An Exploration of the Arithmetic Derivative

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1 . An Exploration of the Arithmetic Derivative Alaina Sandhu Research Midterm Report, Summer 006 Under Supervision: Dr. McCallum, Ben Levitt, Cameron McLeman 1 Introduction The Arithmetic Derivative is a recently defined function of whose properties directly relate to some of the most well known conjectures within Number Theory. The arithmetic derivative of an integer is defined to be the map which sends every prime integer to 1 and that satisfies the Leibnitz rule, for all a, b Z, (ab) = a b + ab. From this definition, we can conjecture that for any a N, there exists a solution to the differential equation n = a. A proof of Goldbach s conjecture would validate this statement, as the derivative of the product of two primes (p 1 p ) is the sum. Our goal is to explore the properties of the arithmetic derivative and propose further conjectures with regard to the nature of the function as well as its implications for the Twin Prime Conjecture and Goldbach Conjecture, among others. Definition The arithmetic derivative n of any natural number is defined as follows: p = 1 for any prime p. (ab) = a b + ab for any a, b N (Leibniz rule). 0 = 0. To illustrate this, 15 = (3 5) = = = 8. We now examine the derived formula and ensure the function is well-defined. 1

2 Theorem 1 For any natural number n, if n = k pn i i n = n k is the prime factorization of n, then n i. (1) First we prove by induction on k that if n = k, then n = n k Proof. Consider n N and express n = k, where all are not necessarily distinct. When k = 1, it is clear that n = 1. Assume that n = k. Then, 1. Now, if we express n = k n = n k n i (np k+1 ) = n p k+1 + n(p k+1 ) pn i = n + np k+1 k k+1 = np k+1 1 i, our formula becomes Now, we check that our formula is consistent with the Leibniz rule for the product of two natural numbers; Let a = k pa i i and b = k pb i i. Then, ( ) ( ) k a i + b i k a i k b i ab = a b + a b. This definition maintains some natural properties, such as (n k ) = kn k 1 n. Also, because 1 = (1 1) = = 1,, the derivative of 1 is defined as 1 = 0..1 Bounds for the Arithmetic Derivative Theorem The only positive integer which satisfies n = 0 is n = 1. Proof. This is a direct result of the definition of our function. Theorem 3 The only solutions to n = 1 in natural numbers are prime numbers. Proof. A composite number can be expressed as the product of prime numbers, of which the derivative of (by the product rule) is the sum of at least two positive integers, which is greater than 1. Theorem 4 [1] For any positive integer n 1 n n log n. ()

3 If n is composite, Furthermore, if n is a product of k factors larger than 1, then (Proof taken directly from [1]) Proof. If n = k pn i i, then n n. (3) n kn k 1 k. (4) n k n i log n k n i. Using Theorem 1, this inequality translates to n = n k n i n k n i If n = n 1 n n 3 n k then, according to the Leibnitz rule, n log n. n = n 1n n 3 n k + n 1 n n 3 n k + n 1 n n 3 n k n 1 n n 3 n k n n 3 n 4 n k + n 1 n 3 n 4 n k + n 1 n n 4 n k n 1 n n k 1 = ( 1 n ) ( 1 1 n k 1 ) 1 k 1 k 1 = k n n k = k n k. n 1 n n k n 1 n n k Here we have replaced the arithmetic mean by the geometric mean. 3 Properties of the Arithmetic Derivative While the arithmetic derivative has only recently been defined, there has been some significant research published on various properties of this function, notably How to Differentiate a Number by Ufnarovski and Ahlander [1]. We will now address the theorems and properties relevant to our research, and discuss the related questions we seek to resolve. 3.1 Solutions to n = n Numerical derivations have shown positive integers of the form p p (where s prime) solve the differential equation n = n. (p p ) = p p p p = pp (5) This unique property of p s useful in the following theorems. Theorem 5 [1] If n = p p m for some prime p and natural m > 1, then n = p p (m + m ) and lim k n (k) =. 3

4 Proof. The derivative of n = p p m, n = (p p ) m + p p m = p p (m + m ) > n and further proof by induction shows that n (k) n + k. Theorem 6 [1] Let p k be the highest power of prime p that divides the natural number n. If 0 < k < p, then p k 1 is the highest power of p that divides n. Furthermore, each derivative n, n, n,..., n (k) is distinct. Proof. Let n = p k m. Then n = kp k 1 m + p k m = p k 1 (km + pm ), and because k < p, the inside term is not divisible by p. We see that indeed, the only solutions to this equation are integers of the form p p. Theorem 7 [1] For n N, n = n if and only n = p p, where s a prime. Furthermore, there is an infinite number of solutions to the equation. Proof. Assume n = n. Then by Theorem 6, if p n at least p p n or else it would contradict n = n. Conversely, assume n = p p. As seen in (5), n = p p = n. These properties lead to the conjecture that n (k) = n has only numbers of the form p p as solutions. As we will discuss later, we do not expect the function n (k) to cycle for any arbitrary n not of the form p p. 3. Solutions to n = a As previously discussed in Section.1, we have specific solutions to the differential equations n = 0 and n = 1. Now we turn our attention to the existence of solutions to the equation n = a, where a > 1. First, we observe that there can only be a finite number of solutions to n = a because all potential solutions are bounded above by a 4. One direct application of n = a concerns the Goldbach Conjecture, which states that every even number larger than 3 is the sum of two distinct prime numbers. In terms of our function, we can restate it as follows: Conjecture 1 For any a N, there exists a solution to the equation n = a. The Goldbach Conjecture allows us to represent a = p 1 + p, so if we take the derivative of the product (p 1 p ) = (p 1 ) (p ) + p 1 (p ) = p + p 1 = a. Additionally, we find some solutions for n = a where a is odd: Theorem 8 For any prime, p, and a N which can be expressed a = p +, s a solution to the n = a. Proof. (p) = p + p = p +. It is useful to explore the properties of the function I(a) which calculates all positive integers n such that n = a and the corresponding function i(n) which denotes the number of such solutions. After implementing the program within Pari, initial explorations supported 4

5 our above conjecture that the each even integer through 100, had at least one solution to n = a. We observed that a substantial amount of prime numbers (through 1000) did not have any anti-derivatives, which may prove useful when considering bounds for the function i(n) and I(n). Theorem 9 [1] The function i(n) is unbounded for n > 1. (This proof was taken directly from [1]) Proof. Suppose that i(n) < C for all n > 1 for some constant C. Then n i(k) < Cn k= for any n. But for any two primes p, q the product pq belongs to I(p + q) thus n i(k) > k= p q n π(n)(π(n) + 1) 1 = > π(n), where means that the sum runs over the primes, and π(n) is the number of primes not exceeding n. This leads to the inequality Cn > π(n) π(n) < Cn, which contradicts the known asymptotic behavior π(n) n ln n. We are going to use this proof as a starting point towards a stronger statement regarding the infinitude of i(n) for n > 1. We would like to yield some asymptotic relationship for the function. 3.3 Dynamics of n (k) as k It appears that for any natural number, the trajectory of its n-th derivative can follow any three paths: Conjecture For any n N, one of the following could happen: n (k) = 0 as k n (k) = as k n = p p for some prime p., and thus n (k) = p p as k We would like to focus on the proofs of the former two points, as well as prove that not only are cycles infinite, but they never repeat themselves. Additionally, we are forming bounds with which to describe integers that go to infinity as well as those which go to zero. A trivial example would be that at least 1 4 of all integers go to infinity (factors of ). The following conjecture is related to positive integers which go to zero: 5

6 Conjecture 3 There exist infinitely many composite numbers n such that n (k) sufficiently large natural k. = 0 for This conjecture is dependent upon the validity of the Twin Prime Conjecture which states that there exists infinitely many prime pairs of the form (p, p + ). This idea is applied to our derivative as follows: As seen in Theorem 8, (p) = + p. Subsequently ( + p) = 1 and 1 = 0. Thus, a proof of the Twin Prime Conjecture would validate this proof. Our current research is mainly concerned with the preceding theorems and conjectures within the boundaries of natural numbers. However it is important to know that the derivative can be extended to integers as well as rational numbers, with which comes many more interesting conjectures and research problems. 4 Codes in Pari The following are codes for various functions with the arithmetic derivative. The comment line above each code describes its function. This code will compute the arithmetic derivative of any integer or rational number: {f(n)=sign(n)*abs(n)*sum(, matsize(factor(abs(n)))[1],} {factor(abs(n))[i,]/factor(abs(n))[i,1])} This code will compute the solutions, a for a = n (the I(n) function in the paper): {I(n) = for(, n^/4+1, if((f(i))==n, print(i)));} This code computes the number of solutions, a, to solve a = n (the i(n) function in the paper): { i(n)= count = 0; k=n; for(, k^/4+1, if((f(i))== k, count = count + 1)); print(count);} [Note: We also have the code to compute the n th derivative of a number, but there is one minor change we want to make to it. It will be on the final report with some additional codes] 6

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8 5 Table of n to the k th derivative, for n 100 and k 10 n n (1) n () n (3) n (4) n (5) n (6) n (7) n (8) n (9) n (10)

9 n n (1) n () n (3) n (4) n (5) n (6) n (7) n (8) n (9) n (10)

10 n n (1) n () n (3) n (4) n (5) n (6) n (7) n (8) n (9) n (10) References [1] Ahlander, Bo and Ufnarovski, Victor. How to differentiate a Number. Journal of Integer Sequences. Vol. 6 (003). [] Barbeau, E.J. Remark on an arithmetic derivative. Canadian Mathematical Bulletin. Vol. 4 (1961): [3] Buium, A. Arithmetic analogues of derivations. Journal of Algebra. Vol. 198 (1997) : [4] Hardy, G.H. and Wright, E.M. An Introduction to the Theory of Numbers 5th ed., Oxford University Press, [5] Niven, Ivan et. al. An Introduction to the Theory of Numbers 5th ed., Wiley Textbooks,