THE FACTORADIC INTEGERS Joshua S. Brinseld Thesis submitted to the Faculty of the Virginia Polytechnic Institute and State University in partial fulllment of the requirements for the degree of Master of Science in Mathematics William J. Floyd, Chair Ezra A. Brown Leonardo C. Mihalcea May 5, 2016 Blacksburg, Virginia Keywords: Distributive Law, Arithmetic Progression, P-adic Number, Factorial Copyright 2016, Joshua Sol Brinseld The Factoradic Integers Joshua S. Brinseld (ABSTRACT) The arithmetic progressions, considered as functions x 7! λx+φ equipped with addition and composition, are examined from an algebraic standpoint as left outer-distributive rings, i.e. objects satisfying the ring axioms with a weakened left-distributive law: namely, x(y + z) = xy + xz + D(x), where D is a function dependent only on x. Under appropriate constraints on λ and φ, the images of these functions give bases for topologies on Z (or any unique factorization domain); the ring of factoradic integers Z¯ is dened as the completion of the topology generated on Z by the constraints λ 2 N, φ 2 Z (i.e. the evenly spaced integer topology) under the metric 1 , maximal such that . It is shown d!(n; m) = N! N N! j (n − m) that ¯ is ring-isomorphic to the direct product of the -adic integers over all primes , i.e. Z p Zp p to the pronite completion of the integers. Analogies between Z¯ and Z are exploited to allow for general factoradic integers to be used as exponents; this results in a unique factorization theorem which completely characterizes multiplication in Z¯, giving the multiplicative group of units as ∼ Q , where the direct product is over all odd primes U = Z=2Z×Z2 × Z=(p−1)Z×Zp p p. Similarly, bounds of summation are extended to take general values in Z¯, and continuous functions are examined in terms of these factoradic series and the nite dierence operator. It is found that a continuous function is equal to its own Newton series if and only if it can be decomposed as a direct product of functions over all primes . The relationship Zp ! Zp p between Z¯ and certain completions of polynomial rings is then examined, and it is shown that ¯ ∼= R=xR, where R = lim [x]=x(x + 1):::(x + n) [x]: Z − Z Z The Factoradic Integers Joshua S. Brinseld (GENERAL ABSTRACT) The arithmetic progressions under addition and composition satisfy the usual rules of arith- metic with a modied distributive law. The basic algebra of such mathematical struc- tures is examined; this leads to the consideration of the integers as a metric space un- der the factoradic metric, i.e., the integers equipped with a distance function dened by d(n; m) = 1=N!, where N is the largest positive integer such that N! divides n − m. Via the process of metric completion, the integers are then extended to a larger set of numbers, the factoradic integers. The properties of the factoradic integers are developed in detail, with particular attention to prime factorization, exponentiation, innite series, and continuous functions, as well as to polynomials and their extensions. The structure of the factoradic in- tegers is highly dependent upon the distribution of the prime numbers and relates to various topics in algebra, number theory, and non-standard analysis. Contents Introduction 1 Motivation 1 Notation and Background 1 Topology 1 Algebra 3 Modular Arithmetic 4 P-Adic Numbers 5 Factoradic Integers 6 Z¯ and Related Constructions 9 Part 1. Left Outer-Distributive Rings 11 1. Denition & General Properties 11 2. Wave Spaces 13 Part 2. The Factoradic Integers 17 3. Construction 17 4. The Factoradic Rationals 22 5. The Digit-Flip Function 23 6. Sequences & Series 25 6.1. Innite Series 25 6.2. Factoradic Limits 26 6.3. Factoradic Sequences in Z¯ 27 7. Exponentiation 29 8. The Units 30 8.1. Units and Zero-Divisors 30 8.2. Topological Z¯-Modules and the Multiplicative Group of Units 32 8.3. The Total Logarithm 35 9. Continuous Functions 35 9.1. Some Useful Functions 35 9.2. Finite Dierences 37 iv 9.3. Direct & Indirect Continuity 39 9.4. Factoradic Series 41 9.5. Geometric Series of Units 45 10. Projective Limits 45 10.1. Projective Limits 45 10.2. Direct Limits & Prüfer p-Groups 47 Part 3. Projective Limits and Polynomial Completions 48 11. Power Series in Z¯ 48 12. F[x] 49 13. lim [x]=x!n [x] 50 −Z Z Bibliography 53 References 53 v Introduction Motivation The prime numbers are easily dened in terms of multiplication, but their additive distri- bution is extremely complex; a similar situation holds for many sets of integers of number- theoretic interest. Indeed, multiplying a number by a sum is trivial in light of the distributive law a(b+c) = ab+ac, but there is no such general formula for adding a number to a product; intuitively, addition is too fundamental to be so easily described. The structure of the inte- gers, however, is entirely determined by the interplay between addition and multiplication. The simplest functions involving both addition and multiplication are the waves, i.e. the functions f(x) = λx + φ for some xed λ, φ 2 Z, whose images are fundamental periodic sets which correspond to the bidirectionally innite arithmetic progressions. It is the object of this paper to examine these waves from two perspectives. First, it is observed that as functions under addition and composition they satisfy a modication of the ring axioms with an altered distributive law, and the basic algebra of such structures is investigated. The case of waves over unique factorization domains is given special attention, resulting in a collection of topologies over any unique factorization domain R whose basis elements are wave images, i.e. sets of the form . Second, the topology generated fλx + φgx2R by all non-singleton wave images over the integers is given a particular metric and completed, extending Z to the larger topological ring of factoradic integers Z¯, and the properties of Z¯ are investigated. In particular, a unique factorization theorem is proven for Z¯ which characterizes multiplication in Z¯ completely, including the structure of the multiplicative group of units, and innite summations over Z¯ are developed in some detail. As a ring, Z¯ is found to be isomorphic to the pronite completion of the integers, i.e. to the direct product of the p-adic integers over all primes p; several alternative constructions are given, including the pronite construction via projective limits and a construction as a quotient of a certain completion of the polynomial ring Z[x]. Notation and Background Notationally, the natural numbers include all positive integers but not ; denotes N 0 N0 N [ f0g. As usual, Z denotes the set of integers, Q the set of rational numbers, and R the set of real numbers. When the meaning is clear, subscripts may be used to modify sets in obvious ways, e.g. denotes the nonzero integers, denotes the negative real numbers, Z6=0 R<0 etc. The symbol P always denotes the set of prime numbers, i.e. the positive integers each of which is divisible only by itself and by 1. The number 1 itself does not count as prime. Basic facts and denitions from algebra, modular arithmetic, and topology are assumed as common knowledge; this section elucidates some standard denitions and basic results from these elds. Topology. Given a set X, a topology is a collection T of subsets of X such that X; ; 2 T , T is closed under nite intersections of elements, and T is closed under arbitrary unions of 1 elements; the sets in T are called open sets, and their complements are called closed sets. If a set is both open and closed, it is called clopen. A basis for a topology T is a set B ⊂ T such that every element of T is a union of elements of B; a set B is a basis for a topology on X whenever 8x 2 X 9b 2 B such that x 2 b, and 8a; b 2 B if x 2 a \ b then 9c 2 B such that x 2 c ⊂ a \ b. A set X equipped with a topology T is called a topological space. Given two topological spaces X and Y , a function f : X ! Y is continuous if preim- ages of open sets are open; that is, if whenever S ⊂ Y is an open set in Y , f −1(S) = fx 2 X : f(x) 2 Sg is an open set in X. If there exists a continuous bijection f : X ! Y which has a continuous inverse, we say f is a homeomorphism between X and Y , which are consequently homeomorphic topological spaces. A metric space is a set together with a function , called a metric, X X d: X × X ! R≥0 which satises the following: (1) (Identity) 8x; y 2 X d(x; y) = 0 if and only if x = y. (2) (Symmetry) 8x; y 2 X d(x; y) = d(y; x). (3) (Triangle Inequality) 8x; y; z 2 X d(x; z) ≤ d(x; y) + d(y; z). An ultrametric is a metric satisfying d(x; z) ≤ max fd(x; y); d(y; z)g, a stronger form of the triangle inequality. If X is a metric space, we can dene the open metric balls as follows: if and , then the open metric ball of radius centered at is r 2 R>0 x 2 X r x Br(x) := fy 2 X : d(x; y) < rg. It is also convenient to consider the closed metric balls, which allow d(x; y) = r; the closed metric ball of radius r centered at x is Br[x] = fy 2 X : d(x; y) ≤ rg.