The Derivative Of A Number

Are you kidding?

Edward Barbeau is now a professor emeritus of mathematics at the University of Toronto. Over the years he has been working to increase the interest in mathematics in general, and enhancing education in particular. He has published several books that are targeted to help both students and teachers see the joys of mathematics: one is called Power Play; another Fallacies, Flaws and Flimflam; and another After Math.

Today I want to discuss his definition of the derivative of a number, yes a number.



We all know the concept of a derivative of a function. It is one of the foundational concepts of calculus, and is usually defined by using limits. For the space of polynomials it can be viewed as a linear operator so

The derivative operator in general satisfies many properties; one is the product law:

This rule is usually credited to Gottfried Leibniz. Somehow the great Issac Newton did not know this rule—at least that is what is claimed by some.

Definition

Barbeau defined the derivative of a natural number in 1961. Define for a natural number by the following rules:

for all primes . . for all numbers .

Here is a picture from his paper:

This proves that he really did it a long time ago. Note the typewriter type face: no LaTeX back then. He proved the basic result that is well defined. This is not hard, is necessary to make the definition meaningful, but we will leave it unproved here. See his paper for details.

A simple consequence of the rules is that for a prime. This follows by induction on . For it is rule (1). Suppose that :

Unfortunately this does not hold in general. Also is not a linear operator: and . This double failure, the derivative of a power is not simple and the derivative is not linear in general, makes difficult to use. One of the beauties of the usual derivative, even just for polynomials, is that it is a linear operator.

Results

The derivative notion of Barbeau is interesting, yet it does not seem to have been intensively studied. I am not sure why—it may be because it is a strange function—I am not sure.

There is hope. Recently there have been a number of papers on his notion. Perhaps researchers are finally starting to realize there may be gold hidden in the derivative of a number. We will see.

Most of the papers on have been more about intrinsic properties of rather than applications. A small point: most of the papers replace by : so if you look at papers be prepared for this notation shift. I decided to follow the original paper’s notation.

The papers have results of three major kinds. One kind is the study of what are essentially differential equations. For example, what can we say about the solutions of

where is a constant? The others are growth or inequality results: how fast and how slow does grow? For example, for not a prime,

A third natural class of questions is: can we extend to more than just the natural numbers? It is easy to extend to integers, a bit harder to rationals, and not so easy beyond that.

Here are two interesting papers to look at:

Investigations on the properties of the arithmetic derivative , which is a paper by Niklas Dahl, Jonas Olsson, and Alexander Loiko.

, which is a paper by Niklas Dahl, Jonas Olsson, and Alexander Loiko. How to Differentiate a Number, which is a paper by Victor Ufnarovski and Bo Åhlander.

An Application

I tried to use to prove something interesting. I think if we could use to prove something not about but about something that does not mention at all, that would be exciting. Tools must be developed in mathematics, but the key test of their power is their ability to solve problems from other areas. One example: the power of complex analysis was manifest when it was used to prove deep theorems from number theory. Another example: the power of the theory of Turing machines was clear when it was used to yield an alternate proof of the Incompleteness Theorem.

The best I could do is use to prove an ancient result: that is not rational. Well I may be able to prove a bit more.

We note that from the product rule: , for any . Recall if were prime this would be .

Now assume by way of contradiction that is rational number. Then for some positive numbers we have

As usual we can assume that are co-prime.

So let’s take derivatives of both sides of the equation—we have to use sometime, might as well start with it.

Note that it is valid to apply to both sides of an equation, so long as one is careful to obey the rules. For example allows but there is no additive rule to make the right-hand side become which would make the equation false.

The result of taking the derviative of both sides is:

Now square both sides and substitute for to get:

This implies that divides . This leads to a contradiction, since it implies that are not co-prime. Whether we also get that divides is possibly circular, but anyway this is enough. The point is that owing to , the derivative removed the problematic factor of .

Note from the original equation, we only get that divides which is too weak to immediately get a contradiction. Admittedly ours is not the greatest proof, not better than the usual one especially owing to the squaring step, but it does use the derivative of an number.

One idea: I believe that this idea can be used to prove more that the usual fact that has no nonzero solutions over the integers. I believe we can extend it to prove the same result in any ring where can be defined, possibly modulo issues about lack of unique factorization. This handles the Gaussian integers, for example.

Open Problems

Can we use this strange function to shed light on some open problem in number theory? Can we use it in complexity theory? A simple question is: what is the complexity of computing ? If where and are primes, then by the rules. But we know that and thus we have two equations in two unknowns and we can solve for and . So in this case computing is equivalent to factoring . What happens in the general case? An obvious conjecture is that computing is equivalent to factoring.

[fixed error in proof that owed to typo , as noted by user “Sniffnoy” and others, and changed some following text accordingly; further fix to that proof; fixed typo p^k to p^{k-1}]