Part 1 - Motivation

Why Number Theory?

Short answer: it’s fun! please don’t ask me why.

Long answer follows.

Is it true that for all integers $ n \geq 2 $, there exist $ x, y, z \in \mathbb{N} $ satisfying,

$$ \frac{4}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z} $$

Is is true that when $ x $ is large enough, then

$$ | {p \leq x : p \text{ is prime} }| - \int_2^x \frac{dt}{\log t} < x^{\frac{1}{2}} (\log x)^{100000} $$

Show that there are infinitely many primes of the form $ 2^p - 1 $ (with $ p $ prime)?

Is it true that the equation $ x^n + y^n = z^n + w^n $ has no integer solutions $ x, y, z, w, n \geq 5 $, other than the “obvious” solutions?

Do there exist odd perfect numbers?

(Perfect number $ n $ has the property that the sum of its proper divisors is equal to $ n $)
- If $ n $ is odd and perfect, then $ n > 10^{1500} $ and has $ \geq $ 101 prime factors and $ \geq $ 10 distinct prime factors.