Day 10: Heart of Mathematics

• Show how to express 1729 in base 2.
• Explain why we can't express 1729 using base 1.
• Show how to add 147 and 173 (both numbers expressed in base 8)
• Bring one thing from nature that illustrates Fibonacci numbers
• Problems from section 2.2: pp. 60-61, #16, 18, 20, 22

• Rules:
  • Two players, n sticks/coins/bottlecaps/etc.
  • First player must take anywhere from 1 to (n-1) sticks
  • From then on, the next player may take from 1 up to twice what the previous player took.
  • Winner is the player who picks up the last stick
• Is there a winning strategy?
• Is there a perfect defensive strategy?

every natural number is either

1. Fibonacci, or
2. can be written as a sum of non-consecutive Fibonacci's in a unique way.

Examples? Can we justify this?

• If a number is Fibonacci, then we're done.
• Assume a number is not Fibonacci:
  • then there is a largest Fibonacci that "fits inside it" -- e.g. 24 = 21 + 3.
  • If the remainder (3, in the example above), after subtraction, is Fibonacci, how do we know that it is not consecutive (that is, that it is not 13)? That is, how do we know that the two Fibonacci numbers are not successive Fibonacci numbers?
  • If it is not Fibonacci, can we not simply repeat the current process of looking for a sum for a non-Fibonacci number, but using the remainder instead of the original number? (i.e. can we not simply "recurse" -- that is, do it again, and so construct a chain of numbers leading down towards 1.

    Example: 33=21+12=21+8+4=21+8+3+1

• We can "decompose" (or factor) natural numbers using prime numbers and products.

• 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)

• 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 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 -- is 1729 prime?)
• Example: #35, p. 80
• Conclusion: gaps without primes of any size exist in the natural numbers!

• 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.

• 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!;)