Richard G. E. Pinch
A number $n$ is said to be economical if the prime power factorisation of $n$ can be written with no more digits than $n$ itself. We show that under a plausible hypothesis, related to the twin prime conjecture, there are arbitrarily long sequences of consecutive economial numbers, and exhibit such a sequence of length 9.
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$α$-numbers, diophantine exponent and factorisations of sturmian words
该研究引入Sturmian字的$α$-数和形式截距概念，通过Rauzy图构建形式体系，推导其一般分解公式，还计算了重复函数与丢番图指数，为Sturmian字分析提供新工具。
The classical congruences satisfied by the Fibonacci and Lucas sequences are reflected with the decomposition of primes in the ring generated by the gold number. This generalizes to establish a correspondence that we hope will be new between Chebotarev's theorem and the congruences satisfied by linear sequences. This link is done into the context of number field extensions. In particular we characterize primes ideals totally decomposed by simple congruences on the terms of linear recurrent sequences. Our results are illustrated by numerous examples, including Padovan sequencesof group $S_3$ an…
A sharp bound is obtained for the number of ways to express the monomial $X^n$ as a product of linear factors over $\mathbb{Z}/p^α\mathbb{Z}$. The proof relies on an induction-on-scale procedure which is used to estimate the number of solutions to a certain system of polynomial congruences. The method also applies to more general systems of polynomial congruences that satisfy a non-degeneracy hypothesis.
Let $\mathbb{F}_q$ be a finite field with $q$ elements, $ψ$ a non-zero element of $\mathbb{F}_q$, and $n$ an integer $\geq 3$ prime to $q$. The aim of this article is to show that the zeta function of the projective variety over $\mathbb{F}_q$ defined by $X_ψ\colon x_1^n+...+x_n^n - n ψx_1... x_n=0$ has, when $n$ is prime and $X_ψ$ is non singular (i.e. when $ψ^n \neq 1$), an explicit decomposition in factors coming from affine varieties of odd dimension $\leq n-4$ which are of hypergeometric type. The method we use consists in counting separately the number of points of $X_ψ$ and of some vari…
We extend our previous computations to show that there are 246683 Carmichael numbers up to $10^{16}$. As before, the numbers were generated by a back-tracking search for possible prime factorisations together with a ``large prime variation''. We present further statistics on the distribution of Carmichael numbers.
Any rational number can be factored into a product of several rationals whose sum vanishes. This simple but nontrivial fact was suggested as a problem on a maths olympiad for high-school students. We completely solve similar questions in all finite fields and in some other rings, e.g., in the complex and real matrix algebras. Also, we state several open questions.
We extend our previous computations to show that there are 585355 Carmichael numbers up to $10^{17}$. As before, the numbers were generated by a back-tracking search for possible prime factorisations together with a ``large prime variation''. We present further statistics on the distribution of Carmichael numbers.
We extend our previous computations to show that there are 1401644 Carmichael numbers up to $10^{18}$. As before, the numbers were generated by a back-tracking search for possible prime factorisations together with a ``large prime variation''. We present further statistics on the distribution of Carmichael numbers.
Let $f(n)$ denote the number of distinct unordered factorisations of the natural number $n$ into factors larger than 1.In this paper, we address some aspects of the function $f(n)$.
The first purpose of our paper is to show how Hooley's celebrated method leading to his conditional proof of the Artin conjecture on primitive roots can be combined with the Hardy-Littlewood circle method. We do so by studying the number of representations of an odd integer as a sum of three primes, all of which have prescribed primitive roots. The second purpose is to analyse the singular series. In particular, using results of Lenstra, Stevenhagen and Moree, we provide a partial factorisation as an Euler product and prove that this does not extend to a complete factorisation.
Factorisation of integers $n$ is of number theoretic and cryptographic significance. The Number Field Sieve (NFS) introduced circa 1990, is still the state of the art algorithm, but no rigorous proof that it halts or generates relationships is known. We propose and analyse an explicitly randomised variant. For each $n$, we show that these randomised variants of the NFS and Coppersmith's multiple polynomial sieve find congruences of squares in expected times matching the best-known heuristic estimates.
An Approximate Version of Vu's Theorem on Economical Subbases For Non-Integer Exponents
该论文证明了 Vu 关于经济基定理在非整数幂下的近似版本，给出了存在集合 𝔛 使得解数满足渐近等价条件，推广了原有结果。
In this article we compute the $q$th power values of the quadratic polynomials $f$ with negative squarefree discriminant such that $q$ is coprime to the class number of the splitting field of $f$ over $\mathbb{Q}$. The theory of unique factorisation and that of primitive divisors of integer sequences is used to deduce a bound on the values of $q$ which is small enough to allow the remaining cases to be easily checked. The results are used to determine all perfect power terms of certain polynomially generated integer sequences, including the Sylvester sequence.
We present a new way to factor the dirichlet convolution for completely multiplicative functions whitch led us to constructing a ring that arise from the operations involved in the factorisation. We will conclude by some identities that was found during this work. An application of the results gives us a generalisation of the following Hardy formula: $$ζ(x)^{2} = ζ(2x)\sum_{m=1}^{+\infty} \frac{2^{ω(m)}}{m^{x}}$$ which is: $$|ζ(z)|^{2} = ζ(2x)\sum_{m=1}^{+\infty}\frac{1}{m^{x}}2^{ω(m)}\prod_{p | m , p \in \mathbb{P}}^{ω(m)}\cos(y\ln(p^{v_{p}(m)}))$$ with: $z$ a complex number with $z = x+iy$ a…
We define a scheme for labelling and ordering integral ideals of number fields, including prime ideals as a special case. The order we define depends only on the choice of a monic irreducible integral defining polynomial for each field $K$, and we start by defining for each field its unique reduced defining polynomial, after Belabas. We define a total order on the set of prime ideals of $K$ and then extend this to a total order on the set of all nonzero integral ideals of $K$. This order allows us to give a unique label of the form $N.i$, where $N$ is its norm and $i$ is the index of the ideal…
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