# Lucky, lucky 2019

Welcome back to Smartodds loves Statistics.

Let’s start the new year with a fun statistic:

2019 is a lucky, lucky year.

Why is that? Well, let’s start with prime numbers. You’ll know that a prime number is a whole number that can’t be written as a multiple of other whole numbers. For example 6 is not a prime number since $6 = 3 \times 2$, but 7 is a prime since it can only be factorised as $7 = 7 \times 1$.

One way of generating the prime numbers is as follows.

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30….

The first number remaining is 2, so remove all multiples of 2 that are bigger than 2:

2, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29…..

The second number remaining is 3, so remove all multiples of 3 that are bigger than 3:

2, 3, 5, 7, 11, 13, 17, 19, 23, 25, 29…..

The third number remaining is 5, so remove all multiples of 5 that are bigger than 5:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29…

And keep going this way. The numbers that remain are the prime numbers.

It’s easy to check that

2, 3, 5, 7, 11, 13, 17, 19, 23, 29

comprise all of the prime numbers that are smaller than 30. To get the bigger prime numbers you just have to apply more steps using the same procedure.

Lucky numbers are generated in much the same way. This time we start with the sequence of all positive whole numbers:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,…..

The second number is 2, so we remove every second number from the sequence, leaving

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, …..

The third number remaining is 5, so we remove every fifth number of the sequence

1, 3, 5, 7, 13, 15, 17, 19, …..

The fourth number remaining is 7, so we remove every 7th number.

1, 3, 5, 7, 13, 15, 19, …..

And so on….

The numbers that remain in this procedure are said to be lucky numbers. And proceeding in this way, it’s easy to check that 2019 is a lucky number. But 2019 isn’t just ‘lucky’, it’s ‘lucky, lucky’. Every whole number can be written uniquely as a multiple of prime numbers. In the case of 2019 the unique prime factorisation is:

$2019 = 3 \times 673$

And… both 3 and 673 are also lucky numbers. So 2019 is doubly lucky in the sense that it is both lucky itself and all of its prime factors are lucky. Moreover, 2019 is the only year this century that has this property, so enjoy it while it lasts.

This post is really more about mathematics than statistics, but how about this? If I take a large number, 2020 say, and pick a number at random from 1 to 2020, what’s the probability that it will be a lucky number? One way to do this would be to identify all of the lucky numbers up to 2020. If there are m such numbers, then the probability a randomly selected number will be lucky is m/2020.  But it turns out there’s a good approximation that can be calculated very easily, and it works for any large number, not just 2020.

A classical result from number theory is that the probability that a randomly selected number in the sequence 1,2,…., N is a prime number, for any large value of N, is approximately

$1/\log(N)$

where  log is the logarithmic function. With N=2020, this is equal to 0.13, so there’s roughly a 13% chance that a number from 1 to 2020 is a prime number. But almost incredibly, this same approximation works also for lucky numbers, so there’s also roughly a 13% chance that a number from 1 to 2020 will be a lucky number. Obviously lucky, lucky numbers are much rarer, and I don’t know of any formula that can be used to calculate the probability of such numbers. The fact that there is just one lucky year this century, though, suggests the probability is pretty low.