofs are short and rest heavily on results whose proofs required Wiles' deep machinery.

Keywords:Diophantine equations

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