Concept

Euclid's theorem

Related publications (8)

Topological Effects in Carbon-based Two-dimensional Nanostructures

Recent advances on low-dimensional and topological materials has greatly inspired the research in condensed matter physics. This thesis is devoted to the computational and theoretical study of topological effects in two-dimensional materials, especially na ...

EPFL2023

A new elementary proof of the Prime Number Theorem

Florian Karl Richter

Let

\Omega(n)

denote the number of prime factors of

n

. We show that for any bounded

f\colon\mathbb{N}\to\mathbb{C}

one has [ \frac{1}{N}\sum_{n=1}^N, f(\Omega(n)+1)=\frac{1}{N}\sum_{n=1}^N, f(\Omega(n))+\mathrm{o}_{N\to\infty}(1). ] This yields a ...

2021

Dynamical generalizations of the Prime Number Theorem and disjointness of additive and multiplicative semigroup actions

Florian Karl Richter

We establish two ergodic theorems which have among their corollaries numerous classical results from multiplicative number theory, including the Prime Number Theorem, a theorem of Pillai-Selberg, a theorem of Erd\H{o}s-Delange, the mean value theorem of Wi ...

2020

Efficient Fail-Stop Signatures from the Factoring Assumption

Khaled Ouafi, Atefeh Mashatan

In this paper, we revisit the construction of fail-stop signatures from the factoring assumption. These signatures were originally proposed to provide information-theoretic-based security against forgeries. In contrast to classical signature schemes, in wh ...

Springer2011

High-Performance Modular Multiplication on the Cell Processor

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This paper presents software implementation speed records for modular multiplication arithmetic on the synergistic processing elements of the Cell broadband engine (Cell) architecture. The focus is on moduli which are of special interest in elliptic curve ...

Springer-Verlag New York, Ms Ingrid Cunningham, 175 Fifth Ave, New York, Ny 10010 Usa2010

High-Performance Modular Multiplication on the Cell Processor

Joppe Willem Bos

Springer Berlin Heidelberg2010

Efficient generation of prime numbers

Serge Vaudenay

The generation of prime numbers underlies the use of most public-key schemes, essentially as a major primitive needed for the creation of key pairs or as a computation stage appearing during various cryptographic setups. Surprisingly, despite decades of in ...

2000

The Factorization of the ninth Fermat Number

Arjen Lenstra

In this paper we exhibit the full prime factorization of the ninth Fermat number F9 = 2(512) + 1. It is the product of three prime factors that have 7, 49, and 99 decimal digits. We found the two largest prime factors by means of the number field sieve, wh ...

American Mathematical Society1993

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A new elementary proof of the Prime Number Theorem

Florian Karl Richter

Let

\Omega(n)

denote the number of prime factors of

n

. We show that for any bounded

f\colon\mathbb{N}\to\mathbb{C}

one has [ \frac{1}{N}\sum_{n=1}^N, f(\Omega(n)+1)=\frac{1}{N}\sum_{n=1}^N, f(\Omega(n))+\mathrm{o}_{N\to\infty}(1). ] This yields a ...

2021