Molly Roberts

Have you ever had a favourite number? Perhaps there’s one that comes up every time something good happens to you, and so you say that’s your lucky number? Or perhaps you know the number of muscles it takes for somebody to smile? You might think of that as a happy number; after all, people normally smile when they are happy. Perhaps there is a number which you think is simply the perfect number – of people at a table, maybe, or of books in a series. In fact, for something to officially be a happy number, a lucky number or indeed a perfect number, it has to satisfy certain conditions, which is exactly what we’ll be discussing in this article… Maybe you’ll find out that your lucky number isn’t really very lucky at all!

Happy Numbers

A happy number is defined as follows:

The sum of the squares of the digits eventually reaches 1.

What does this mean? Well, for example, we might take the number 10. 10 has digits 1 and 0, and so the sum of the squares of the digits is 1² + 0² = 1. So 10 is happy.

You’ll notice that we only had to do one sum here to discover that 10 is a happy number. Sometimes, you will have to do a lot more. For example, 7 is a happy number. This is shown by the following calculations:

7² = 49

4² + 9² = 97

9² + 7² = 130

1² + 3² + 0² = 10

1² + 0² = 1

Here, we repeatedly take the sum of the squares of the digits of the previous result. After five steps, we reach the number 1. So 7 is happy.

Are there numbers that don’t eventually give 1 when put through this process? Let us try the number 16:

1² + 6² = 37

3² + 7² = 58

5² + 8² = 89

8² + 9² = 145

1² + 4² + 5² = 42

4² + 2² = 20

2² + 0² = 4

4² = 16

As you can see, after eight steps, we return to the number 16, without ever having passed through 1. This means that no matter how many more steps you try, this process will just repeat itself over and over. So 16 cannot be a happy number, and we would therefore call it unhappy.

Now, as an activity, can you work out which of the numbers from 1 to 9 are happy? How about the number 1913091205? Can you guess the significance of this number?

[Scroll down to reveal the answer – and to read the next section on Lucky Numbers]

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Only 1 and 7 are happy. The rest of the numbers end up giving one of the values which is in the same loop that 16 ends up in above, and so are unhappy.

The number 1913091205 is what you get if you write out the word SMILE using the positioning in the alphabet for the respective letters (S is the 19th letter, M is the 13th, I is the 9th, etc.). Rather nicely, it turns out to be happy: 1² + 9² + 1² + 3² + 0² + 9² + 1² + 2² + 0² + 5² = 203 then 2² + 0² + 3² = 13 then 1² + 3² = 10 and finally 1² + 0² = 1.

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Lucky Numbers

To be lucky, a number must survive a specific example of something called a sieving process. This is essentially where you take a sequence of numbers, and then remove some of them according to a set of rules.

Finding the lucky numbers involves using a sieving process detailed in a 1956 paper titled ‘On Certain Sequences of Integers Defined by Sieves’ by Gardiner, Lazarus, Metropolis and Ulam. A preview of this paper can be found here.

Their sieving process is as follows:

Step 1: Take a list of all the whole numbers (integers) and delete every second-one to create a new sequence.

Step Two: Look at the next surviving number n in the new sequence (in this case it is 3) and delete every n^th (in this case third) integer to create a new sequence.

Step Three: Look at the next surviving number n in the new sequence (in this case it is 7) and delete every n^th (in this case seventh) integer to create a new sequence.

The sieving process continues like this for an infinite number of steps. As you can see, the next surviving number would be 9, and so you would delete every 9^th number of the new sequence, and then the 13^th and then the 15^th and so forth. The lucky numbers are defined as the ones which do not get deleted. The first 20 lucky numbers are:

Taylor Swift will be extremely relieved to see this list, since 13 has been her lucky number for her entire career. But how about you – is your lucky number on the list? What are some other patterns for sieving you can think of? For example, what numbers from 1 to 20 remain when you remove every 2^nd number, then every 3^rd number, then every 4^th number, and so on? How about when you remove every 2^nd number, then every 4^th number, then every 6^th number, and so on?

[Scroll down for the answers – and the next section on Perfect Numbers]

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Removing every 2^nd number, then every 3^rd number, then every 4^th number, and so on leaves 1, 3, 7, 13, and 19.

Removing every 2^nd number, then every 4^th number, then every 6^th number, and so on leaves 1, 3, 5, 9, 11, 17, and 19.

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Perfect Numbers

The final type of number which we’ll look at in this article is the perfect number. This is perhaps the most pleasing and intuitive of the three types discussed here – there is no random squaring or sieving of anything! Instead we are just dealing with good old fashioned divisors.

Let us first discuss what we mean by divisors. A divisor for an integer is a whole number which exactly divides said integer into another whole number. So for example, 3 is a divisor of 6 because 3 divides 6 into 3 lots of 2. Or 17 is a divisor of 68, because 17 divides 68 into 17 lots of 4. Another way of thinking about it is that a divisor of a certain integer is a number which has that integer as a multiple.

Now, a perfect number is a number whose proper divisors – that is, the divisors other than the number itself – sum to equal the number you started with. For example, 6 and 28 are both perfect.

How do we know this? Well 6 has proper divisors 1, 2, and 3, and 28 has proper divisors 1, 2, 4, 7, and 14.

1 + 2 + 3 = 6

1 + 2 + 4 + 7 + 14 = 28

Hence by definition they are both perfect.

If you are born on the 6th of the month, or on the 28th (like me!) then congratulations. You are literally perfect. Everybody else, bad luck. There are no other perfect numbers between 1 and 31. In fact, the next perfect number isn’t until 496. These numbers are few and far between, and you might think it would be nearly impossible to find more of them. In fact, perfect numbers have been studied for a very long time, and there is a useful theorem for finding even perfect numbers called the Euclid-Euler Theorem.

The theorem states that all numbers of the form 2^p-1(2^p-1) are perfect numbers if both p and 2^p-1 are prime. In this case, 2^p-1 is what’s known as a Mersenne prime. Furthermore, according to this theorem, every even perfect number can be written in this form! How very handy. Can you find the next two even perfect numbers after 496 using this formula and a calculator? What about any odd perfect numbers? I’ll leave you to investigate further…

[Scroll down for the answers]

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The next two even perfect numbers are 2⁶(2⁷-1) = 8128 and 2¹²(2¹³-1) = 33550336. In the first case, p=7, and in the second p=13. Using p=11 does not give a perfect number as 2¹¹ – 1 = 2047 = 23 x 89 and so is not prime, hence the conditions of the theorem do not hold.

As for odd perfect numbers, there have not been any found so far. There are a set of rules which any odd perfect number is proven to have to obey if such a number existed, but so far this existence has not been proven or disproven.

If you enjoyed this article on numbers with cool names, be sure to check out the next two in the series:

Numbers With Cool Names Part 2 – Amicable, Sociable, Friendly

Numbers With Cool Names Part 3 – Weird, Sexy, Untouchable