Something, admittedly, strange about me when it comes to numbers, is that generally, I don't like even numbers. I'm not sure what it is, but I just don't like them. I like odd numbers. I think I will always have a group of many, changing, favourite numbers (which we can explore in due course) but as the favourite of all my favourite numbers it will be the number 1. That is why my profile picture is a number 1. 1 is so simple. So lovely. 1 is, as the first positive odd number, almost more odd than any other number! And though other "favourite numbers" can be favourite for this cool reason or that, there will always be another number with another cool reason to be a favourite number - whereas there will never be another number to start off the whole counting system!
So anyway, it is strange that I would be writing about the number 86 - it is even, contains only even digits, and it's not even part of many of the categories which make up lots of my favourite numbers. It's clearly not prime (by far the best grouping of numbers!!! > which brings me on to the mixed feelings I have for the number 2 - it's like the most even of even numbers - using the same logic as for 1 being the most odd of odd numbers - and yet it is also prime; what are you supposed to do with that kind of situation?!), it's not a smith number, nor a perfect number, it's not a primary pseudo-perfect number, it's not a harshad number, it's not narcissistic, or amicable, and there are many other things which the number 86 is not. So why then, write a post about the number 86???
Well, when I arrived at Cambridge University - well about half a week later when I'd finally worked out how the intranet worked here - I had been issued with an email address of the form ...@cam.ac.uk. But it doesn't stop there, this ... section of the address is also my log in for every single possible website, computer or ANYTHING to do with my time here in Cambridge at all. And part of this, featured the number 86 - to my utter disdain when I found this out.
Eventually, after trying all of the number types above (and many more) to find 86 is none of those either :( I was almost at the point of leaving the university altogether {DISCLAIMER: that may be a slight exaggeration}. So I paid a visit to the Zoo (the Zoo of Numbers, that is)!!! And this is a pretty good website - it has interesting facts (some not mathematical) about practically every single number you could possibly require an interesting fact for! And here, I discoverd that 86 is a happy number!
Now happy numbers were something I vaguely remembered having seen a +Numberphile video on ages ago - so I looked this up. Happy numbers are numbers that if you square each digit and then add them together, and repeat this to that answer and so on, you eventually reach the number 1 (told you the number 1 is great)! So, turns out, 86 is actually VERY happy as it only takes 2 iterations to reach 1:
86: 8^2 + 6^2 = 64 + 36 = 100
100: 1^2 + 0^2 + 0^2 = 1 + 0 + 0 = 1
So I thought, well that's great! But to be honest, it's probably not that rare - and numbers have much better reasons for being good when those reasons make them rare! So I set about working out, for all the numbers up to 100, how many were happy. I found that, surprisingly, only 20 numbers from 1 - 100 (including both 1 and 100) are happy numbers. That's 1 in 5 - and my 86 happened to be a part of that! Things were starting to look up for my relationship with my ID.
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| The happy numbers are in the boxes at the top; the other numbers all feed into the red circle in the middle! [16, 37, 58, 89, 145, 42, 20, 4, 16...] |
A fitting name for this cycle, I feel, was that of the "melancoil" as named by Matt Parker in the Numberphile video on that! In my diagram, the cycle starts with 16 (circled) and goes like this:
16: 1^2 + 6^2 = 1 + 36 = 37
37: 3^2 + 7^2 = 9 + 49 = 58
58: 5^2 + 8^2 = 25 + 64 = 89
89: 8^2 + 9^2 = 64 + 81 = 145
145: 1^2 + 4^2 + 5^2 = 1 + 16 + 25 = 42
42: 4^2 + 2^2 = 16 + 4 = 20
20: 2^2 + 0^2 = 4 + 0 = 4
4: 4^2 = 16
and there is starts again!
So if any number, at any point in doing this to it, contains one of these 8 numbers; it will feed into this cycle. And when using the power of 2 and working in base 10, these are the ONLY 2 possibilities - it either feeds down to 1 or it gets trapped in this cycle. That's pretty cool. As far as mathematical importance goes, it's considered "recreational number theory"; but hey, it's made my identification at uni bearable - so I'll say it's pretty important right now!!!
But it does make you wonder... Why does this happen? Where could it lead? If anyone knows of any mathematical research articles or anything on these numbers I would be interested to find out - I conducted a brief search of the web and only found the Wikipedia page. It would be odd if NO-ONE thought to just see where it takes them!!!
This is why I love number theory - sometimes, it can just be so abstract and unrelated to the real world that it makes you stand back and just appreciate it. It stops your mind from running a hundred miles a minute just trying to keep up with all the practicalities of life - with this, you just experience and discover; in your own time how you like to. It's nice to do something without an agenda sometimes!!!
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