Normally, when X is progress on a problem in number theory, n is a non-negative integer, strictly less than unity. Therefore it was with ebullience that, on the same day, I read of the proof of a centuries-old problem, and of admirable, and completely unexpected, progress towards a millenia-old problem. This short note attempts to explain the two problems and to give a brief outline of the meth...

numeration systems The motivation for the introduction of abstract numeration systems stemsfrom the celebrated theorem of Cobham dating back to 1969 about the so-called recognisable sets of integers in any integer base numeration system.An abstract numeration system is simply an infinite genealogically ordered(regular) language. In particular, this notion extends the usual integ...

Divisibility and primes Unless mentioned otherwise throughout this lecture all numbers are non-negative integers. We say that A divides B, denoted A|B if there’s a K such that KA = B. We say that P is prime if for A > 0, A|P only for A = 1 and A = P . Modulu For every two numbers A and B there is unique K and R such that 0 ≤ R ≤ B − 1 and A = KB + R. In this case we say that R = A (mod P ). Cle...

This is a collection of formalized proofs of many results of number theory. The proofs of the Chinese Remainder Theorem and Wilson’s Theorem are due to Rasmussen. The proof of Gauss’s law of quadratic reciprocity is due to Avigad, Gray and Kramer. Proofs can be found in most introductory number theory textbooks; Goldman’s The Queen of Mathematics: a Historically Motivated Guide to Number Theory...

I survey some of the connections between formal languages and number theory. Topics discussed include applications of representation in base k, representation by sums of Fibonacci numbers, automatic sequences, transcendence in nite characteristic , automaticreal numbers, xed points of homomorphisms, automaticity, and k-regular sequences.

Algebraic Number Theory: • What is it? The goals of the subject include: (i) to use algebraic concepts to deduce information about integers and other rational numbers and (ii) to investigate generalizations of the integers and rational numbers and develop theorems of a more general nature. Although (ii) is certainly of interest, our main point of view for this course will be (i). The focus of t...

Let ζ > e be arbitrary. In [31], the main result was the computation of p-adic topoi. We show that there exists an almost surely co-characteristic number. It was Hermite who first asked whether moduli can be classified. The work in [31] did not consider the bijective case.

An arithmetic function is a function f : N → C; there are many interesting and natural examples in analytic number theory. To begin with we consider what is perhaps the best known π(n) := xn: 1 P (x), the usual counting function of the primes. Various heuristic arguments suggest that one should expect x to be prime with probability 1/ log x, and coupled with a body of numerical evidence this pr...

Un domaine arithmétique où l’unité semble faire absolument défaut, c’est la théorie des nombres premiers ; on n’a trouvé que des lois asymptotiques et l’on n’en doit pas espérer d’autres; mais ces lois sont isolées et l’on n’y peut parvenir que par des chemins différents qui ne semblent pas pouvoir communiquer entre eux. Je crois entrevoir d’où sortira l’unité souhaitée, mais je ne l’entrevois ...