Day 09: Heart of Mathematics

Here's Competitive Nim: The game of Fibonacci Nim (modified on-line version)

• Start from the idea of decomposing (or factoring) natural numbers into products.

  Last time (in Fibonacci Nim) we decomposed natural numbers using Fibonacci numbers and sums.

• We can find them using the Sieve of Eratosthenes

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

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

• Even though primes appear to appear irregularly amongst the natural numbers, the relative number of primes in the first n natural numbers appears to approach a known value:
  Number of primes in first n natural numbers = n/Ln(n)

  Again, we see that as we head off to infinity (wherever that is!) a quantity is tending toward some nice limiting value.

• Exercises: ??