There are many different types of numbers and
classifications of numbers, some of which you will have heard of such as the
obvious integers, negatives, evens and the like. However, there are also some rarer,
more interesting types a few examples of which I will give now.

__Vampire Numbers__
Vampire Numbers are whole numbers which can be factorised
into two numbers which together have the same digits as the original number.
The two factors must have half the number of digits of the original number, and
the number itself must therefore have an even number of digits (the factors
must also not both have trailing zeroes). It is easier to explain these with an
example, so I will give one. 1395 is a vampire number because it is 15
multiplied by 93 and 1395 has the same digits as 15 and 93.Similarly 1260 is a
vampire number as it is 21 times 60 and 1435 is 35 multiplied by 41.

There are also different subsets of vampire numbers such as
numbers which have two or more possible pairs of ‘fangs’ (as the factors are
known). The first one of these is 125460 which is 204 × 615 and 246 × 510, the
first number with three possible pairs of factors is 13078260, which equals
1620 × 8073, 1863 × 7020 and 2070 × 6318. I could go on. Also there are
Pseudovampire numbers whose fangs are not equal to half the number of digits of
the original number. There are lots of these, well infinitely many, such as 126
which equals 6×21. Finally, there are Prime vampire numbers whose fangs are its
(only) prime factors as in 117067 which is 167*701.

__Perfect Numbers__
Perfect numbers are numbers whose divisors (not including
itself) sum to the original number. The first example of this is the number 6,
whose factors are 1, 2 and 3, which clearly add up to six, and the second
perfect number is 28, which is 1 + 2 + 4 + 7 + 14. What is interesting about
Perfect Numbers is that they are indefinitely linked with Mersenne Primes,
which are primes which are 1 less than a power of 2 such as 3 which is 1 less than
4, 7 (1 less than 8) and 31 (1 less than 32). For every Mersenne Prime, m,
m(m+1)/2 is a Perfect Number. For example, the first Mersenne Prime is 3 and
3(3+1)/2 = 12/2 = 6, the first Perfect Number. Leonhard Euler, one of my
favourite mathematicians, proved that all perfect numbers are like this. As a
concept, they are very old. The Greeks knew the first four (6, 28, 496, 8128),
and the fifth was first recorded in the mid 15

^{th}century.**Amicable and Sociable Numbers**

Perfect numbers are linked with Amicable and Sociable
numbers, which both involving adding up the factors. Amicable numbers come in
pairs, such as 220 and 284, where the sum of the factors of the first number is
the second number and vice versa. Other pairs include 1184 and 1210 as well as
2620 and 2924. Sociable numbers are groups of numbers that form a chains so
that the sum of the factors of A equals B, the sum of B equals C and the sum of
C equals A. One such group is of four numbers; 1,264,460 -> 1,547,860 ->
1,727,636 -> 1,305,184 which goes back to 1,264,460. One delightful use of
amicable numbers is as a nerdy, romantic gesture as shown by the people at
Maths Gear - http://mathsgear.co.uk/products/amicable-numbers-pair-of-keyrings-nerd-romance.

## No comments:

## Post a Comment