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

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