Arithmetic partial derivative | number theory | advanced level
Episode 107.
Arithmetic partial derivative | number theory | advanced level.
Branch of mathematics: number theory.
Difficulty level: advanced.
The arithmetic derivative is a function $D$ from natural numbers to natural numbers defined by the 2 properties:
1. For any prime number $p$, we have $D(p)=1$.
2. For any 2 natural numbers $m$ and $n$, we have $D(m \cdot n) = D(m) \cdot n + m \cdot D(n)$.
For a fixed prime number $p$, the arithmetic partial derivative is a function $D_p$ from natural numbers to natural numbers defined by the 2 properties:
1. We have $D_p(p) = 1$. And, for any prime number $q$, $q \neq p$, we have $D_p(q)=0$.
2. For any 2 natural numbers $m$ and $n$, we have $D(m \cdot n) = D(m) \cdot n + m \cdot D(n)$.
Theorem. If $n = p^k \cdot q$, where $q$ is not divisible by $p$, then $D_p(n)=n \cdot \frac{k}{p}$.
Theorem. For any natural number $n$, we have $D(n) = \sum_{p, \text{$p$ is prime}}{D_p(n)}$.
Mathematics. Number theory.
#Mathematics #NumberTheory
The video on YouTube:
https://youtu.be/Z_knP4RNuME
The same video on Telegram:
https://t.me/mathematical_bunker/132
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