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 $a^{k} + a^{l} + p^{α}$ ”

Fermat $k$ -Fibonacci and $k$ -Lucas numbers

Jhon J. Bravo, Jose L. Herrera (2020)

Mathematica Bohemica

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Using the lower bound of linear forms in logarithms of Matveev and the theory of continued fractions by means of a variation of a result of Dujella and Pethő, we find all $k$ -Fibonacci and $k$ -Lucas numbers which are Fermat numbers. Some more general results are given.

Lucas factoriangular numbers

Bir Kafle, Florian Luca, Alain Togbé (2020)

Mathematica Bohemica

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We show that the only Lucas numbers which are factoriangular are $1$ and $2$ .

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

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

Acta Arithmetica

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Erratum to the paper "On the disc theorem" (Ann. Polon. Math. 55 (1991), 1-10)

Cabiria Andreian Cazacu (1992)

Annales Polonici Mathematici

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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)

Acta Arithmetica

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

Karolina Krawciów (2011)

Colloquium Mathematicae

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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)

Časopis pro pěstování matematiky a fysiky

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Diophantine approximations with Fibonacci numbers

Victoria Zhuravleva (2013)

Journal de Théorie des Nombres de Bordeaux

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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.