Monday, 28 July 2014

Types of Numbers - Vampire Numbers

This week I thought I would write about something slightly more fun, and less hard work to understand. I came across these numbers when I was on a website about maths careers, although apparently I had read about them first in Simon Singh's book - The Simpsons and Their Mathematical Secrets. I don't remember, but my mum has been reading it, so I will trust her! The numbers I am going to talk about are Vampire Numbers.

The Wikipedia page says that a vampire number is:

"a composite natural number v, with an even number of digits n, that can be factored into two integers x and y each with n/2 digits and not both with trailing zeroes, where v contains precisely all the digits from x and from y, in any order, counting multiplicity."

In English, this means that a vampire number is any whole number with an even number of digits (so it can be a 2 digit number, a 4 digit number, a 108 digit number, you get the idea...), call it v, such that there are two whole numbers with half the number of digits of v, call them x and y, where the digits of x and y are exactly the same as the digits of v in any order, and they multiply together to make v. We call x and y the fangs of the vampire number. The fact that the fangs do not have trailing zeroes just means that the fangs can't both be divided by a factor of 10 to give a whole number. For example, 600 and 740 have trailing zeroes because you can divide both by 10 to give 60 and 74.

I think it is easier to see what I mean with an example of a vampire number. Take v = 1260. This is actually the first vampire number. You can see that it has an even number of digits (4), so the fangs will both have 2 digits. If you have x = 21 and y = 60, you can check that 21 x 60 = 1260. Also, 21 and 60 use the exact same digits as 1260. They don't have trailing zeroes (60 can be divided by 10, but 21 cannot), so this satisfies all of the conditions for 1260 to be a vampire number.

The list of vampire numbers is infinitely long, meaning you can always find another one, your list will never be complete. I don't know how this has been proved, but I would be interested to find out. Vampire numbers are a relatively new concept in maths, only being written about for the first time in 1994 by Clifford A. Pickover.

Some vampire numbers have more than one set of fangs. For example, 125460 has two sets of fangs: 204 and 615 or 246 and 510. I will leave it to you to check that they satisfy the conditions!

There are also different types of vampire numbers, such as prime vampire numbers where the fangs are both prime numbers.

As far as I'm aware, vampire numbers have no mathematical significance, other than the fact they are rather beautiful things. Numbers are weird, and I love all the patterns, constructions and strange coincidences you can find within them.

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