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

H. G. Grundman; L. L. Hall-Seelig

Displaying similar documents to “Solutions to xyz = 1 and x + y + z = k in algebraic integers of small degree, II”

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.

On the Lebesgue-Nagell equation

Andrzej Dąbrowski (2011)

Colloquium Mathematicae

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We completely solve the Diophantine equations $x ² + 2^{a} q^{b} = y ⁿ$ (for q = 17, 29, 41). We also determine all $C = p ₁^{a ₁} \dots p_{k}^{a_{k}}$ and $C = 2^{a ₀} p ₁^{a ₁} \dots p_{k}^{a_{k}}$ , where $p ₁, . . ., p_{k}$ are fixed primes satisfying certain conditions. The corresponding Diophantine equations x² + C = yⁿ may be studied by the method used by Abu Muriefah et al. (2008) and Luca and Togbé (2009).

Complete solutions of a Lebesgue-Ramanujan-Nagell type equation

Priyanka Baruah, Anup Das, Azizul Hoque (2024)

Archivum Mathematicum

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We consider the Lebesgue-Ramanujan-Nagell type equation $x^{2} + 5^{a} 13^{b} 17^{c} = 2^{m} y^{n}$ , where $a, b, c, m \geq 0, n \geq 3$ and $x, y \geq 1$ are unknown integers with $gcd (x, y) = 1$ . We determine all integer solutions to the above equation. The proof depends on the classical results of Bilu, Hanrot and Voutier on primitive divisors in Lehmer sequences, and finding all $S$ -integral points on a class of elliptic curves.

A note on the article by F. Luca “On the system of Diophantine equations $a ² + b ² = {(m ² + 1)}^{r}$ and $a^{x} + b^{y} = {(m ² + 1)}^{z}$ ” (Acta Arith. 153 (2012), 373-392)

Takafumi Miyazaki (2014)

Acta Arithmetica

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Let r,m be positive integers with r > 1, m even, and A,B be integers satisfying $A + B \sqrt (- 1) = {(m + \sqrt (- 1))}^{r}$ . We prove that the Diophantine equation ${| A |}^{x} + {| B |}^{y} = {(m ² + 1)}^{z}$ has no positive integer solutions in (x,y,z) other than (x,y,z) = (2,2,r), whenever $r > 10^{74}$ or $m > 10^{34}$ . Our result is an explicit refinement of a theorem due to F. Luca.

On the diophantine equation $x^{2} + 2^{a} 3^{b} 73^{c} = y^{n}$

Murat Alan, Mustafa Aydin (2023)

Archivum Mathematicum

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In this paper, we find all integer solutions $(x, y, n, a, b, c)$ of the equation in the title for non-negative integers $a, b$ and $c$ under the condition that the integers $x$ and $y$ are relatively prime and $n \geq 3$ . The proof depends on the famous primitive divisor theorem due to Bilu, Hanrot and Voutier and the computational techniques on some elliptic curves.

Diophantine equations involving factorials

Horst Alzer, Florian Luca (2017)

Mathematica Bohemica

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We study the Diophantine equations ${(k!)}^{n} - k^{n} = {(n!)}^{k} - n^{k}$ and ${(k!)}^{n} + k^{n} = {(n!)}^{k} + n^{k},$ where $k$ and $n$ are positive integers. We show that the first one holds if and only if $k = n$ or $(k, n) = (1, 2), (2, 1)$ and that the second one holds if and only if $k = n$ .

On some Diophantine equations involving balancing numbers

Euloge Tchammou, Alain Togbé (2021)

Archivum Mathematicum

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In this paper, we find all the solutions of the Diophantine equation $B_{1}^{p} + 2 B_{2}^{p} + \dots + k B_{k}^{p} = B_{n}^{q}$ in positive integer variables $(k, n)$ , where $B_{i}$ is the $i^{t h}$ balancing number if the exponents $p$ , $q$ are included in the set ${1, 2}$ .

Finiteness results for Diophantine triples with repdigit values

Attila Bérczes, Florian Luca, István Pink, Volker Ziegler (2016)

Acta Arithmetica

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Let g ≥ 2 be an integer and $_{g} \subset ℕ$ be the set of repdigits in base g. Let $_{g}$ be the set of Diophantine triples with values in $_{g}$ ; that is, $_{g}$ is the set of all triples (a,b,c) ∈ ℕ³ with c < b < a such that ab + 1, ac + 1 and bc + 1 lie in the set $_{g}$ . We prove effective finiteness results for the set $_{g}$ .

An $a b c d$ theorem over function fields and applications

Pietro Corvaja, Umberto Zannier (2011)

Bulletin de la Société Mathématique de France

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We provide a lower bound for the number of distinct zeros of a sum $1 + u + v$ for two rational functions $u, v$ , in term of the degree of $u, v$ , which is sharp whenever $u, v$ have few distinct zeros and poles compared to their degree. This sharpens the “ $a b c d$ -theorem” of Brownawell-Masser and Voloch in some cases which are sufficient to obtain new finiteness results on diophantine equations over function fields. For instance, we show that the Fermat-type surface $x^{a} + y^{a} + z^{c} = 1$ contains only finitely many rational or elliptic...

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 systems of diophantine equations with a large number of solutions

Jerzy Browkin (2010)

Colloquium Mathematicae

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We consider systems of equations of the form $x_{i} + x_{j} = x_{k}$ and $x_{i} \cdot x_{j} = x_{k}$ , which have finitely many integer solutions, proposed by A. Tyszka. For such a system we construct a slightly larger one with much more solutions than the given one.

A remark on a Diophantine equation of S. S. Pillai

Azizul Hoque (2024)

Czechoslovak Mathematical Journal

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S. S. Pillai proved that for a fixed positive integer $a$ , the exponential Diophantine equation $x^{y} - y^{x} = a$ , $min (x, y) > 1$ , has only finitely many solutions in integers $x$ and $y$ . We prove that when $a$ is of the form $2 z^{2}$ , the above equation has no solution in integers $x$ and $y$ with $gcd (x, y) = 1$ .

The Diophantine Equation X³ = u+v over Real Quadratic Fields

Takaaki Kagawa (2011)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let k be a real quadratic field and let $_{k}$ and $_{k}^{\times}$ be the ring of integers and the group of units, respectively. A method of solving the Diophantine equation X³ = u+v ( $X \in_{k}$ , $u, v \in_{k}^{\times}$ ) is developed.