Premium Only Content

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
-
1:31:26
Glenn Greenwald
3 hours agoAaron Maté on More Russiagate Fallout, Protests in Ukraine, and Israel's Strikes on Syria With Special Guests John Solomon, Marta Havryshko, and Joshua Landis | SYSTEM UPDATE #491
83.6K10 -
LIVE
Melonie Mac
2 hours agoGo Boom Live Ep 56!
298 watching -
LIVE
RalliedLIVE
8 hours ago $1.85 earned10 WINS WITH THE SHOTTY BOYS
258 watching -
1:33:40
Mally_Mouse
9 hours agoLet's Hang!! - P.O. Box & Chill - WE HIT 10,000!!!
9.98K2 -
31:34
Friday Beers
3 hours ago $0.36 earnedOur Horrifying Night Drunk Ghost Hunting the Manson Murders
11K6 -
40:48
The Kevin Trudeau Show Limitless
11 hours agoHow Karma Really Works (And How the Elites Use it)
4.64K -
1:08:42
BonginoReport
4 hours agoDems Exposed For Hiding Hillary Clinton’s Poor Health - Nightly Scroll w/ Hayley Caronia (Ep.96)
129K59 -
53:21
ZeeeMedia
4 hours agoAustralia's Abortion Rewards, Crypto Surge & Organ Harvest Horror | Daily Pulse Ep 72
12.3K4 -
DVR
Playback Request Live
1 hour agoFrom Myspace to Rumble: Supmikecheck in the Creator Spotlight
2.8K1 -
48:36
The Mel K Show
2 hours agoLive Q&A From Planet Chaos with Mel K 7-23-25
20K9