In Chester, where I live in Northwest England, we are preparing for a by-election on the 1^{st} December where we will elect a new Member of Parliament (MP). Our previous MP resigned and it has been a rush to nominate candidates and canvass for all parties. The MP for Chester will be one of a total of 650 which cover the whole of the UK. As this by-election is such a prominent part of my life at the moment – I’m literally out every day distributing leaflets – then the number 650 will be the focus of this blog post.

In my posts I select a number and look at mathematical properties that I find particularly interesting. The posts don’t require any special mathematical ability – no more than GCSE-level.

We know that 650 is evil and polite from earlier posts (the relevant details are at the end of each post) – this seems an odd combination but I suppose an evil person can be polite. I’m sure that Hitler would have been polite had you met him. I hope that I don’t know any evil people so wouldn’t know directly!

There are many more mathematical properties of 650 but I’ll just concentrate on those that made me smile and which I think will be of greatest interest to you.

# Apocalyptic Numbers

Firstly, I love the fact that 650 is described as an apocalyptic number and hope that this doesn’t reflect the UK parliament in any way – a number that contains the digits “666” (the number of the beast). This really makes me smile.

2 to the power 650 (2^{650}) is

46719391924451279038885597242370983166906659914225253689130931383288577712216856437820547187889883133733297250033606730861452336346884485102107251913955473287700425465449113088848631728605222** 666**24

You can see that I’ve highlighted “666” towards the end of the number. Such a large number is easy to calculate with an online calculator that gives you all the digits of the calculation. This is the calculator that I used.

The first apocalyptic powers of 2 are 157, 192, 218, 220, 222, 224, 226, 243, 245, 247, 251, 278, 285, 286, 287, 312, 355, 361, 366, 382, 384, 390, 394, 411, 434, 443, 478, 497, 499, 506 and so on.

Interestingly, an apocalypse number is one with 666 digits. I doubt we’ll be meeting many of these!

# Admirable Numbers

A number is called admirable if it is equal to the sum of its proper divisors, where one of them has minus sign.

For example, 12 is admirable because its proper divisors are 1, 2, 3, 4, 6, and 12 = 1 + 2 + 3 – 4 + 6.

In the case of 650 its proper divisors are:

1, 2, 5, 10, 13, 25, 26, 50, 65, 130, 325

When we add all the divisors of 650 together you get:

1 + 2 + 5 + 10 + 13 + 25 + 26 + 50 + 65 + 130 + 325= 652

Which is 2 too many. If we put a minus sign in front of the 1 we get

– 1 + 2 + 5 + 10 + 13 + 25 + 26 + 50 + 65 + 130 + 325 = 650

and you can see that 650 is admirable.

The first admirable numbers are 12, 20, 24, 30, 40, 42, 54, 56, 66, 70, 78, 84, 88, 102, 104, 114, 120, 138, 140, 174, 186, 222 and so on.

Unlike apocalyptic numbers I don’t see a connection between the properties of an admirable number and its name. I think that number theorists can throw these adjectives around rather randomly. But I find it interesting nevertheless.

# Interprime

In my last post I wrote about the number 19 which is a weak prime as it is closer to the previous prime (17) than the one after it (23). 650 is not prime but is described as an interprime number because it is at equal distance from the previous prime (647) and next prime (653).

# Wasteful numbers

Wasteful numbers, in my mind, do have some logic to their name. At the moment most of us are doing our best to spend our money carefully. In the UK inflation has gone over 11%, the highest it has been for over forty years. I find it interesting that numbers, like humans, can be described as wasteful. A number n is called wasteful if the number of digits in its prime factorisation (including powers) uses more digits than the number of digits in the number itself. The first few wasteful numbers are 4, 6, 8, 9, 12, 18, 20, 22, 24 and so on.

The prime factorisation of 650 = 2 x 5 x 5 x13 (i.e. 4 digits) whereas 650 has only 3 digits.

In my head this definition is saying “come on 650 you could use fewer factors and be less wasteful”.

# Hoax Numbers

A hoax number is one where the sum of its digits coincides with the sum of the digits of its ** distinct** prime factors.

In the case of 650:

- the sum of its digits is 6 + 5 = 11
- its distinct prime factors are 2, 5, 13 and 2 + 5 + 1 + 3 = 11

I imagine that this is a hoax as there is no logical reason for it.

The first few hoax numbers are 22, 58, 84, 85, 94, 136, 160, 166, 202, 234 and so on.

# Final Thoughts

I enjoy writing these posts and I hope you enjoy reading them. 650 turned out to be a number with some very interesting properties. When I first considered it there didn’t seem to be anything of immediate interest. But its properties are really quite wide-ranging.pr