https://web.evanchen.cc/napkin.html If that’s down, please see backup here.

What’s above complex numbers? Quaternions. Then octonians. Further higher number systems can be invented (or discovered).

Abstract Algebra

Emmy Noether

Fields

Galois Theory

Rings

Modules

Representation Theory

Categories and Functors

Number Theory

Prime numbers cannot be negative

“Why Isn’t 1 a Prime Number?”

1 is not a prime number; see the Fundamental Theorem of Arithmetics for why (has to do with uniqueness)

Mersenne prime

Perfect number

Fermat’s Last Theorem

Goldbach’s conjecture

  • Arithmetic function or number theoretic function is any function f(n) whose domain is the positive integers and whose range is a subset of the complex numbers
  • Divisor function is an example of arithmetic function whose value at a positive integer n is equal to the number of divisors n
  • There is another class of function of number theoretic functions that do not fit the above definition
  • Multiplicative function is an arithmetic function f(n) of a positive n with the property that f(1) = 1 and whenever a and b are coprime, then f(ab) = f(a)f(b)
  • Euler’s totient function counts the positive integers up to n, that are relatively prime to n.
  • \(\varphi(mn)\) = \(\varphi(m) \varphi(n)\) if m and n are relatively prime

“The sum of its digits will never by divisible by 3. If the sum of the digits is a multiple of 3, then the number is a multiple of 3 (i.e. not prime).

For example, 349: this ends in 9, so it passes the 2 and 5 tests immediately. Digit sum isn’t divisible by 3, so it passes the 3 test.

“Final digit must be 1, 3, 7, or 9” is equivalent to saying that it’s not divisible by 2 or by 5.

It’s 300+49, so not divisible by 7; 330+19, so it’s not divisible by 11; 310+39, so not divisible by 13; 340+9 so not divisible by 17.
I did all of those in forms that don’t even require a carry digit, so they’re pretty fast to do by inspection.

And the number is greater than 19^2, so that covers all cases – it’s prime.” karlo_

p-adic

Group Theory

Category Theory

“Category Theory for Neuroscience (pure math to combat scientific stagnation)” https://www.youtube.com/watch?v=4GJ4UQZvCNM

Language of relationships that focuses on how mathematical structures relate to one another rather specific details (numbers, shapes, equations, etc.)

Basically like learning the rules of grammar rather than just individual words

Categories

Morphisms

Compositions

An example:

  • Category: category of sets, “think of this as a family where each member is a set”
  • Objects: each object is a set, set A, set B, etc.
  • Morphisms: arrows, morphisms are functions that take elements from one set and relates them to another, for example, f(1) = a, f(2) = b, etc.
  • Composition of morphisms: we can do function compositions