On rough and smooth neighbors.

William D. Banks; Florian Luca; Igor E. Shparlinski

Displaying similar documents to “On rough and smooth neighbors.”

.121221222... is not quadratic.

Florian Luca (2005)

Revista Matemática Complutense

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In this note, we show that if b > 1 is an integer, f(X) ∈ Q[X] is an integer valued quadratic polynomial and K > 0 is any constant, then the b-adic number ∑ (a / b), where a ∈ Z and 1 ≤ |a| ≤ K for all n ≥ 0, is neither rational nor quadratic.

When the group-counting function assumes a prescribed integer value at squarefree integers frequently, but not extremely frequently

Claudia A. Spiro-Silverman (1992)

Acta Arithmetica

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Power-free values, large deviations, and integer points on irrational curves

Harald A. Helfgott (2007)

Journal de Théorie des Nombres de Bordeaux

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Let $f \in ℤ [x]$ be a polynomial of degree $d \geq 3$ without roots of multiplicity $d$ or $(d - 1)$ . Erdős conjectured that, if $f$ satisfies the necessary local conditions, then $f (p)$ is free of $(d - 1)$ th powers for infinitely many primes $p$ . This is proved here for all $f$ with sufficiently high entropy. The proof serves to demonstrate two innovations: a strong repulsion principle for integer points on curves of positive genus, and a number-theoretical analogue of Sanov’s theorem from the theory of large deviations. ...

Variants of the Brocard-Ramanujan equation

Omar Kihel, Florian Luca (2008)

Journal de Théorie des Nombres de Bordeaux

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In this paper, we discuss variations on the Brocard-Ramanujan Diophantine equation.

Restriction theory of the Selberg sieve, with applications

Ben Green, Terence Tao (2006)

Journal de Théorie des Nombres de Bordeaux

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The Selberg sieve provides majorants for certain arithmetic sequences, such as the primes and the twin primes. We prove an $L^{2}$ – $L^{p}$ restriction theorem for majorants of this type. An immediate application is to the estimation of exponential sums over prime $k$ -tuples. Let $a_{1}, \dots, a_{k}$ and $b_{1}, \dots, b_{k}$ be positive integers. Write $h (θ) : = \sum_{n \in X} e (n θ)$ , where $X$ is the set of all $n \leq N$ such that the numbers $a_{1} n + b_{1}, \dots, a_{k} n + b_{k}$ are all prime. We obtain upper bounds for ${∥ h ∥}_{L^{p} (𝕋)}$ , $p > 2$ , which are (conditionally on the Hardy-Littlewood prime tuple conjecture) of the correct...

Equations involving arithmetic functions of factorials.

Luca, Florian (2000)

Divulgaciones Matemáticas

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Some solved and unsolved problems in combinatorial number theory, ii

P. Erdős, A. Sárközy (1993)

Colloquium Mathematicae

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In an earlier paper [9], the authors discussed some solved and unsolved problems in combinatorial number theory. First we will give an update of some of these problems. In the remaining part of this paper we will discuss some further problems of the two authors.