On the distribution of Hawkins’ random “primes”

Tanguy Rivoal

Displaying similar documents to “On the distribution of Hawkins’ random “primes””

On an arithmetic function considered by Pillai

Florian Luca, Ravindranathan Thangadurai (2009)

Journal de Théorie des Nombres de Bordeaux

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For every positive integer $n$ let $p (n)$ be the largest prime number $p \leq n$ . Given a positive integer $n = n_{1}$ , we study the positive integer $r = R (n)$ such that if we define recursively $n_{i + 1} = n_{i} - p (n_{i})$ for $i \geq 1$ , then $n_{r}$ is a prime or $1$ . We obtain upper bounds for $R (n)$ as well as an estimate for the set of $n$ whose $R (n)$ takes on a fixed value $k$ .

An arithmetic function arising from Carmichael’s conjecture

Florian Luca, Paul Pollack (2011)

Journal de Théorie des Nombres de Bordeaux

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Let $φ$ denote Euler’s totient function. A century-old conjecture of Carmichael asserts that for every $n$ , the equation $φ (n) = φ (m)$ has a solution $m \neq n$ . This suggests defining $F (n)$ as the number of solutions $m$ to the equation $φ (n) = φ (m)$ . (So Carmichael’s conjecture asserts that $F (n) \geq 2$ always.) Results on $F$ are scattered throughout the literature. For example, Sierpiński conjectured, and Ford proved, that the range of $F$ contains every natural number $k \geq 2$ . Also, the maximal order of $F$ has been investigated by Erdős and Pomerance....

Mythical numbers.

Bázlik, Miro (2009)

Acta Mathematica Universitatis Comenianae. New Series

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On Robin’s criterion for the Riemann hypothesis

YoungJu Choie, Nicolas Lichiardopol, Pieter Moree, Patrick Solé (2007)

Journal de Théorie des Nombres de Bordeaux

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Robin’s criterion states that the Riemann Hypothesis (RH) is true if and only if Robin’s inequality $σ (n) : = \sum_{d | n} d < e^{γ} n log log n$ is satisfied for $n \geq 5041$ , where $γ$ denotes the Euler(-Mascheroni) constant. We show by elementary methods that if $n \geq 37$ does not satisfy Robin’s criterion it must be even and is neither squarefree nor squarefull. Using a bound of Rosser and Schoenfeld we show, moreover, that $n$ must be divisible by a fifth power $> 1$ . As consequence we obtain that RH holds true iff every natural number divisible by...

On the ternary Goldbach problem with primes in arithmetic progressions having a common modulus

Karin Halupczok (2009)

Journal de Théorie des Nombres de Bordeaux

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For $ε > 0$ and any sufficiently large odd $n$ we show that for almost all $k \leq R : = n^{1 / 5 - ε}$ there exists a representation $n = p_{1} + p_{2} + p_{3}$ with primes $p_{i} \equiv b_{i}$ mod $k$ for almost all admissible triplets $b_{1}, b_{2}, b_{3}$ of reduced residues mod $k$ .

Non-degenerate Hilbert cubes in random sets

Csaba Sándor (2007)

Journal de Théorie des Nombres de Bordeaux

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A slight modification of the proof of Szemerédi’s cube lemma gives that if a set $S \subset [1, n]$ satisfies $| S | \geq \frac{n}{2}$ , then $S$ must contain a non-degenerate Hilbert cube of dimension $⌊ {log}_{2} {log}_{2} n - 3 ⌋$ . In this paper we prove that in a random set $S$ determined by $\Pr {s \in S} = \frac{1}{2}$ for $1 \leq s \leq n$ , the maximal dimension of non-degenerate Hilbert cubes is a.e. nearly ${log}_{2} {log}_{2} n + {log}_{2} {log}_{2} {log}_{2} n$ and determine the threshold function for a non-degenerate $k$ -cube.