On June 19th 2019 I made a post titled Arithmetic Derivative and in this post I'll return to the topic. I was prompted to do so by one of the properties of the number associated with my diurnal age today. The number is 27265 and it has the property that its arithmetic derivative, 11448, has no digits in common with the number itself. I've listed these numbers in an entry in my Bespoken for Sequences.
I'm not going to pursue that topic in this post but I was reminded of arithmetic derivatives and got to thinking about records being set by the size of arithmetic derivatives as the natural numbers are traversed. It didn't take long to develop a SageMath algorithm to explore this topic (permalink). Detailed results are shown below in Table 1 with derivatives up 40000 in size, although the algorithm lists the records for numbers up to one million.
Table 1
The record setting sizes of the arithmetic derivatives form OEIS A131116 and the initial values are as follows:
As can be seen from Table 1, all the numbers associated with these records contain all powers of 2 together with multiples of these powers. The initial numbers are:
I spotted this little problem on Twitter (see Figure 1):
Figure 1
The proximity of \(+31\) to \(+32\) should alert us to the possibility that either \(x=+2\) or \(y=+2\) is a solution. After that, the corresponding \(x\) and \(y\) values fall into place. However, let's suppose that we didn't spot this and I have to confess I didn't. In that case, we follow the conventional method.
At first sight, it looks like you will end up with a quintic equation after substituting \(y=1-x\) into \(x^5+y^5=31\) and there is no general formula for solving quintic equations. However, the \(x^5\) terms cancels out nicely leaving us with a quartic equation. Let's see how:$$\begin{align} x^5+(1-x)^5&=31\\x^5+(1-5x+10x^2-10x^3+5x^4-x^5)&=31\\5x^4-10x^3+10x^2-5x-30&=0\\5x^3(x-2)+5(2x^2-x-6)&=0\\5x^3(x-2)+5(2x+3)(x-2)&=0\\(x-2)(5x^3+10x+15)&=0\\5(x-2)(x^3+2x+3)&=0\\5(x-2)(x+1)(x^2-x+3)&=0 \end{align}$$Thus \(x=2\) and \(y=-1\) or \(x=-1\) and \(y=2\). Using GeoGebra, this can be confirmed visually as shown in Figure 2:
Figure 2
Differentiation reveals a little more about the graph of \(x^5+y^5=31\):$$\begin{align} y'&=\frac{-x^4}{y^4}\\&=-\frac{x^4}{\sqrt[5]{31-x^5}} \end{align} $$It can be seen that \(y'\) is undefined when \(x=\sqrt[5]{31} \approx 1.9873 \) and thus the graph is vertical at this point. If we zoom in a little, this becomes clearer (see Figure 3). There is a point of horizontal inflection when \(x=0\).
Figure 3
This was a useful little exercise as I got to revisit the binomial expansion for \( (a+b)^5\) and the factor theorem in order to factorise \(x^3+2x+3\). Using 1 rather than 31, I was also prompted to consider the area under the curve for \( x^5+y^5 =1\) between 0 and 1. See Figure 4.
Figure 4
In this graph, the vertical tangent occurs at \(x=1\) and this got me thinking about the area under the curve between 0 and 1. In other words, what is the value of:$$ \int_0^1 \sqrt[5]{1-x^5} \, dx$$Using SageMathCell, the integral evaluates to:$$ 0.2 \cdot \beta \! \left (0.2, 1.2 \right )$$Now I vaguely remember coming across a so-called \( \beta \) function that is in some way, I think, related to the \(\Gamma \) function. Further investigation revealed that they are indeed related. In fact:$$ 0.2 \cdot \beta(0.2, 1.2)= 0.2 \cdot \frac{ \Gamma(0.2) \cdot \Gamma(1.2)}{ \Gamma(0.2 +1.2)} \approx 0.950150138988437 $$This result looks about right but what is going on with this \( \beta \) function? Well, the beta function is actually an integral between 0 and 1 involving two different values, let's say \(a\) and \(b\), so that we have:$$\beta(a,b)=\int_0^1 t^{a-1} \cdot (1-t)^{b-1} dt$$If we can get our integral into that form then we can integrate it. Firstly, we make a substitution. Let \(t=x^5\) and so \(dt/dx=5x^4\) and \(dx=0.2 x^{-4}\). We can write \(x^{-4}=t^{-0.8}\) and the integral now transforms into:$$ \begin{array} \displaystyle \displaystyle \int_0^1 \sqrt[5]{1-x^5} \, dx &=\text{0}.2 \displaystyle \int_0^1 t^{-0.8} (1-t)^{0.2} \, dt \\&=0.2 \displaystyle \int_0^1 t^{0.2-1} (1-t)^{1.2-1} \, dt\\&=0.2 \, \beta(0.2,1.2) \end{array} $$So this was quite an interesting post that led me in some unexpected directions. I also was interested in the area between the curves \(x^2+y^2=1\) and \(x^5+y^5=1\) in the range from 0 to 1. See Figure 5 where eq2 is \(x^2+y^2=1\) and the curve \(x^5+y^5=1\) is a tangent to the circle at \(x=0\) and \(x=1\).
Figure 5
Clearly, the area between the two curves from \(x=0 \text{ to }1\) is given by \( \beta(0.2,1.2) - \dfrac{\pi}{4} \). What I've learned from all this is that the \(\beta\) function is a very useful tool to have in one's integration armoury.
With Mathematics, there's always something new under the Sun and today it happened to be the concept of the arithmetic derivative that I can't recall ever hearing of before.
Today I turned 25637 days old and this number is a member of OEIS A189639: numbers n such that n'' = n'+1 where n' and n'' are respectively the first and the second arithmetic derivative of n. The members of this sequence, up to and including 25637, are:
Not knowing anything about such derivatives, I set out to learn more about them. Here is a quote from Wikipedia (quotation in blue):
For natural numbers the arithmetic derivative is defined as follows: * \(p' \;=\; 1 \) for any prime \(p \). * \((pq)'\;=\;p'q\,+\,p q' \) for any \(p \textrm{,}\, q \;\in\; \mathbb{N}\). E. J. Barbeau was most likely the first person to formalize this definition. He also extended it to all integers by proving that \((-x)' \;=\; -(x')\) uniquely defines the derivative over the integers. Barbeau also further extended it to rational numbers, showing that the familiar quotient rule gives a well-defined derivative on Q: $$\left(\frac{p}{q}\right)' = \frac{p'q-p q'}{q^2} \ .$$Victor Ufnarovski and Bo Åhlander expanded it to certain irrationals. In these extensions, the formula above still applies, but the exponents \(e_i \) are allowed to be arbitrary rational numbers. Elementary properties The Leibniz rule implies that \(0'=0 \) (take \(p = q = 0 \)) and \(1'=0 \) (take \(p = q = 1 \)). The ''power rule'' is also valid for the arithmetic derivative. For any integers \(p \) and \(n'' >= 0\): $$(p^n)' = np^{n-1} p'.$$This allows one to compute the derivative from the prime factorisation of an integer, \(x = p_1^{n_1}\cdots p_k^{n_k}\): $$x' = \sum_{i=1}^k n_i p_1^{n_1} \cdots p_{i-1}^{n_{i-1}} p_i^{n_i-1} p_{i+1}^{n_{i+1}}\cdots p_k^{n_k} = \sum_{i=1}^k \frac {n_i} {p_i}x.$$ For example: $$60' = (2^2 \cdot 3 \cdot 5)' = \left(\frac{2}{2} + \frac{1}{3} + \frac{1}{5}\right) \cdot 60 = 92$$ $$81' = (3^4)' = 4\cdot 3^3\cdot 3' = 4\cdot 27\cdot 1 = 108.$$The sequence of number derivatives for \(k = 0, 1, 2, ... \) begins OEIS A003415:$$0, 0, 1, 1, 4, 1, 5, 1, 12, 6, 7, 1, 16, 1, 9, \ldots$$It is only numbers of the form \(p^p\) where \(p\) is prime that have arithmetic derivatives equal to themselves \(4 = 2^2, 27 = 3^3, 3125 = 5^5, 823543 = 7^7 \) etc.
I came across an interesting paper published about the properties of these derivatives. Here is the abstract:
The notion of the arithmetic derivative, a function sending each prime to 1 and satisfying the Leibnitz rule, is extended to the case of complex numbers with rational real and imaginary parts. Some constraints on the solutions to some arithmetic differential equations are found. The homogeneous arithmetic differential equation of the k-th order is studied. The factorization structure of the antiderivatives of natural numbers is presented. Arithmetic partial derivatives are defined and some arithmetic partial differential equations are solved.
The paper is too heavy for me to digest but it certainly shows that arithmetic derivatives (sometimes called Lagarias arithmetic derivatives or number derivatives), are serious mathematical functions and to quote from the conclusion of the Wikipedia article:
Victor Ufnarovski and Bo Åhlander have detailed the function's connection to famous number-theoretic conjectures like the twin prime conjecture, the prime triples conjecture, and Goldbach's conjecture. For example, Goldbach's conjecture would imply, for each k > 1 the existence of an n so that n' = 2k. The twin prime conjecture would imply that there are infinitely many k for which k'' = 1.
In the article mentioned, the arithmetic derivatives is extended to rational, real and complex numbers. Getting back to basics however, I should add that the product rule for derivatives can be used to find a value for the arithmetic derivative of 1:
1' = (1.1)' = 1'.1+1.1' = 2*1' and the only way the equation holds is if 1'=0.
Here is a YouTube video that introduces the arithmetic derivative:
I owe to the presenter of the video the following line of enquiry. In some cases, repeated "differentiation" leads to 1 and thus 0. 25637 is an example of this where 25637 --> 858 --> 859 --> 1 because 859 is prime. However, for other numbers, the results increase at a rapid rate. Take 88 as a case in point. Here:
Whether there is, eventually, a derivative that is a prime number and will bring the sequence to an end, I don't know. It would be an interesting idea to explore. Here is what the repeated differentiation of the natural numbers from 2 to 100 produces. I stopped after ten steps and marked these numbers in bold.
Notice that 74 and 75 both produce the same sequence of derivatives after the first derivative. It can be seen that these two numbers, and many others, join the sequence 8 12 16 32 80 176 368 752 1520 ... However, there are other sequences as well such as 20, 24 44 48 112 240 608 1552 3120 8144 16304 ... Lots of food for thought here.
On a final note, mention should be made of Giuga numbers that are conjectured to be solutions of the equation n' = n + 1. They are rather rare. In fact, the numbers 30, 858, 1722 and 66198 are the only such numbers below one million.
