Day 06 in Mat115

Today:

• (2 and 8 graded).

• No problems to collect.

• Return 1.4 problems: graded 2 and 8.
  • Use complete sentences! Write well!
  • If your solutions were one page long, you certainly didn't do enough work.
  • Don't say "I tried lots of things" without telling me what.
  • Don't introduce hypotheses of convenience -- "Suppose that the missionaries have guns...."
  • Did you read the hints? For many of you, there was no indication that you had. Kudos to those who said "I checked the hint, and tried...."
  • #2: Politicians
  • #8: Cannibals

• Questions/Comments on old stuff?

• Section 2.3: Prime Numbers:
  • Decomposing numbers:
    • Last time (in Fibonacci Nim) we decomposed natural numbers using Fibonacci numbers and sums (every natural number is either Fibonacci, or can be written as a sum of non-consecutive Fibonacci's in a unique way).
    • Now we will decompose (or factor) natural numbers using prime numbers and products.

  • What is a prime number?
    • A natural number that can be divided evenly by only two distinct natural numbers: 1 and itself.
    • We can find them using the Sieve of Eratosthenes

  • Prime Factorization Theorem: Every natural number greater than 1 is either prime, or it can be expressed as a product of prime numbers (in one and only one way, order of the product aside).

    Example: #2, p. 77

  • If natural number n may or may not divide natural number m evenly, but there's always a unique way of writing the attempt:
    m=qn+r

    where 0 ≤ r ≤ n-1.

    Obviously, if r=0 then n divides m.

  • How many primes are there? There are infinitely many -- they just don't stop! (But how do we know? We prove this theorem!)

  • How do we know that there are primes bigger than 100? The sieve produced 23 -- we're missing the rest of the infinite primes, and they're all bigger than 100!

  • The distribution of primes
    • The first 100 Primes (primes come around a lot less often, the higher up you go in the natural numbers)

    • #35, p. 80

    • Conclusion: gaps of any size exist in the natural numbers!