In my previous post I mentioned that we were lucky enough to spend Christmas and New Year in Barbados. During that time I wasn’t short of numbers to think about as well as appreciate the beautiful sights, sounds and the joy of meeting many delightful people. One thing that affected us constantly was the currency. You can use US dollars and, if you do, you’re given change in Barbadian (or Bajan) dollars. There are two Bajan dollars to every US dollar. This would be straightforward enough but, as we’re from the UK, it’s easiest for us to gauge the cost of things by converting into pound sterling. The rate was approximately 83p to the US dollar. Thus, the currency conversion led to a few thought processes before being comfortable with the price of anything. For example, we bought supper one night for 108 Bajan dollars. This sounded a lot until I divided in two (54 US dollars) and then roughly took 80% (£41ish). And this was extremely reasonable given that I’d had a rum punch, Howard (my husband) had had several beers and we’d had some delicious barbequed fish. In fact if you avoided hotels most places were very reasonably priced.

One good thing about getting home was that I no longer needed to carry out sums in my head before buying anything!

Given my experience with exchange rates in Barbados I thought that 83 would make a good number to write about for this blog post. If this is the first of my posts you have read, the idea is to take a number that has arisen naturally in my daily life and to find an interesting property that it holds. You don’t need any specialist knowledge to read these posts – nothing more than you received in ordinary classes at school.

As I was looking into 83 the most interesting property that I found, which I hadn’t covered before anywhere else, was that 83 is a magnanimous number. It is also evil and polite – both of which I’ve mentioned in earlier posts so please follow the links and scroll toward the end if you’d like to read more.

# Magnanimous Numbers

A magnanimous number is a number with at least two digits such that the sum obtained by inserting a “+” amongst its digit in any position gives a prime. For example, 170 is not itself prime but it is a  magnanimous umber because the numbers 1 + 70 = 71 and 17 + 0 = 17 are both prime.

As all the prime numbers are odd, except for 2, all the magnanimous numbers, except for 11, are either a sequence of odd digits followed by an even digit (such as 152 where 1 + 52 = 53 and 15 + 2 = 17 are both prime), or a sequence of even digits followed by an odd digit (e.g. 203 where 20 + 3 = 23 and 2 + 03 = 5 are both prime).

It is believed that the magnanimous numbers are finite and that probably the largest one is 97393713331910. The constraints of being magnanimous become very restrictive as numbers become larger so it is not a surprise that the sequence is finite. Even in the table below, which covers numbers 11 to 200, you can see that there are 33 magnanimous numbers between 11 and 99 and only 11 between 100 and 199 (i.e. the first 100 numbers have three times as many magnanimous numbers as the next 100 numbers). The numbers between 100 and 199 have two constraints to meet and so it makes sense that there are significantly fewer of them.

There are many ways of looking at this table. With magnanimous numbers of two digits which are not multiples of 10 or repeated digits (i.e., 11), their mirror images are also magnanimous – which makes sense when we are looking at the sum of the digits – the order of the sum doesn’t matter – a property known as the commutative property. We have (33 – 5) / 2 = 14 pairs as follows:

12 is magnanimous (and so is 21)

23 is magnanimous (and so is 32)

34 is magnanimous (and so is 43)

and so on up to

89 is magnanimous (and so is 98)

I wonder why these numbers are called magnanimous which, as I’m sure you know, means “generous or forgiving, especially towards a rival or less powerful person”. The property of being magnanimous is enveloping more numbers than the property “is prime”. From the above table numbers only 11, 23, 29, 41, 43, 47, 61, 67, 83, 89, 101 are prime. By applying the magnanimous property – that is a property relating to prime numbers – a further 33 numbers are included under 200 to the property “is prime”.