Second Draft NeutralMichael Thomas DeDigits Vlieger Abstract n r (n does not divide r evenly, or n is not a divisor of r), and n r (n is coprime to r, or n is relatively prime to r) is also Elementary number theorists are familiar with two major important to this investigation. Note that in this paper, we will classes of digits n of a number base r. The divisor counting use the term “coprime” adjectivally when discussing a digit of function enumerates all divisors n r and the Euler totient r, and use the noun “totative” for a digit which is coprime to r. function counts all totatives n r. A set of “neutral digits,” nei- We will use the term “factor” at times exclusively for divisors ther divisors of nor coprime to r are investigated. Two types of digits when contrast needs to be made between factors of a of neutral digits are proved to exist. Methods of construction digit and divisors of r. Numbers considered in this paper are and quantification of such digits are introduced. generally positive integers, except the number r½, the square Keywords: digit, number base, radix, divisor, totative, coprime, root of r. neutral digit, fundamental theorem of arithmetic, divisor count- Some observations regarding prime numbers should be ing function, Euler totient function, regular digit, regular num- kept in mind. Consider an arbitrary prime p and a larger ar- ber, semidivisor, semi-coprime number, semitotative. bitrary positive integer r. The prime digit p must either be a divisor of r or coprime to r [15]. A prime base p is the product Mathematics Subject Classification (msc-2010) 11A99, 11N25. of its trivial divisors {1, p} and is coprime to all numbers n < p, including n = 1. By these observations, two things are clear: 1. Introduction 1. Prime digits p must either be divisors or totatives of This work regards possible relationships between digits n an arbitrary number base r. We need not examine of a number base r. This paper is the first of a series of papers prime digits to determine if they are neutral. and articles examining the nature and practical application of number bases. Thus, we will employ the variables mentioned 2. No neutral digit can exist for a prime base p. We need rather than the usual variables seen in elementary number not consider the existence of neutral digits for prime theory to convey the equivalent of digits and number bases. bases; they do not exist. A digit is a positive integer 0 < n ≤ r, where the symbol Therefore, neutral digits must be composite and must exist “0” (i.e., zero) symbolizes congruence with r. Observe that in only in composite bases. the decimal number 20, the digit zero in the rightmost place Three formulæ prove essential in this investigation. The simply signifies that the quantity twenty is congruent with first is the standard form of prime decomposition denoting 101. When the digit 0 stands alone, the symbol “0” stands for the distinct prime divisors of r and their exponents α [16]. Let actual zero. We will ignore the case where “0” connotes actual i and k be positive integers with 1 ≤ i ≤ k. Let pk be the k-th zero, since this produces complications. Thus in this paper, distinct prime divisor of r, and the integer αi ≥ 1 be the multi- [1] [17] the digit 0 r. The digit ω = (r – 1) is the greatest and final plicity of the distinct prime divisor pi in r. digit of base r. Thus the “digit range” of base r is {0, …, ω}, α1 α2 αk includes all possible digits of r, and numbers r elements. (1.1) r = p1 p2 …pk (α1 > 0, α2 > 0, …, αk > 0, p1 < p2 < … < pk) 0 1 2 3 4 5 6 7 8 9 The second key formula relates a divisor d with a divisor’s complement d′, both the divisor and its complement being [18] Figure 1.1. The range of digits n of base r = 10. positive integers. (1.2) r = d · d′ A number base or radix is a positive integer r ≥ 2. [2] To be sure, a radix need not be confined to positive integers, but for Let the integer 1 ≤ δi ≤ αi be the multiplicity of each of the base the sake of this study, we’ll focus on the case where r is strictly r’s distinct prime divisors pi in the divisor d. Each divisor d a positive integer ≥ 2. thus has a standard form prime decomposition Let’s review some basic aspects of elementary number δ1 δ2 δr (1.3) d = p1 p2 …pr theory before exploring the relationship of digits and number (p < p < … < p ) bases. We will presume a knowledge of primes [3, 4, 5, 6], com- 1 2 k posites [7, 8, 9, 10, 11], and units [12, 13, 14]. Familiarity with the three Formulæ specific to certain aspects of this paper will be intro- relationships n r (n divides r evenly, or n is a divisor of r), duced later. Neutral Digits • 1 2. The Existence of Neutral Digits. Proof. We can rewrite formula (3) as k Let D be the set of divisors of r, and T be the set of totatives k (2.3) σ0(ri) = (αi + 1) = 2 of r. Let d D be a positive integer d r and t T be a positive Πi = 1 Each prime divisor p contributes the factor (α + 1) to integer t r. There are two well-known functions which count i i the equation. Since all multiplicities α = 1, all the factors the number of divisors in the set D and totatives in the set T. i (α + 1) = 2. Lemma 2.1.1 shows that a prime r yields 2 divi- Through examination of the behavior of these functions, we i can see that there exist integers 0 < s < r for some values of r sors. Each additional distinct prime divisor contributes a fac- that are neither divisors of nor coprime to r. In the case of r as tor 2 to formula (2.3). a number base, these numbers s are digits, because these are Lemma 2.1.3. The quantity of divisors for all primorials r# = k less than or equal to r, yet still positive. Thus, we will refer to p1 · p2 · … · pk is 2 . such numbers s as “neutral digits”. The primorial r# is a special case of Lemma 2.1.2. Let j ≥ 1 Figures 1.2 and 1.3 show the digits d r and t r respec- be a positive integer. Let π be a prime number such that πj is tively, given r = 10. the j-th prime, with π1 = 2. If the prime decomposition of the primorial r# is in standard form, then i = j. The least prime divisor p1 = π1 = 2. Further, each prime divisor pi is the j-th 0 1 2 3 4 5 6 7 8 9 prime number πj. Figure 1.2. The divisors d of base r = 10, shown in red. (2.4) pi = πj Since r# is squarefree, we can use formula (2.3) to compute 0 1 2 3 4 5 6 7 8 9 its divisor counting function. A primorial r# and a squarefree r differ in two possible ways. The primorial will have its least Figure 1.3. The totatives t of base r = 10, shown in blue. prime divisor p1 = 2, while the squarefree r may have an arbi- trary prime as its least prime divisor. The primorial r# will have The divisor counting function. its i-th prime divisor pi = πj, the j-th prime number, and i = j for all prime divisors of r#. The prime divisors p of a squarefree The divisor counting function, σ0(r), counts the number i of positive divisors of the integer r [19, 20]. Let p denote a prime r may be arbitrary, with i not necessarily matching j. Thus a squarefree r is a primorial r# if and only if the prime decom- number. Then 0σ (p) = 2, since all prime numbers possess the [21] position is in standard form and i = j for all p . An example of a trivial divisors {1, p} . Because σ0 is a multiplicative func- i [22] primorial r# is 30 = 2 · 3 · 5. Each p = π and p = 2. An example tion , we can produce σ0 for any number r given its prime i j 1 decomposition in standard form given by formula (1.1). Let i of a squarefree r which is not a primorial is 165 = 3 · 5 ·11. The and k be positive integers with 1 ≤ i ≤ k. least prime divisor p1 ≠ 2 and all prime divisors pi ≠ πj. The k primorial r# is a special case of a squarefree r. Additionally, the (2.1) σ0(r) = (αi + 1) = (α1 + 1)(α2 + 1)…(αk + 1). primorial r # is the smallest squarefree r . Πi = 1 k k The divisor counting function counts an increasingly small Proof of Theorem 2.1. Theorem 2.1 can be broken into two proportion of the digits of base r as r increases. Observe that cases, with one case consisting of two situations. every distinct prime divisor p contributes a factor (α + 1) to i i Case 1. The divisor counting function σ (r) = r for r = 2. The formula (2.1), regardless of the magnitude of p itself. The val- 0 i number 2 is prime, thus by Lemma 2.1.1, σ (r) = 2. The divisor ue of each factor depends on the multiplicity α of each prime 0 i counting function σ (r) = r, since r = 2.