Fermat numbers and integers of the form $a^k+a^l+p^{α}$

Yong-Gao Chen; Rui Feng; Nicolas Templier

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Displaying similar documents to “Fermat numbers and integers of the form $a^{k} + a^{l} + p^{α}$ ”

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Zhi-Wei Sun, Mao-Hua Le (2001)

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Cabiria Andreian Cazacu (1992)

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Due to a technical error, part of a sentence was omitted on the top of page 8. The first line should read: “where $f_{p}^{k}$ , $p = a_{l}$ or $b_{l}$ , means the number of folds of the covering $(δ_{k}^{''}, T |, Δ_{l}^{''})$ ending at p, i.e. covering a neighbourhood of p in $a_{l} b_{l}$ without covering p itself”.

Integers not of the form $c (2^{a} + 2^{b}) + p^{α}$

Pingzhi Yuan (2004)

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

Karolina Krawciów (2011)

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This paper focuses on the Diophantine equation $x ⁿ + p^{α} y ⁿ = M z ³$ , with fixed α, p, and M. We prove that, under certain conditions on M, this equation has no non-trivial integer solutions if $n \geq ℱ (M, p^{α})$ , where $ℱ (M, p^{α})$ is an effective constant. This generalizes Theorem 1.4 of the paper by Bennett, Vatsal and Yazdani [Compos. Math. 140 (2004), 1399-1416].

An application of the method wherein the $p h i$ - and $c h i$ - methods are combined for the determination of the grating constant. [II.]

Swami Jnanananda (1936)

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Diophantine approximations with Fibonacci numbers

Victoria Zhuravleva (2013)

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Let $F_{n}$ be the $n$ -th Fibonacci number. Put $ϕ = \frac{1 + \sqrt{5}}{2}$ . We prove that the following inequalities hold for any real $α$ : 1) ${inf}_{n \in ℕ} | | F_{n} α | | \leq \frac{ϕ - 1}{ϕ + 2}$ , 2) ${lim inf}_{n \to \infty} | | F_{n} α | | \leq \frac{1}{5}$ , 3) ${lim inf}_{n \to \infty} | | ϕ^{n} α | | \leq \frac{1}{5}$ . These results are the best possible.

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K. Orlov (1981)

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A. Makowski (1964)

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А.М. Вершик (1972)

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P. Srivastava, K. K. Azad (1981)

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Ralph McKenzie (1971)

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Janusz Matkowski (1989)

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Stephan Baier (2004)

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The existence of $P (α)$ -points of N* for $ℵ_{0} α < 𝔠$

A. Szymański (1977)

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Jan A. Rempała (1985)

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M. K. Sen (1971)

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Stephan Baier (2005)

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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.