EUDML

This paper investigates the system of equations $x^{2} + a y^{m} = z_{1}^{2}, x^{2} - a y^{m} = z_{2}^{2}$ in positive integers $x$ , $y$ , $z_{1}$ , $z_{2}$ , where $a$ and $m$ are positive integers with $m \geq 3$ . In case of $m = 2$ we would obtain the classical problem of congruent numbers. We provide a procedure to solve the simultaneous equations above for a class of the coefficient $a$ with the condition $gcd (x, z_{1}) = gcd (x, z_{2}) = gcd (z_{1}, z_{2}) = 1$ . Further, under same condition, we even prove a finiteness theorem for arbitrary nonzero $a$ .

Diophantine triples with values in binary recurrences

Clemens Fuchs; Florian Luca; Laszlo Szalay — 2008

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

In this paper, we study triples $a, b$ and $c$ of distinct positive integers such that $a b + 1, a c + 1$ and $b c + 1$ are all three members of the same binary recurrence sequence.

On the Diophantine equation $\sum_{j = 1}^{k} j F_{j}^{p} = F_{n}^{q}$

Gökhan Soydan; László Németh; László Szalay — 2018

Archivum Mathematicum

Let $F_{n}$ denote the $n^{t h}$ term of the Fibonacci sequence. In this paper, we investigate the Diophantine equation $F_{1}^{p} + 2 F_{2}^{p} + \dots + k F_{k}^{p} = F_{n}^{q}$ in the positive integers $k$ and $n$ , where $p$ and $q$ are given positive integers. A complete solution is given if the exponents are included in the set ${1, 2}$ . Based on the specific cases we could solve, and a computer search with $p, q, k \leq 100$ we conjecture that beside the trivial solutions only $F_{8} = F_{1} + 2 F_{2} + 3 F_{3} + 4 F_{4}$ , $F_{4}^{2} = F_{1} + 2 F_{2} + 3 F_{3}$ , and $F_{4}^{3} = F_{1}^{3} + 2 F_{2}^{3} + 3 F_{3}^{3}$ satisfy the title equation.

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On the resolution of simultaneous Pell equations.

Experimental results on probable primality.

Lucas Diophantine triples.

Unimodal rays in the regular and generalized Pascal pyramids.

On a geometrical interpretation of a sampling.

Unimodality of certain sequences connected with binomial coefficients.

Superelliptic Equations of the form y^p=x^kp+a_(kp-1)x^(kp-1)+...+a_0

Congruent numbers with higher exponents

Diophantine triples with values in binary recurrences

On the Diophantine equation $\sum_{j = 1}^{k} j F_{j}^{p} = F_{n}^{q}$