Here are some things to think about:
- Can you find an interesting set S so that |Sn| is finite but Pr(S) doesnt
exist? Can you find an interesting set S so that |Sn| is bounded but Pr(S)
- Whats the probability that three integers chosen at random are relatively
prime? You can do a numerical simulation and even set up an infinite
product, but no one knows the exact values for the infinite product youll
create. It is known that the value is irrational.
- Learn about the Bernoulli numbers Bn. These are numbers that are defined by
Yes, these numbers are from Pascals triangle. It turns out that
This generalizes the formula
to sums of reciprocals of even powers. Do you see how? Very little is known
about odd powers.
|B0 || = 1 || ||
|B0 + 2B1 || = 0 || ||
|B0 + 3B1 + 3B2 || = 0 || ||
|B0 + 4B1 + 6B2 + 4B3 || = 0 || |
- Whats the probability that a prime number will leave a remainder of 3 when
you divide it by 4?