On the binary expansions of algebraic numbers

David H. Bailey; Jonathan M. Borwein; Richard E. Crandall; Carl Pomerance

Displaying similar documents to “On the binary expansions of algebraic numbers”

On the greatest prime factor of $n^{2} + 1$

Jean-Marc Deshouillers, Henryk Iwaniec (1982)

Annales de l'institut Fourier

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There exist infinitely many integers $n$ such that the greatest prime factor of $n^{2} + 1$ is at least $n^{6 / 5}$ . The proof is a combination of Hooley’s method – for reducing the problem to the evaluation of Kloosterman sums – and the majorization of Kloosterman sums on average due to the authors.

On the multiplicities of binary complex recurrences

F. Beukers, R. Tijdeman (1984)

Compositio Mathematica

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New bounds on the length of finite pierce and Engel series

P. Erdös, J. O. Shallit (1991)

Journal de théorie des nombres de Bordeaux

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Every real number $x, 0 < x \leq 1$ , has an essentially unique expansion as a Pierce series : $x = \frac{1}{x^{1}} - \frac{1}{x^{1} x^{2}} + \frac{1}{x^{1} x^{2} x^{3}} - \dots$ where the $x_{i}$ form a strictly increasing sequence of positive integers. The expansion terminates if and only if $x$ is rational. Similarly, every positive real number $y$ has a unique expansion as an Engel series : $y = \frac{1}{y^{1}} - \frac{1}{y^{1} y^{2}} + \frac{1}{y^{1} y^{2} y^{3}} + \dots$ where the $y_{i}$ form a (not necessarily strictly) increasing sequence of positive integers. If the expansion is infinite, we require that the sequence yi...

On the sequence of numbers of the form $ε ₀ + ε ₁ q + . . . + ε_{n} q^{n}$ , $ε_{i} \in 0, 1$

Paul Erdős, István Joó, Vilmos Komornik (1998)

Acta Arithmetica

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The complex sum of digits function and primes

Jörg M. Thuswaldner (2000)

Journal de théorie des nombres de Bordeaux

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Canonical number systems in the ring of gaussian integers $ℤ [i]$ are the natural generalization of ordinary $q$ -adic number systems to $ℤ [i]$ . It turns out, that each gaussian integer has a unique representation with respect to the powers of a certain base number $b$ . In this paper we investigate the sum of digits function $ν_{b}$ of such number systems. First we prove a theorem on the sum of digits of numbers, that are not divisible by the $f$ -th power of a prime. Furthermore, we establish an Erdös-Kac type...

Linear forms in the logarithms of three positive rational numbers

Curtis D. Bennett, Josef Blass, A. M. W. Glass, David B. Meronk, Ray P. Steiner (1997)

Journal de théorie des nombres de Bordeaux

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In this paper we prove a lower bound for the linear dependence of three positive rational numbers under certain weak linear independence conditions on the coefficients of the linear forms. Let $Λ = b_{2} log α_{2} - b_{1} log α_{1} - b_{3} log α_{3} \neq 0$ with $b_{1}, b_{2}, b_{3}$ positive integers and $α_{1}, α_{2}, α_{3}$ positive multiplicatively independent rational numbers greater than $1$ . Let $α_{j 1} = α_{j 1} / α_{j 2}$ with $α_{j 1}, α_{j 2}$ coprime positive integers $(j = 1, 2, 3)$ . Let $α_{j} \geq max {α_{j 1}, e}$ and assume that gcd $(b_{1}, b_{2}, b_{3}) = 1 .$ Let $b^{'} = (\frac{b_{2}}{log α_{1}} + \frac{b_{1}}{log α_{2}}) (\frac{b_{2}}{log α_{3}} + \frac{b_{3}}{log α_{2}})$ and assume that $B \geq max {10, log b^{'}} .$ We prove that either ${b_{1}, b_{2}, b_{3}}$ is $(c_{4}, B)$ -linearly dependent over $ℤ$ (with respect to $(a_{1}, a_{2}, a_{3})$ )...

On a theorem of Erdös-Kac

A Rényi, P Turán (1958)

Acta Arithmetica

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$P_{2}$ in short intervals

Henryk Iwaniec, M. Laborde (1981)

Annales de l'institut Fourier

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For any sufficiently large real number $x$ , the interval $[x, x + x^{0, 45}]$ contains at least one integer having at most two prime factors .