Recently my husband, Howard, was in hospital for a few nights. The day he went in began reasonably normally although he didn’t feel great and wasn’t eating much. Howard visited his GP expecting some antibiotics, was referred to the Same Day Emergency Care Unit at the local hospital and was then admitted that night with a kidney infection. Although we were extremely grateful to receive such quick and efficient care, I went home on my own that night feeling rather worried and upset. The next morning was spent packing a bag for Howard and then, later, taking the bus to the hospital to see him. This was Valentine’s Day so not very romantic, although at least Howard seemed much better. He was more concerned that he hadn’t bought me a card than anything else. I bought him in Turkish delight, and we sat munching these. Howard came out two days later.

Howard was in ward number 47 (The Acute Medical Unit) and so I thought this would be a good number for my blog post. For those of you who don’t know, for each post I choose a number that has occurred naturally in my everyday life and find an interesting mathematical property to consider.

The staff in Ward 47 were amazing and made me feel welcome every day. But it wasn’t a pleasant place to be: one of the patients died there during one visit. Anyway, on a more cheerful note, when he’s well Howard eats a great deal. When he went into hospital he hadn’t eaten properly for about four days and the staff were fantastic with this too – they were happy to provide him with five Weetabix and several rounds of toast plus porridge and numerous cups of tea for breakfast! He had pleasant pals on the ward too and sometimes I found myself falling asleep in the chair as he chatted away to them.

The number 47 is a pernicious number. I thought this was rather apt as, if you don’t know, pernicious means having a harmful effect particularly in a gradual or subtle way. This was the nature of Howard’s illness which had been gradually growing worse for months.

Pernicious Numbers

# Pernicious Numbers

A pernicious number is a positive integer such that there is a prime number of 1s when it is written in its binary form.

In base 10 we work in powers of 10 e.g. 10^{3}, 10^{2}, 10^{1}, 10^{0} (or 1000, 100, 10, 1)

The number 47 is 4×10^{1} plus 7×10^{0} where 10^{0}=1.

Putting the powers of 10 as column headings we can represent 47 as follows in base 10:

10^{3} 10^{2} 10^{1} 10^{0}

4 7

In binary the column headings are powers of 2 rather than powers of 10 as follows:

2^{7} 2^{6} 2^{5} 2^{4 }2^{3} 2^{2} 2^{1} 2^{0}

These correspond to 128, 64, 32, 16, 8, 4, 2, 1

To convert 47 to binary first find the largest power of 2 less than 47 – that’s 2^{5} = 32. We put a 1 in the 32 column:

2^{7} 2^{6} 2^{5} 2^{4 }2^{3} 2^{2} 2^{1} 2^{0}

1

We have 47-32=15 left to distribute. We find that 2^{4} = 16 is too large but 2^{3} = 8 is fine. So we put a 1 in the 2^{3} column as follows:

2^{7} 2^{6} 2^{5} 2^{4 }2^{3} 2^{2} 2^{1} 2^{0}

1 0 1

We now have 47 – (32 + 8) = 40 so have 7 left to find. If we look at 2^{2} = 4, 2^{1} = 2 and 2^{0} = 1 their sum is 4 + 2 + 1 = 7. So we can put 1s in the last three columns on the right to make up 47:

2^{7} 2^{6} 2^{5} 2^{4 }2^{3} 2^{2} 2^{1} 2^{0}

1 0 1 1 1 1

The binary representation of 47 is therefore 101111 which has five 1s and, as 5 is prime, 47 is a pernicious number.

The first few pernicious numbers are:

3, 5, 6, 7, 9, 10, 11, 13, 14, 17, 18, 19, 20, 21, 22, 24 and so on.

A square value of 2 cannot be a pernicious number. The reason is that the binary representation will just require a single 1 in one of the columns: for example, 64 is 2^{6} and its binary representation is 1000000. As I’m sure you know, 1 is not a prime number.

A number that can be expressed 2^{n} – 1 has n 1s. If you think about it, 2^{5} = 32 which is 100000: if you subtract 1 (32 – 1 = 31) then you will get a 1 in each column after 2^{5} to the right giving 11111 – i.e. five 1s. Clearly, if n is prime the number of 1s will be prime and the original number will be pernicious.

The thing about pernicious numbers is that there seem to be rather a lot of them! But this is on top of the fact that all base 10 numbers with a binary representation with an even number of 1s are considered evil and all those with an odd number of 1s are considered odious (47 is odious) as seen in my post here (scroll to the end). I find all this rather depressing – surely numbers aren’t this bad! Fortunately, the number 47 has some more endearing properties. For example, it is polite as it can be written as the sum of consecutive integers: 23 + 24. We met polite numbers in this post.