On the Brocard-Ramanujan problem and generalizations

Andrzej Dąbrowski

Displaying similar documents to “On the Brocard-Ramanujan problem and generalizations”

On a number theoretic conjecture on positive integral points in a 5-dimensional tetrahedron and a sharp estimate of the Dickman–De Bruijn function

Ke-Pao Lin, Xue Luo, Stephen S.-T. Yau, Huaiqing Zuo (2014)

Journal of the European Mathematical Society

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It is well known that getting the estimate of integral points in right-angled simplices is equivalent to getting the estimate of Dickman-De Bruijn function $ψ (x, y)$ which is the number of positive integers $\leq x$ and free of prime factors $> y$ . Motivating from the Yau Geometry Conjecture, the third author formulated the Number Theoretic Conjecture which gives a sharp polynomial upper estimate that counts the number of positive integral points in n-dimensional ( $n \geq 3$ ) real right-angled simplices. In this...

Results related to Huppert’s $ρ$ - $σ$ conjecture

Xia Xu, Yong Yang (2023)

Czechoslovak Mathematical Journal

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We improve a few results related to Huppert’s $ρ$ - $σ$ conjecture. We also generalize a result about the covering number of character degrees to arbitrary finite groups.

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 coarse embeddability into $ℓ_{p}$ -spaces and a conjecture of Dranishnikov

Piotr W. Nowak (2006)

Fundamenta Mathematicae

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We show that the Hilbert space is coarsely embeddable into any $ℓ_{p}$ for 1 ≤ p ≤ ∞. It follows that coarse embeddability into ℓ₂ and into $ℓ_{p}$ are equivalent for 1 ≤ p < 2.

Non-Wieferich primes in number fields and $a b c$ -conjecture

Srinivas Kotyada, Subramani Muthukrishnan (2018)

Czechoslovak Mathematical Journal

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Let $K / ℚ$ be an algebraic number field of class number one and let $𝒪_{K}$ be its ring of integers. We show that there are infinitely many non-Wieferich primes with respect to certain units in $𝒪_{K}$ under the assumption of the $a b c$ -conjecture for number fields.

Around the Littlewood conjecture in Diophantine approximation

Yann Bugeaud (2014)

Publications mathématiques de Besançon

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The Littlewood conjecture in Diophantine approximation claims that $inf_{q \geq 1} q \cdot ∥ q α ∥ \cdot ∥ q β ∥ = 0$ holds for all real numbers $α$ and $β$ , where $∥ \cdot ∥$ denotes the distance to the nearest integer. Its $p$ -adic analogue, formulated by de Mathan and Teulié in 2004, asserts that $inf_{q \geq 1} q \cdot ∥ q α ∥ \cdot {| q |}_{p} = 0$ holds for every real number $α$ and every prime number $p$ , where ${| \cdot |}_{p}$ denotes the $p$ -adic absolute value normalized by ${| p |}_{p} = p^{- 1}$ . We survey the known results on these conjectures and highlight recent developments. ...

On a conjecture of Dekking : The sum of digits of even numbers

Iurie Boreico, Daniel El-Baz, Thomas Stoll (2014)

Journal de Théorie des Nombres de Bordeaux

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Let $q \geq 2$ and denote by $s_{q}$ the sum-of-digits function in base $q$ . For $j = 0, 1, \dots, q - 1$ consider $# {0 \leq n < N : s_{q} (2 n) \equiv j (mod q)} .$ In 1983, F. M. Dekking conjectured that this quantity is greater than $N / q$ and, respectively, less than $N / q$ for infinitely many $N$ , thereby claiming an absence of a drift (or Newman) phenomenon. In this paper we prove his conjecture.

The generalized Hodge and Bloch conjectures are equivalent for general complete intersections

Claire Voisin (2013)

Annales scientifiques de l'École Normale Supérieure

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We prove that Bloch’s conjecture is true for surfaces with $p_{g} = 0$ obtained as $0$ -sets $X_{σ}$ of a section $σ$ of a very ample vector bundle on a variety $X$ with “trivial” Chow groups. We get a similar result in presence of a finite group action, showing that if a projector of the group acts as $0$ on holomorphic $2$ -forms of $X_{σ}$ , then it acts as $0$ on $0$ -cycles of degree $0$ of $X_{σ}$ . In higher dimension, we also prove a similar but conditional result showing that the generalized Hodge conjecture for general $X_{σ}$ ...

A variety of Euler's sum of powers conjecture

Tianxin Cai, Yong Zhang (2021)

Czechoslovak Mathematical Journal

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We consider a variety of Euler’s sum of powers conjecture, i.e., whether the Diophantine system $\{\begin{matrix} n = a_{1} + a_{2} + \dots + a_{s - 1}, \\ a_{1} a_{2} \dots a_{s - 1} (a_{1} + a_{2} + \dots + a_{s - 1}) = b^{s} \end{matrix}$ has positive integer or rational solutions $n$ , $b$ , $a_{i}$ , $i = 1, 2, \dots, s - 1$ , $s \geq 3 .$ Using the theory of elliptic curves, we prove that it has no positive integer solution for $s = 3$ , but there are infinitely many positive integers $n$ such that it has a positive integer solution for $s \geq 4$ . As a corollary, for $s \geq 4$ and any positive integer $n$ , the above Diophantine system has a positive rational solution. Meanwhile, we give conditions...

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.

Characterization of the alternating groups by their order and one conjugacy class length

Alireza Khalili Asboei, Reza Mohammadyari (2016)

Czechoslovak Mathematical Journal

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Let $G$ be a finite group, and let $N (G)$ be the set of conjugacy class sizes of $G$ . By Thompson’s conjecture, if $L$ is a finite non-abelian simple group, $G$ is a finite group with a trivial center, and $N (G) = N (L)$ , then $L$ and $G$ are isomorphic. Recently, Chen et al. contributed interestingly to Thompson’s conjecture under a weak condition. They only used the group order and one or two special conjugacy class sizes of simple groups and characterized successfully sporadic simple groups (see Li’s PhD dissertation)....

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.