Let $P$ be the set of positive prime numbers. Let $P^k = P\cdot P \cdots P$ ($k$ times, elementwise). In other words $pqr \in P^3$ whenever $p,q,r \in P$. Let $P^0 = \{1\}$. Now $\Bbb{N} = P^0 \uplus P^1 \uplus P^2 \uplus \cdots$. Notice the union is disjoint and fully covers $\Bbb{N}$. Thus...

Mathematicians like to ask covering various sets with open intervals and the answers to these riddles have strange tendencies to become strange lemma's or theorems Heine-Borel Theorem, lebesgue's Number Lemma, Vitali Covering Lemma, Besicovitch Covering Theorem, (etc) -3blue1brown By...

The Definite Integral Problem (with a twist) In the Riemann integral one essentially calculates the area by splitting the area into $N$ rectangular strips and then taking $N \to \infty$. Here's something I asked myself related to the Riemann integral. Let's say I split the area into say $3$ st...