3. Introductory Material

Created Sunday 16 February 2014

https://class.coursera.org/maththink-004/lecture/2

Proof that there is an infinite number of prime numbers:

  1. Let's say p1, p2, ..., pn are the first prime numbers
  2. Let's say that N = p1 * p2 * ... * pn + 1
  3. If N is the prime number then the pn is not the last prime number
  4. If N isn't the prime number, then it's dividable by some prime number P.
    1. If P is one of p1, p2, ..., pn then N/P = some whole number + remain of 1 — but it should be just whole, so
    2. P isn't one of the first primes p1, ..., pn — as they are the first primes, P > p1, ..., pn, so
    3. pn is not the last prime number
  5. In either way, for any number of the first prime numbers there is always at least one greater prime number, so the first prime numbers list is extendable infinitely

Almost every key statement in maths (axioms, conjectures, hypotheses, theorems) is a positive or negative verison of 1 of 4 linguistic forms:

  1. Object a has property P
  2. Every object of type T has property P
  3. There is an object of type T having property P
  4. If statement A, then statement B

Combinators that connect these simple statements' forms:

  1. And
  2. Or
  3. Not
  4. Implies
  5. For all
  6. There exists



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