ALGEBRAIC ENTROPIES OF NATURAL NUMBERS WITH ONE OR TWO PRIME FACTORS

Abstract

We formulate the additive entropy of a natural number in terms of the additive partition function, and show that its multiplicative entropy is directly related to the multiplicative partition function. We give a practical formula for the multiplicative entropy of natural numbers with two prime factors. We use this formula to analyze the comparative density of additive and multiplicative entropy, prove that this density converges to zero as the number tends to infinity, and empirically observe this asymptotic behavior.

1. INTRODUCTION

In statistical physics, entropy S is a measure of the size of all the possible microstates of a system, and S can be expressed as k · ln Ω, where k is Boltzmann’s constant and Ω is the number of microstates [1]. In information theory, following Shannon [2], entropy H(X) measures the amount of information in a message, and H(X) can be expressed as , where p is the probability mass function of a discrete random variable X [2]. It is well known that physical and information entropy are equivalent.

In mathematics, the set of natural numbers, ℕ, conceals information in the form of arithmetic relations. Each natural number is endowed with such information by the two fundamental algebraic operations on ℕ, addition and multiplication. This information can be extracted by using mathematical logic to elucidate the relations between natural numbers. Can the concept of entropy be used to measure the amount of mathematical information (or the number of microstates) inherent in natural numbers? In this paper, we answer this question.

Since Patrick Billingsley’s lectures in 1973 [3], there have been several efforts [4, 5, 6] to apply the idea of entropy, and other information-theoretic concepts, to number theory. Shannon’s entropy function H was recently applied [6] to ℕ, yielding the concept of the entropy of a natural number. However, this form of entropy does not measure the amount of information inherent in natural numbers, but is rather an aid to understanding effects such as the distribution of prime numbers. In this paper we will consider natural numbers as repositories of quantitative information originating from algebraic operations, and explicitly measure the amount of quantitative information stored in these numbers.

The mathematical properties of ℕ primarily stand on two algebraic operations, addition and multiplication. Addition gives ℕ a structure of order, in the sense that it arranges natural numbers as follows: for any a, b ∈ ℕ, a < b ⇔ if ∃ k ∈ ℕ such that b = a + k. Multiplication provides a cladogram of numbers, in the sense that factorization reveals how a natural number is made up of its most elementary factors, which are prime numbers. Thus we can characterize each natural number n in ℕ in terms of the information incorporated in:

(i) the possible ways of generating n by adding elements of ℕ, and (ii) the possible ways of generating n by multiplying elements of ℕ.

We call the amount of this information, the additive and multiplicative entropy respectively. We will show that additive entropy is easily measured by the additive partition function, and that multiplicative entropy is directly related to the multiplicative partition function. This project is strongly connected to previous work on the additive partition function [7, 8] and on the multiplicative partition function [9, 10, 11, 12, 13]. In particular, our approach is closely related to previous work [11, 12, 13] on the unordered factorization of natural numbers, in the sense that the multiplicative entropy of a natural number has to be computed by such factorization. In this paper we will present a computable formula for the multiplicative entropy of natural numbers with two prime factors, and introduce the comparative density of two algebraic entropies, and use this to analyze their asymptotic behavior. Moreover, we will prove that this density asymptotically converges to zero as the number approaches infinity, and empirically analyze the speed of this convergence.

The remainder of this paper is organized as follows: In Section 2, we introduce the additive entropy and multiplicative entropy of a natural number and their comparative density. In Section 3, we compute the multiplicative entropy and the comparative density of natural numbers with two prime factors. We also prove that this density converges to zero as the number approaches infinity. We demonstrate that additive and multiplicative entropy have a logarithmic relationship, and observe the asymptotic behavior of their comparative density. Finally in Section 4, we summarize our results and propose some directions for further study.

 

2. TWO FUNDAMENTAL ENTROPIES OF A NATURAL NUMBER

For a natural number n in ℕ, we can introduce two algebraic entropies, additive entropy and multiplicative entropy.

Let p(n) be the partition function representing the number of possible partitions of a natural number n by addition, which is to say the number of distinct ways of producing n by adding natural numbers. Then we can define additive entropy:

Definition 1. For a natural number n, A(n) is given by

Let 𝓜(n) be the number of distinct ways of producing n by multiplying natural numbers. Then we can define multiplicative entropy P(n) as follows:

Definition 2. For a natural number n, P(n) is given by

The comparative density drc(n) of the algebraic entropies A(n) and P(n) can be expressed as follows:

Definition 3. For a natural number n, drc(n) is given by

Remark 1. From now on, following Landau’s treatment of the constant k in the definition of thermodynamic entropy k ln Ω, we will assume that ka = km = 1.

 

3. ENTROPIES OF NATURAL NUMBERS WITH FEW PRIME FACTORS

3.1. Entropies of natural numbers with one prime factor

Lemma 1. Let N = qm, where q is a prime number and m ∈ ℕ. Then the multiplicative entropy of N is given by

Proof. N = N1 · N2 = qk1 · qk2, with k1 + k2 = m, which implies that 𝓜(N) = p(m). Consequently, we obtain P(N) = ln p(m).                                                             ☐

Using Definition 1, it follows from Eq. (4) that P(N) = A(m), when N = qm. Moreover, since , we additionally obtain . This implies that, in the simplest case where N = qm, our two algebraic entropies are logarithmically related.

Theorem 1. Let N = qm, where q is a prime number and m ∈ ℕ. Then the comparative density of N is given by

Proof. It is apparent from Lemma 1 and Definition 3.                                                             ☐

Theorem 2. Let N = qm, where q is a prime number and m ∈ ℕ. Then for any prime number q, and

Proof. From Eq. (5) we have

Since for a sufficiently large s ∈ ℕ, we can compute , which further implies that , and ln for a sufficiently large m.                                                             ☐

Figure 1 shows the behavior of drc(N) and ln(drc(N)) when q = 2, which provides numerical confirmation of Theorem 2.

Figure 1.Behavior of drc and ln(drc), when q = 2.

We can also deduce from Eq. (6) that, for a sufficiently large m,

which relates the additive and multiplicative entropy for the class of natural numbers given by N = qm.

3.2. Entropies of natural numbers with two prime factors

Now we consider the more complicated case of a natural number , where q1, q2 are prime numbers and m, n ∈ ℕ.

Definition 4. Let N be a natural number given by , where q1, q2 are prime numbers and m, n ∈ ℕ. A factor of N, denoted by , is said to be hybrid if i ≠ 0 and j ≠ 0.

Definition 5. Consider the sets of natural numbers A = {a1, a2, · · · , am−1, am} and B = {b1, b2, · · · , bn−1, bn}, and let k be a number. Then we define the product k · (A × B) as follows:

k · (A × B) = {k · as1 · bs2}, where 1 ≤ s1 ≤ m, 1 ≤ s2 ≤ n.

Lemma 2. Let , where q1 and q2 are prime numbers and m, n ∈ ℕ, and let 𝓗0(N) be the number of distinct ways of producing N by multiplying non-hybrid factors of N. Then

Proof. Let Q1 and Q2 be sets of distinct ways of producing and by multiplication. Then, by Lemma 1, we have 𝓜(N1) = |Q1| = p(m) and 𝓜(N2) = |Q2| = p(n). Since there are Q1 × Q2 distinct ways of producing N = N1 · N2 by multiplying non-hybrid factors of N, we obtain 𝓗0(N) = |Q1 × Q2| = |Q1| · |Q2| = p(m) · p(n).                                                             ☐

Theorem 3. For a natural number , we define

Then, , and moreover

for sufficiently large m and n.

Proof. Since

and using with a sufficiently large s, we obtain

In addition, for sufficiently large value of m and n, we can also write

And hence, from Eq. (10), we finally obtain

                                                             ☐

can be considered as a partial comparative density of algebraic entropies of N, because is the ratio of the multiplicative entropy of a particular subset of all the distinct ways of producing N by multiplication to the logarithm of the additive entropy of N. Figure 2 illustrates that is a natural extension of , and we can formalize this:

Figure 2.Behavior of when q1 = 2 and q2 = 3. The black curve is the graph of drc from Figure 1 (a).

Remark 2. If the magnitude of m dominates that of n, then we can rewrite Eq. (11) as follows:

which is the result established by Theorem 2. (Conversely, when n dominates m, we obtain Furthermore, Eq. (11) also implies

which is an extension of Eq. (8). These results signify that Eq. (11) is a natural extension of Eq. (6).

Lemma 3. For a natural number , let be the number of distinct ways of producing N by multiplication with only one hybrid factor of N. Then is given by

Proof. Let and be sets of distinct ways of producing and by multiplication. Then we have and . Note that the set of distinct ways of producing N by multiplying non-hybrid factors of and together with the hybrid factor is given by . Thus, we obtain .                                                             ☐

Theorem 4. For a natural number , let be the number of possible ways of producing N distinctly by multiplication with only one hybrid factor of N. Then is given by

Proof. Since , by the previous lemma, we have

                                                             ☐

We will now introduce some necessary new notation.

Definition 6. Given two sets of indices I = {i1, i2, · · · , il} and J = {j1, j2, · · · , jl}, we define the operator ☉l such that

I ☉l J = {(i1, j1; i2, j2; · · · ; il, jl)}.

In addition, consider λ1, λ2 ∈ I ☉l J given by λ1 = (i1, j1; i2, j2; · · · ; il, jl) and . Then λ1 = λ2 if and only if and for all i = 1, 2, · · · , l.

Lemma 4. Let λ = (i1, j1; i2, j2) ∈ I ☉2 J. Then, the number of distinct ways of producing by multiplication, with no hybrid factor other than and , , is given by

Proof. Let and be sets of distinct ways of producing and by multiplication. Then we have and . Note that the set of distinct ways of producing N by multiplying non-hybrid factors of and together with the hybrid factors and is given by . Thus we can write

                                                             ☐

From now on, we will assume that the elements of the set are well ordered through constraint by the inequality i1 < i2, or by j1 ≤ j2 if i1 = i2. Then, we will describe two sets and , with and , as mutually disjoint, if . We will use ΣI☉2J|Aλl| to denote Σλl|Aλl| when any two sets Aλl and are mutually disjoint for λl, .

Theorem 5. Let be the number of possible ways of producing a natural number by the multiplication of just two hybrid factors. Then is given as follows;

Proof. Since , by the previous lemma, we have

In addition, since the factors in the disjoint sets are well-ordered, we can rewrite Eq. (17) as follows:

Note that the first and second terms of Eq. (18) denote the number of all the elements in the mutually disjoint sets, when i1 = i2 and i1 < i2 respectively. This completes the proof.                                                             ☐

Lemma 5. For a natural number , let λ = (i1, j1; i2, j2; · · · ; ik, jk) ∈ I ☉k J. Then, the number of distinct ways of producing N by multiplication only the hybrid factors , · · · , and , , is given by

Proof. Let and be sets of distinct ways of producing and by multiplication. Then we have and . Note that the set of distinct ways of producing N by multiplying non-hybrid factors of and together with the hybrid factors , · · · , is given by . Thus we obtain

                                                             ☐

Theorem 6. For a natural number , let be the number of possible ways of producing N by multiplication with no hybrid factor other than the k hybrid factors of N. Then is given as follows:

Proof. Since , b the previous lemma, we have

where λ = (i1, j1; i2, j2; · · · ; ik, jk) ∈ I ☉k J. In addition, since the factors in the disjoint sets are well-ordered, according to the cases whether ik = ik−1 or ik > ik−1, we can rewrite Eq. (20) as follows:

This completes the proof.                                                             ☐

Theorem 7. The multiplicative entropy P(N) of a natural number , is given by

Proof. It is apparent from the definition .                                                             ☐

Tables 1 and 2 show the multiplicative entropies of N = 2m · 3n for 1 ≤ m ≤ 20 and 1 ≤ n ≤ 20, obtained from Eq. (22). Note that P(2m · 3n) = P(3m · 2n), and the symmetric entries are omitted.

Table 1.Multiplicative entropies of , when q1 = 2 and q2 = 3, for 1 ≤ m ≤ 20 and 1 ≤ n ≤ 10.

Table 2.Multiplicative entropies of , when q1 = 2 and q2 = 3, for 11 ≤ m ≤ 20 and 11 ≤ n ≤ 20.

From Tables 1 and 2, we observe that multiplicative entropy increases very slowly with m and n, whereas the additive entropy ln p(N) increases quickly with N. This suggests that → 0 as N tends to ∞, a conjecture supported by Figure 3.

Figure 3.Behavior of , when q1 = 2 and q2 = 3, for 1 ≤ m ≤ 20 and 1 ≤ n ≤ 20.: each row of dots of the same color shows the behavior of when m is fixed.

Next, we compute the comparative density of algebraic entropies of N, and show that this density converges to zero as m, n approach ∞.

Let , n ∈ ℕ; and we now introduce aTb to denote the number of distinct ways of producing a natural number a by adding b natural numbers. Then we have the following lemma:

Lemma 6. For a natural number ,

Proof. We can rewrite Eq. (21) as follows:

Let Λk = maxt1=k,··· ,m {λ∗(t1)} and ∆k = maxt2=k,··· ,n {δ∗(t2)}. Then we can write

Moreover, since Λk ≤ mTk and ∆k ≤ nTk, we obtain

                                                             ☐

Theorem 8. For a natural number ,

Proof. From Eq. (23) we obtain

In addition, since

where σ(k) is the sum of the divisors of k, and so

Note that

Combining Eqs. (25) , (26) and (27), we can write:

Consequently we obtain

For sufficiently large s, we can write , and hence

                                                             ☐

 

4. CONCLUDING REMARKS AND SUGGESTIONS FOR FURTHER STUDY

We have introduced the additive and multiplicative entropy of a natural number, and defined their comparative density. We have showed that, for natural numbers with one prime factor, the additive and multiplicative entropy are logarithmically related, by computing the relative density of the algebraic entropies of these numbers and analyzing its asymptotic behavior. We have also presented a practical formula for computing the relative density of natural numbers with two prime factors. We observed that the logarithmic relation between additive and multiplicative entropy applies to larger numbers in a special subclass of natural numbers which have no hybrid factor in any of the possible multiplicative partitions. We also proved that their relative density converges to zero as the number tends to infinity. In addition, we empirically showed that the asymptotic behavior of the relative density of numbers with two prime factors is more complicated and quite different from the behavior of numbers with a single prime factor, as shown in Figures 1 and 3. We proved that this relative density also converges to zero as the number tends to infinity.

We propose two avenues for further study: firstly, to complete the analysis of the asymptotic behavior of the relative density for natural numbers with two prime factors, and explore this behavior empirically in more detail; and, secondly, to extend our work to natural numbers with more than two prime factors. In particular, a fuller understanding of the multiplicative entropy of an arbitrary natural number might provide useful information on the distribution of prime numbers, which must be related to the distribution of the multiplicative entropies of natural numbers.