Arithmetic derivative: an introduction | number theory | advanced level

Episode 100.

Arithmetic derivative: an introduction | 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 propeties:
1. For any prine number $p$, we have $D(p)=1$.
2. For any 2 natural numbers $m$ and $n$, we have $D(m \cdot n) = D(m) \cdot n + m \cdot D(n)$.

There is also an explicit formula. If $n = \prod_{i}{p_i}$ is the representation of $n$ as a product of its prime divisors (some of them could be duplicated), then $D(n)=n \sum_{i}{\frac{1}{p_i}}$.

Mathematics. Number theory.
#Mathematics #NumberTheory

tags:
mathematics,number theory,prime,prime number,arithmetic derivative,derivative,differentiate,differentiation,product rule

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