My husband, Howard, and I help to run a history society in our village hall which is the rather impressive building shown in the photo. Usually nine talks are given per year on topics that cover the local area. Due to COVID recent talks have been spread over the period of time from January 2020 until now. We have had talks in January and February 2020, October 2021 and March, April, May, June, October and November 2022. Our talks have ranged from a description of local pubs, a history of Ness Gardens which was famous for selling seeds, local windmills and watermills. We have some people who pay a £12 membership to cover the nine talks and others who pay £3 / talk and come when the talk is of particular interest to them – e.g. we have one member who is particularly interested in anything to do with the Second World War. We had a total of 25 members since January 2020 and 34 guests. Our group is small but it’s lively and it’s fun to get together after the talk and catch up over tea, coffee and biscuits. We’ve discovered that Aldi sell delicious chocolate biscuits at a very affordable price so these are always made available: for November’s talk we had mince pies as it’s the last talk before Christmas.
Our committee are concerned about rising prices and keeping the society going. We’re planning to raise the membership tee to £15/year from January 2023. Howard and I currently buy all the milk and biscuits and we’ll start asking for a donation for these. We’ll also look for members who can give talks.
Thus, the number 25 has occurred naturally in my everyday life (being the number of members of the history society since January 2020) and will form the focus of this blog post. If this is the first of my posts that you have read then the idea is to identify a number and to find a few interesting mathematical properties that it holds. These posts are written for everyone and you do not need any specialist mathematical knowledge to understand them – nothing beyond standard schooling.
There are many properties of the number 25 but I will pick out the ones that I find most interesting. My hope is that these posts will widen your appreciation of numbers.
I particularly enjoy learning about numbers which are described by adjectives that are recognisable. My first thought on coming across the fact that 25 is an aspiring number was “what is 25 aspiring to be?” as 25 seems a perfectly good number as it is. As you’ll see below, aspiring numbers such as 25, aspire to be perfect numbers where a perfect number is a positive integer that is equal to the sum of its positive divisors, except the number itself (its proper divisors). For example, the proper divisors of 6 are 1, 2, 3 and
1 + 2 + 3 = 6
Aspiring numbers are closer at reaching the ‘perfect’ property than other numbers.
A number is called an aspiring number if its aliquot sequence converges to a perfect number, and it is not a perfect number itself. An aliquot sequent is one made up of positive integers in which each term is the sum of the proper divisors of the previous term.
The proper divisors of 25 are 1 and 5 and 1 + 5 = 6
The proper divisors of 6 are 1, 2, 3 and 1 + 2 + 3 = 6 illustrating that that 6 is a perfect number as we saw before.
The corresponding aliquot sequence is 25, 6, 6, 6, 6, 6 and so on.
To reiterate, a number such as 25 that is not perfect but whose aliquot sequence becomes constant with a perfect number is known as an aspiring number.
This existence of perfect numbers explains why the aliquot sequence becomes constant.
The first few aspiring numbers are : 25, 95, 119, 143, 417, 445, 565, 608, 650, 652 and so on.
Try another number, say 20.
The proper divisors of 20 are 1, 2, 4, 5 and 10.
Their sum is 1 + 2 + 4 + 5 + 10 = 22
Thus 20 isn’t perfect as it is not the sum of its proper divisors. So far the aliquot sequence is:
The proper divisors of 22 are 1, 2, 11 and 1 + 2 + 11 = 14
The proper divisors of 14 are 1, 2, 7 and 1 + 2 + 7 = 10
After these two steps we have aliquot sequence:
20, 22, 14, 10
The proper divisors of 10 are 1, 2, 5 and 1 + 2 + 5 = 8 giving
20, 22, 14, 10, 8
You can see that this sequence is not converging on a perfect number.
Let’s have a look at another number that we know to be aspiring: 95 is the next on the list. The proper divisors of 95 are 1, 5, 19. Their sum is:
1 + 5 + 19 = 25
So we’re back to the stage we were at before:
The sum of 25’s proper divisors is 1 + 5 = 6
The sum of 6’s proper divisors is 1 + 2 + 3 = 6
The aliquot sequence for 95 is
95, 25, 6, 6, 6, 6, 6 and so on
I’ll try one more: 119.
The proper divisors of 119 are 1, 7, 17
Their sum is 1 + 7 + 17 = 25
And then we get the sequence
119, 25, 6, 6, 6, 6 and so on.
25 is not perfect as it is not the sum of its proper divisors. But its aliquot sequence becomes constant – i.e. reaches a perfect number – and, in this sense, 25 aspires to be a perfect number. I love this about the number 25. Many humans aspire to be something they are not at the present time and I find it touching that some numbers do the same! One thing that 25 has achieved is that it is the smallest of all aspiring numbers.
Interesting properties of squared numbers
I’m sure we’re all aware that 25 is a squared number:
5 x 5 = 25
But 25 has far more unusual and, as far as I’m concerned, interesting properties related to its squares. It is one of only two automorphic numbers – where automorphic means patterned after self. An automorphic number is one where the square and higher powers of the number also end in the same last two digits. I needed a scientific calculator to show all the significant figures for some of these calculations. I used this one as I did in my previous post. Use xy as follows: type 25 (or 76) then xy and then the power you want. (I’m criticised sometimes for explaining things in this amount of detail – some feel it’s obvious – but if you haven’t used a calculator since you were a child then you may well have forgotten. These posts are for everyone!)
252 = 625, 253 = 5625, 254 = 390625, 255 = 9765625 and so on. Let’s try a much larger power:
2550 = 7888609052210118054117285652827862296732064351090230047702789306640625
The only other number with this property is 76 – I find it fascinating that there are only two numbers with this property and that they look so different from each other. Here are the first higher powers of 76:
762 = 5776, 763 = 438976, 764 = 33362176, 765 = 2535525376
I thought I’d try a really big power of 76 just in case you need further convincing:
7675 = 1150851852325435496965212183375947079594790621601577391596662822693657268477725262660679935558021705907335658634674691093065886890773893349376
The last two digits of the result of any power of 25 will be 25 and the last two digits of the result of any power of 76 will be 76. How cool is that?! And we’re back to a discussion of very large numbers, just like the number of settings of the Enigma machine, which always makes me happy.
The next property of 25 and its squares is no doubt one you’ve met at school. But, for completeness, I’ll mention it anyway.
25 is the smallest square that is the sum of two (non-zero) squares: 32 + 42 = 52. For this reason it often appears in illustrations of the Pythagorean theorem. Here is a link if you wish to refresh your memory.
I have grown quite fond of the number 25 – I find it touching that it aspires to be perfect! To me it has a lot going for it anyway, being one of only two automorphic numbers and the smallest square that is the sum of two non-zero squares.
If you have a number that you’d like me to consider in a blog post then please let me know either below or via email at email@example.com And if you have any questions then feel free to post comments below.