3 is a very lucky number indeed. Well, at least mathematically. Firstly, it's the first odd prime number. It's also the first Fermat prime, Marsenne prime, Gaussian prime, Genocchi prime and lucky prime.

The last of these is pretty interesting. So for those who love the big M like me, the sequence of lucky numbers, first discussed by Stanisław Ulam in 1955, is obtained as follows:

First consider the entire set of positive integers, i.e.

1, 2, 3, 4, 5, 6, 7, 8, ...

Then eliminate every second term from the list, leaving

1, 3, 5, 7, 9, 11, 13, 15, ...

The second surviving term is 3, so now we eliminate every third term from the list, leaving

1, 3, 7, 9, 13, 15, 19, 21, ...

The third surviving term is 7, so we eliminate every seventh term from the list, leaving

1, 3, 7, 9, 13, 15, 21, 25, ...

Repeat this process indefinitely and we will have the sequence of lucky numbers:

1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, 43, 49, 51, 63, 67, 69, 73, 75, ...

A lucky prime is simply a lucky number that also happens to be a prime. These are

3, 7, 13, 31, 37, 43, 67, 73, 79, 127, 151, 163, 193, ...

The implication of the lucky primes cannot be discussed in brief here, so i wouldn't even attempt to. But it's curious to see that besides the much loved 7, 13 is actually a very lucky number as well. :)