Molly Roberts

Act One Scene 1: (Thunder and lighting. Three weird numbers enter.)

70: When shall we three meet again? In thunder, lighting, or in rain?

836: When the hurly-burly’s done, when the battle’s lost and won.

4030: That will be ere the set of sun.

70: Where the place?

836: Upon the heath.

4030: There to meet with Macbeth.

For those of you unfamiliar with Shakespeare’s Macbeth, this is the opening scene, where the three weird sisters – or witches – meet to discuss the fate of the plays titular character. The joke here is that 70, 836, and 4030 are the first, second, and third weird numbers respectively. In this third article of the numbers with cool names series, we’ll be investigating what it means for a number to be weird, what it means for a number to be a sexy prime, and how a number can reach celebrity status by being untouchable. Among numbers, 2 is practically Beyoncé.

Weird Numbers

Before we start to learn some new types of number, I would recommend checking out the previous two articles in this series as I will be using some terminology previously explained in greater detail. The links to these can be found at the bottom of the page.

Okay, onto the weird and wonderful. A weird number is a number which satisfies two conditions:

1. The sum of the proper divisors is greater than the number itself
2. No subset of the proper divisors sums to the number itself

Now, this might seem like quite a confusing definition, so I think we’d better begin with an example. As I mentioned above, 70 is the first of the weird numbers. Let’s consider the proper divisors of 70 (whole numbers less than 70 which divide it without leaving any remainder): 1, 2, 5, 7, 10, 14, and 35. These add up to 74, which is clearly greater than 70, and so the first condition of being weird is satisfied. In order to satisfy the second condition, we need to make sure we can never find a subset of the proper divisors which add together to make exactly 70. In other words, if we take some, but not all, of the proper divisors and add them together, this total must never be 70.

Suppose this is not the case. Since the proper divisors sum to 74, we need to be able to subtract one or more of them from 74 to get 70. Clearly all the candidates that are bigger than or equal to 5 will not work. But then the only proper divisors left are 1, and 2, and these sum only to 3. So subtracting any number of the proper divisors from 74 will give a number either strictly bigger or strictly less than 70. 

Therefore, we conclude that we can never find a subset of the proper divisors which add together to make exactly 70, and so condition 2 is satisfied: 70 is weird.

Let’s try a second example with a number that isn’t weird. Pick 12. This has proper divisors 1, 2, 3, 4, and 6. Now, these sum to 16, which is greater than 12, and so the first condition of weirdness is satisfied. However, if we take 1, 2, 3, and 6, which is a subset of the proper divisors, we can see that these sum to exactly 12. Hence the second condition of weirdness is violated and 12 is not weird.

In fact, every number between 1 and 69 is not weird. Why don’t you investigate this! Have a go at finding the proper divisors for 18, 30, and 60. In each case, can you find the sum of the proper divisors? And then can you find subsets which sum to give the number itself? Hint: there is more than one subset for each example.

[Scroll down to reveal the answers and the next section on Sexy Numbers!]

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Answers

• 18 has proper divisors 1, 2, 3, 6, and 9. The sum of these is 21. The subsets of these which sum to 18 are {1, 2, 6, 9} and {3, 6, 9}.
• 30 has proper divisors 1, 2, 3, 5, 6, 10, and 15. The sum of these is 42. The subsets of these which sum to 30 are {1, 3, 5, 6, 15} and {5, 10, 15}.
• 60 has proper divisors 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30. The sum of these is 108. The subsets of these which sum to 60 are {1, 3, 5, 6, 15, 30} and {5, 10, 15, 30}.

Sexy Primes

Which numbers do you think are sexy? I think everyone will be disappointed to learn that 69 is not a sexy prime – as it’s simply not a prime number. In fact, sexy primes are fairly boring. They are simply prime numbers which differ from another prime number by exactly six. Examples of sexy primes include 5, which differs from 11 by 6, or 23, which differs from 29 by 6. The name is derived from the Latin ‘sex’ meaning six, but mathematicians, having a great sense of humour, decided they considered every prime number differing from another by six to be sexy.

In fact, you can find lots of sequences of sexy primes throughout the integers. The longest possible sequence is of length five, and is unique: 5, 11, 17, 23, 29. Can you see why this is the longest possible? And can you see why it is unique?

The sequence above is the first one to occur since 2 + 6 = 8 which is not prime, and 3 + 6 = 9 which is also not prime. The next prime number after 2 and 3 is 5 and this gives rise to our sequence. We continue to add 6 – moving between primes at each stage – until we reach 29, but then we are forced to stop since 29 + 6 = 35, which is not prime. So this sequence has length 5 and can’t be extended.

Now let’s suppose we have a longer sequence, say of length 6, somewhere else along the numberline. If such a sequence exists, then it also contains a sequence of length 5 (for example just take the first 5 numbers out of the 6 available). This means that if we can show our sequence is the only one of length 5, we automatically get that it is the longest.

So, let’s try to do exactly that. Consider any sequence of sexy primes of length five. Call the first number in the sequence ‘n’. Then our sequence can be written:

n, n + 6, n + 12, n + 18, n + 24 (where all of these values are prime)

Now there are five options:

1. If n is a multiple of 5, it must be exactly 5 as we have assumed that n is prime. This gives our sequence above.
   
2. If n is one more than a multiple of 5, then n + 24 is 25 more than a multiple of 5. But this is then also a multiple of 5, which isn’t allowed as n + 24 must be prime. Therefore option 2 cannot give a sequence.
   
3. If n is two more than a multiple of 5, then n + 18 is 20 more than a multiple of 5, which means it is therefore also a multiple of 5. Once again, this can’t happen since n + 18 must be prime. Therefore option 3 fails to generate a sequence.
   
4. If n is three more than a multiple of 5, then n + 12 is 15 more than a multiple of 5, which is itself also a multiple of 5. As above, this needs to be prime to be in the sequence and so once again option 4 cannot generate a sequence.
   
5. Finally, if n is four more than a multiple of 5, then n + 6 is 10 more than a multiple of 5 which means it is itself also a multiple of 5. Again, nonsense.

In conclusion, given that any number n has to fall into one of the 5 options above, we can see from our working that only option 1 makes any sense, which gives exactly the sequence we claimed was unique.

Now, can you find any pairs of sexy primes larger than 29? How about triplets? Quadruplets?

[Scroll down for the answers and for the next section on Untouchable Numbers!]

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Answers

All the options for primes under 100:

• Pairs: {31, 37}, {37, 43}, {41, 47}, {47, 53}, {53, 59}, {61, 67}, {67, 73}, {73, 79}, {83, 89}
• Triplets: {31, 37, 43}, {41, 47, 53}, {47, 53, 59}, {61, 67, 73}, {67, 73, 79}
• Quadruplets: {41, 47, 53, 59}, {61, 67, 73, 79}

Untouchable Numbers

Ah, the untouchables. Revered, worshipped, elevated beyond human status. Or… Integer status. These numbers stand out from the rest, part of an exclusive club of integers so special that they don’t associate with any others. But what actually makes a number untouchable?

Again, we think about divisors. Untouchable numbers are numbers which cannot be written as the sum of all the proper divisors of any integer. In other words, no matter which integer we pick, if we take all the proper divisors of this integer and add them up, we can guarantee that this sum will not equal our untouchable number. As usual, let’s look at an example to make this clearer.

Consider the number 5. Every integer (except 1) has 1 as a proper divisor. So in order to reach 5 by summing proper divisors, we need a set of distinct numbers that includes 1 which must sum to 5. Now, let’s think of all the ways of summing distinct integers to reach 5:

• 5 = 5
• 1 + 4 = 5
• 2 + 3 = 5

These are the only ways, because summing the smallest three distinct natural numbers gives 1 + 2 + 3 = 6, which is larger than 5 and so immediately we may discount all other combinations of three or more. From the argument above, we need the combination to include a 1 if we are to reach 5 by summing proper divisors. Hence the proper divisors must be exactly 1 and 4, since the other two ways of summing to 5 do not include 1. But if 4 is a proper divisor of a number, then so is 2, and hence the set {1, 4} cannot be the only proper divisors of any number. This means 5 is untouchable.

Now consider the number 3. We may reach 3 by summing 1 and 2, and these are the only proper divisors of the number 4. Hence we conclude that 3 is not untouchable.

There is a very interesting article from 1991 by J. Sesiano, which discusses the theory around untouchable numbers, although the author does not use the term ‘untouchable’. In fact, the article states that 2 and 5 ‘stand among the numbers like bastards among people’. A rather less favourable view of what it means to be untouchable. You can find a preview of the article here.

Now, can you determine which of the numbers from 2 to 10 are untouchable? I’ll give you a hint: consider the products and squares of prime numbers…

[Scroll down for the answer!]

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Answers

• Untouchable: 2, 5 (given above)
• Not untouchable: all the rest
  • 3 is given above
  • 4 is 1 + 3, and {1, 3} are the only proper divisors of 9
  • 6 is 1 + 5 and {1, 5} are the only proper divisors of 25
  • 7 is 1 + 2 + 4 and {1, 2, 4} are the only proper divisors of 8
  • 8 is 1 + 7 and {1, 7} are the only proper divisors of 49
  • 9 is 1 + 3 + 5 and {1, 3, 5} are the only proper divisors of 15
  • 10 is 1 + 2 + 7 and {1, 2, 7} are the only proper divisors of 14

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

Numbers With Cool Names Part 1 – Happy, Lucky, Perfect

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