The arithmetic derivative

Apr 21, 2024

Can you differentiate a number and get a nonzero answer? You can't with the usual kind of differentiation, but you can with the “arithmetic derivative”. The basic version of the arithmetic derivative takes positive integers as inputs. The only requirements are the two shown in the picture: (1) the derivative of a prime number is 1, and (2) the product rule holds. It can be proved fairly easily that there is only one function satisfying these requirements. The table shows the values of the arithmetic derivatives of the first few positive integers; these values will always be other nonnegative integers.

If one knows how to factorize a number n as a product of primes, it is relatively easy to calculate its arithmetic derivative, n’. The picture above shows how to do this in the case n=360=2³×3²×5¹. The first person to investigate the arithmetic derivative systematically seems to have been E.J. Barbeau in 1961, although the function had previously appeared in 1950 as a question on a Putnam Prize competition. The first known reference to the function seems to have been in a conference talk in June 1911 by José Mingot Shelly.

The arithmetic derivative has a number of interesting properties, as well as close connections to some famous problems in number theory. It also allows for the study of differential equations involving natural numbers, using the arithmetic derivative in place of the usual derivative. For example, the only solution of the differential equation n’=0 is n=1, and the only solution of n'=1 is for n to be prime. Both of these results are easy exercises.

An excellent introduction to this area is the 2003 paper How to Differentiate a Number by Victor Ufnarovski and Bo Åhlander, which builds on Barbeau's work. The paper contains the above two results but proves many others and offers several conjectures. Some of these conjectures relate to famous open problems in number theory, such as the Goldbach conjecture and the twin prime conjecture.

The Goldbach conjecture is the hypothesis that every even integer greater than 2 can be expressed as the sum of two primes. Ufnarovski and Åhlander conjecture that the (arithmetic) differential equation n'=2b has a positive integer solution for any natural number b > 1; in other words, that any even number greater than 2 shows up as the arithmetic derivative of some natural number. They point out that the Goldbach conjecture implies this conjecture; this is because if 2b is the sum of the primes p and q, then the arithmetic derivative of the number n=pq is 2b. Note that this is a one-way implication, and it does not mean that a proof of this differential equation identity would prove the Goldbach conjecture.

The twin prime conjecture is the hypothesis that there are infinitely many primes p such that p+2 is also prime. This is still an open problem, although there have been breakthroughs towards proving this since 2013, thanks to the work of Yitang Zhang, Terence Tao, James Maynard, and others. Ufnarovski and Åhlander conjecture that the differential equation n''=1 has infinitely many solutions, and they point out that the twin prime conjecture would imply this. The reason for this is that if p and p+2 are both primes, and we define n=2p, then we have n’=p+2 and n’’=1. As in the case of the other conjecture, this is a one-way implication.

Although the arithmetic derivative obeys the product rule, it does not obey the more basic sum rule, (m+n)’ = m’ + n’. Another function with this combination of properties is the function F(n)=n log(n), which is a constituent of the Gibbs entropy formula in physics. A short calculation (shown above) shows that F(ab)=F(a)b + aF(b), which is how the product rule would behave if F were the differentiation operator. This suggests that the arithmetic derivative can be thought of as a kind of “arithmetic entropy”, in which prime numbers have low entropy and highly composite numbers have high entropy.

There is another way to define the arithmetic derivative so that it satisfies an analogue of both the sum and the product rule, if we are prepared to adopt a strange kind of multiplication called umbral multiplication. One way to think about this is to turn every natural number n into a polynomial with positive integer coefficients, in which the coefficient k of x^m is given by the number of times that n can be divided by the m-th prime number, p_m. The picture above shows what happens if the polynomial has degree at most 2, where p₀=2, p₁=3, and p₂=5.

There is already an obvious way to differentiate polynomials. This induces a way to differentiate the associated numbers, if we associate xⁿ with the n-th prime number, p_n. The rules for this umbral arithmetic derivative are: (a) 2 differentiates to 1; (b) the n-th prime number p differentiates to qⁿ, where q is the (n-1)-st prime number; and (c) the derivative of a product is given by (mn)’=m’n’. Roughly speaking, umbral multiplication is to usual multiplication what multiplication is to addition. This is why the rule for the umbral derivative of a product resembles the sum rule for ordinary differentiation.

More precisely, as shown above, the umbral product distributes over the usual product, just as multiplication distributes over addition. There is also a product rule for umbral differentiation in which the roles of sum and product are played by product and umbral multiplication, respectively. Curiously, the size of the n-th prime p_n is approximately n log(n), where n is the natural logarithm, and this again suggests a link with entropy. The umbral product derivatives of the first few natural numbers are shown in the table below.

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