Day 06 in Mat115

Today:

• Hand in problems from 2.1

• Keep checking for your assignments!

• Return 1.4 problems: graded 2 and 8.
  • I've already had an astonishing number (10 of 29) of people not turn in a homework. You're digging yourself a hole....
  • Please use complete sentences. Write well: your writing may be the first thing a potential employer may know about you, so make a good impression.
  • Avoid run-on sentences. Use more semi-colons!
  • 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. Be specific.
  • Did you read the hints? Kudos to those who said "I checked the hint, and tried...."
  • a "~" on your homework just means something wasn't quite right -- it doesn't count against you. I didn't actually take off for your writing (e.g. spelling, punctuation) this time, but I will as of the third homework you hand in (Wednesday).
  • #2: Politicians
  • #8: Cannibals

• Section 2.2: Fibonacci numbers
  • If you looked at some special cases, what can we conclude?

    Number of sticks	1	2	3	4	5	6	7	8
    Winner	X	P2	P2	P1	P2	P1	P1	P2

  • What's the winning strategy in Fibonacci Nim?
  • The strategy is based on an important fact about numbers -- what is it?

• Section 2.3: Prime Numbers:
  • This concept must be understood for you to understand the upcoming sections! If you want to be a good Spy, and secret coder, pay attention!

  • Decomposing numbers, part II:
    • We can "decompose" (or factor) natural numbers using prime numbers and products.

  • What is a prime number?
    • A natural number that can be divided evenly (that is, without remainder) by only two distinct natural numbers: 1 and itself.
    • Note: therefore 1 is not a prime number.
    • We can find the primes using the Sieve of Eratosthenes (here's an animation)

  • 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

  • The distribution of primes
    • The first 100 Primes (primes come around a lot less often, the higher up you go in the natural numbers -- here are the the primes under 10000)
    • Example: #35, p. 80
    • Conclusion: gaps without primes of any size exist in the natural numbers!

  • How many primes are there? There are infinitely many -- they just don't stop! (But how do we know? We prove this theorem!)
    • A natural number n may or may not divide natural number m evenly, but there's always a unique way of writing the attempt (that is, exactly one way -- no more):
      m=qn+r

      where 0 ≤ r ≤ n-1.

      Obviously, if r=0 then n divides m.

    • We're going to show that there is a prime number bigger than any number you can give.

  • Unanswered questions about prime numbers:
    • Goldbach Question: Can every even natural number greater than 2 be written as the sum of two primes?

    • Twin Prime Question: are there infinitely many pairs of prime numbers that differ from one another by two? (3,5; 5,7; 11,13; etc. -- can you find another pair in the sieve?)

    Mathematicians don't know the answers yet!;)