Lucky Numbers: They exist!

We all heard how there are no such things as lucky numbers, right? Well actually… they exit!

I mean, if you throw a pair of dice and adding them, having 7 as your pet number has nothing to do with how often you are going to be correct. Each dice give you 1 out of 6 chance to get any one of its figures, but if you wish for a 2 or a 12, there are not as many ways to get that sum. Seven, on the other hand, can be obtained with many combinations. See yourself:
Ok, so… no lucky numbers then.
Except there are! Here’re a few, in case you need some:
1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, 43, 49, 51, 63…
Lucky Numbers are actually a very well-defined mathematical concept*. They are simply a group of numbers sharing some properties, like Prime Numbers (a number that can be divided evenly only by 1 or itself). What’s lucky about them is essentially how they “survive” to a process of elimination similar to the one we can use to find Prime Numbers (it’s called a sieve).

This is probably a little bit disappointing, I know… But it still makes an awesome conversation starter!
And in case your date is indeed impressed by your mathematical cleverness, don’t let him/her down yet. You can further demonstrate your skills by showing how to find the first few of these lucky b***.
Write a list of integers (start with 1):

1 	2 	3 	4 	5 	6 	7 	8 	9 	10 	11 	12 	13 	14

Eliminate every second number in the list, you’ll get this:

1 		3 		5 		7 		9 		11 		13

The next term in the sequence is now 3, so you eliminate every third number in the list:

1 		3 				7 		9 				13

The next surviving number is 7, so every seventh remaining number is eliminated… and so on. There you go!
In a dating context, be careful not to confuse Lucky Numbers with Fortunate Numbers: They are something totally different (and actually simply named after a guy)! And well, if you are still upset that Lucky Numbers won’t bring you chance, you are out of luck: You won’t even be able to complain to the mathematician’s quartet Gardiner, Lazarus, Metropolis & Ulam who gave them that name back in 1956. None of them is still alive!
But maybe you can soothe away the pain by having a look at Happy Numbers! They are actually much more interesting in terms of properties (so far), and not really difficult to find. You start with a positive integer, like 49, and replace the number by the sum of the squares of its digits, e.g. 4^2+9^2=16+81=97, and repeat. You can, of course, start with a one digit number:

7^2=49 ; 4^2+9^2=97 ; 9^2+7^2=103 ; 1^2+0^2+3^2=10 ; 1^2+0^2=1If you finish on 1, you got yourself a nice list of Happy Numbers (yep, 7 is both Lucky and Happy, and 1 as well)! Sadly, it is also possible that you will hit a bunch of Unhappy Numbers, and get caught in an infinite loop (always the same one: …4, 16, 37, 58, 89, 145, 42, 20, 4 …). Turns out the chances are actually pretty high since there are only 143 Happy Numbers smaller than 1000, and it doesn’t get better after that. But hey! I’m sure you can come up with a clever line for your date, something like turning your Unhappy phone number into a Happy one… by calling you back!

*Actually, there is TWO kinds of Lucky Numbers! We also have the Lucky Numbers of Euler. They are, however, a little less easy to play with. We define them as the positive integers n such that for all integers k with 1 ≤ k < n, the polynomial k^2 − k + n produces a prime number. And guess what: there are only 6 of them: 2, 3, 5, 11, 17 and 41. How lucky is that?