On generalized Fermat equations of signature (p,p,3)

Karolina Krawciów

Displaying similar documents to “On generalized Fermat equations of signature (p,p,3)”

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.

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 ${1, 2}$ .

On the Diophantine equation $(2^{x} - 1) (p^{y} - 1) = 2 z^{2}$

Ruizhou Tong (2021)

Czechoslovak Mathematical Journal

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Let $p$ be an odd prime. By using the elementary methods we prove that: (1) if $2 ∤ x$ , $p \equiv \pm 3 (mod 8),$ the Diophantine equation $(2^{x} - 1) (p^{y} - 1) = 2 z^{2}$ has no positive integer solution except when $p = 3$ or $p$ is of the form $p = 2 a_{0}^{2} + 1$ , where $a_{0} > 1$ is an odd positive integer. (2) if $2 ∤ x$ , $2 ∣ y$ , $y \neq 2, 4,$ then the Diophantine equation $(2^{x} - 1) (p^{y} - 1) = 2 z^{2}$ has no positive integer solution.

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

Gökhan Soydan, László Németh, László Szalay (2018)

Archivum Mathematicum

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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.

Integral points on the elliptic curve $y^{2} = x^{3} - 4 p^{2} x$

Hai Yang, Ruiqin Fu (2019)

Czechoslovak Mathematical Journal

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Let $p$ be a fixed odd prime. We combine some properties of quadratic and quartic Diophantine equations with elementary number theory methods to determine all integral points on the elliptic curve $E : y^{2} = x^{3} - 4 p^{2} x$ . Further, let $N (p)$ denote the number of pairs of integral points $(x, \pm y)$ on $E$ with $y > 0$ . We prove that if $p \geq 17$ , then $N (p) \leq 4$ or $1$ depending on whether $p \equiv 1 (mod 8)$ or $p \equiv - 1 (mod 8)$ .

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 $gcd (x, y) = 1$ .

Lucas sequences and repdigits

Hayder Raheem Hashim, Szabolcs Tengely (2022)

Mathematica Bohemica

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Let ${(G_{n})}_{n \geq 1}$ be a binary linear recurrence sequence that is represented by the Lucas sequences of the first and second kind, which are ${U_{n}}$ and ${V_{n}}$ , respectively. We show that the Diophantine equation $G_{n} = B \cdot (g^{l m} - 1) / (g^{l} - 1)$ has only finitely many solutions in $n, m \in ℤ^{+}$ , where $g \geq 2$ , $l$ is even and $1 \leq B \leq g^{l} - 1$ . Furthermore, these solutions can be effectively determined by reducing such equation to biquadratic elliptic curves. Then, by a result of Baker (and its best improvement due to Hajdu and Herendi) related to the bounds of the integral...

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.

Complete solution of the Diophantine equation $x^{y} + y^{x} = z^{z}$

Mihai Cipu (2019)

Czechoslovak Mathematical Journal

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The triples $(x, y, z) = (1, z^{z} - 1, z)$ , $(x, y, z) = (z^{z} - 1, 1, z)$ , where $z \in ℕ$ , satisfy the equation $x^{y} + y^{x} = z^{z}$ . In this paper it is shown that the same equation has no integer solution with $min {x, y, z} > 1$ , thus a conjecture put forward by Z. Zhang, J. Luo, P. Z. Yuan (2013) is confirmed.

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.

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 $_{g}$ .