Congruent numbers with higher exponents

Florian Luca; László Szalay

Displaying similar documents to “Congruent numbers with higher exponents”

Common terms in binary recurrences

Erzsébet Orosz (2006)

Acta Mathematica Universitatis Ostraviensis

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The purpose of this paper is to prove that the common terms of linear recurrences $M (2 a, - 1, 0, b)$ and $N (2 c, - 1, 0, d)$ have at most $2$ common terms if $p = 2$ , and have at most three common terms if $p > 2$ where $D$ and $p$ are fixed positive integers and $p$ is a prime, such that neither $D$ nor $D + p$ is perfect square, further $a, b, c, d$ are nonzero integers satisfying the equations $a^{2} - D b^{2} = 1$ and $c^{2} - (D + p) d^{2} = 1$ .

The diophantine equation $x^{2} + 2^{a} \cdot 17^{b} = y^{n}$

Su Gou, Tingting Wang (2012)

Czechoslovak Mathematical Journal

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Let $ℤ$ , $ℕ$ be the sets of all integers and positive integers, respectively. Let $p$ be a fixed odd prime. Recently, there have been many papers concerned with solutions $(x, y, n, a, b)$ of the equation $x^{2} + 2^{a} p^{b} = y^{n}$ , $x, y, n \in ℕ$ , $gcd (x, y) = 1$ , $n \geq 3$ , $a, b \in ℤ$ , $a \geq 0$ , $b \geq 0 .$ And all solutions of it have been determined for the cases $p = 3$ , $p = 5$ , $p = 11$ and $p = 13$ . In this paper, we mainly concentrate on the case $p = 3$ , and using certain recent results on exponential diophantine equations including the famous Catalan equation, all solutions $(x, y, n, a, b)$ of the equation $x^{2} + 2^{a} \cdot 17^{b} = y^{n}$ , $x, y, n \in ℕ$ , $gcd (x, y) = 1$ , $n \geq 3$ , $a, b \in ℤ$ , $a \geq 0$ ,...

A note on the diophantine equation $x^{2} + b^{Y} = c^{z}$

Maohua Le (2006)

Czechoslovak Mathematical Journal

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Let $a$ , $b$ , $c$ , $r$ be positive integers such that $a^{2} + b^{2} = c^{r}$ , $min (a, b, c, r) > 1$ , $gcd (a, b) = 1, a$ is even and $r$ is odd. In this paper we prove that if $b \equiv 3 (m o d 4)$ and either $b$ or $c$ is an odd prime power, then the equation $x^{2} + b^{y} = c^{z}$ has only the positive integer solution $(x, y, z) = (a, 2, r)$ with $min (y, z) > 1$ .

Jeśmanowicz' conjecture with congruence relations

Yasutsugu Fujita, Takafumi Miyazaki (2012)

Colloquium Mathematicae

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Let a,b and c be relatively prime positive integers such that a²+b² = c². We prove that if $b \equiv 0 (m o d 2^{r})$ and $b \equiv \pm 2^{r} (m o d a)$ for some non-negative integer r, then the Diophantine equation $a^{x} + b^{y} = c^{z}$ has only the positive solution (x,y,z) = (2,2,2). We also show that the same holds if c ≡ -1 (mod 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

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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 spacing between terms of generalized Fibonacci sequences

Diego Marques (2014)

Colloquium Mathematicae

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For k ≥ 2, the k-generalized Fibonacci sequence $(F ₙ^{(k)}) ₙ$ is defined to have the initial k terms 0,0,...,0,1 and be such that each term afterwards is the sum of the k preceding terms. We will prove that the number of solutions of the Diophantine equation $F ₘ^{(k)} - F ₙ^{(ℓ)} = c > 0$ (under some weak assumptions) is bounded by an effectively computable constant depending only on c.

Solutions to xyz = 1 and x+y+z = k in algebraic integers of small degree, I

H. G. Grundman, L. L. Hall-Seelig (2014)

Acta Arithmetica

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Let k ∈ ℤ be such that $|_{k} (ℚ) | = 3$ , where $_{k} : y ² = 1 - 2 k x + k ² x ² - 4 x ³$ . We determine all solutions to xyz = 1 and x + y + z = k in integers of number fields of degree at most four over ℚ.

Multiplicative relations on binary recurrences

Florian Luca, Volker Ziegler (2013)

Acta Arithmetica

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Given a binary recurrence ${u_{n}}_{n \geq 0}$ , we consider the Diophantine equation $u_{n_{1}}^{x_{1}} \dots u_{n_{L}}^{x_{L}} = 1$ with nonnegative integer unknowns $n_{1}, . . ., n_{L}$ , where $n_{i} \neq n_{j}$ for 1 ≤ i < j ≤ L, $m a x | x_{i} | : 1 \leq i \leq L \leq K$ , and K is a fixed parameter. We show that the above equation has only finitely many solutions and the largest one can be explicitly bounded. We demonstrate the strength of our method by completely solving a particular Diophantine equation of the above form.

On the Brocard-Ramanujan problem and generalizations

Andrzej Dąbrowski (2012)

Colloquium Mathematicae

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Let $p_{i}$ denote the ith prime. We conjecture that there are precisely 28 solutions to the equation $n ² - 1 = p ₁^{α ₁} \dots p_{k}^{α_{k}}$ in positive integers n and α₁,..., $α_{k}$ . This conjecture implies an explicit description of the set of solutions to the Brocard-Ramanujan equation. We also propose another variant of the Brocard-Ramanujan problem: describe the set of solutions in non-negative integers of the equation n! + A = x₁²+x₂²+x₃² (A fixed).