Eikonal Blog

2011.07.06

Derivatives of numbers

Filed under: mathematics, number theory — Tags: , , — sandokan65 @ 23:45

Definitions:

• $p'=1$ for every prime number $p$
• $(a b)' = a' b + a b'$ (Leibnitz rule) for every two natural numbers $a, b \in {\Bbb N}$

Concequences:

• $1' = 0$.
• $(p^n)' = n \cdot p^{n-1}$
• for any natural number $n= \prod_{i=1}{k} {p_i}^{n_i}$ one has $n' = n \sum_{i=1}^{k} \frac{n_i}{p_i}$.
• Eg: $40' = (2^3 \cdot 5)' = 3 \cdot 2^2 \cdot 5 + 2^3 = 68$.
• in general $(a+b)' \ne a' + b'$
• $\left(\frac{a}{b}\right)' = \frac{a' b - a b'}{b^2}$
• $(p^p)' = p^p$ for any prime number $p$ is equivalent of the exponential function’s property that its derivative is itself.
• If $n = p^p \cdot m$ for prime $p$ and natural $m>1$, then $n' = p^p (m+m') > n$, $n^{(k)} \ge n+k$, and $\lim_{k\rightarrow \infty} n^{(k)} = \infty$.
• For infinitely many natural numbers $n$ there exist suitable $k$ s/t $n^{(k)}=0$
• Ufnarovski and Åhlander give following conjecture: for every natural $n$, as we observe its derivatives $n^{(k)}$ (as $k$ grows to infinity), the limit will be either $0$, $\infty$, or $n$ itself (if $n=p^p$ for some prime $p$).
• If $n'=0$, then $n=1$.
• If $n'=1$, then $n = p$ (for all possible primes).

Sources:

• “Deriving the Structure of Numbers” by Ivars Peterson (Ivars Peterson’s MathTrek) – http://www.maa.org/mathland/mathtrek_03_22_04.html
• “How to differentiate a number” by Ufnarovski, V., and B. Åhlander (Journal of Integer Sequences 6; 2003.09.17) – http://www.cs.uwaterloo.ca/journals/JIS/VOL6/Ufnarovski/ufnarovski.html
Abstract: We define the derivative of an integer to be the map sending every prime to 1 and satisfying the Leibnitz rule. The aim of the article is to consider the basic properties of this map and to show how to generalize the notion to the case of rational and arbitrary real numbers. We make some conjectures and find some connections with Goldbach’s Conjecture and the Twin Prime Conjecture. Finally, we solve the easiest associated differential equations and calculate the generating function.
• “Investigations of the number derivative” by Linda Westrick – http://web.mit.edu/lwest/www/intmain.pdf