Confined to a particularly geeky section of all of our minds, is part us of that thinks about very large numbers. As children, we tried to think of the largest numbers we could. It seems that sometimes there is a point where numbers become so arbitrarily large that we find them difficult to comprehend, and need to use analogies. But this number, has a claim to be the largest number ever used for constructive purposes. It’s so large, that it’s almost impossible to describe how large it is. This is Graham’s number.

The first question you are probably asking is “Who’s Graham?”. This is Ronald Graham, an American mathematician born in 1935 from California. He began looking into Ramsey Theory, a field in Combinatorics, which generally poses problems in the following style.

Last week, the news broke that Shinichi Mochizuki, a Japanese mathematician, has claimed a proof of the abc conjecture, otherwise known as the Oesterlé-Masser conjecture, causing widespread excitement among the mathematical community. But what exactly is the conjecture, and what does its proof mean?

The abc conjecture was, until last week, seen as one of mathematics’ greatest unsolved problems, and sometimes seen as the “Holy Grail” of Number Theory: so many open problems in Number Theory follow as nearly a direct consequence of the abc conjecture or can be reduced to it. Like many Number Theory problems, it can be stated in (relatively) simple terms, and you don’t need to be a highly specialised and educated mathematical genius to understand what the problem is. However, its proof is far from simple, and is only understandable to a handful of mathematicians in the world, and at the moment most likely only to Mochizuki himself. It may take a considerable amount of time for the 500 page document to be fully verified and confirmed correct, but today I’m going to discuss the problem and some of its consequences.

Today, as requested by Oscar Smith (@F15hb0wl) on twitter, we’re going to have a look at Tupper’s self-referential formula: what is it, how does it work, and is it really as amazing as it seems?

Let’s just start by saying what the formula is. It was devised by Jeff Tupper, and goes as follows:

And when this is plotted as a graph starting with N on the Y axis (N is a very large number printed below!), it produces the following graph.

N = 9609393799189588849716729621278527547150043396601293066515055192717028023952664246896428421743 507181212671537827706233559932372808741443078913259639413377234878577357498239266297155171737 1699516523289053822161240323885586618401323558513604882869333790249145422928866708109
6184496091705183454067827731551705405381627380967602565625016981482083418783163849115590225610 003652351370343874461848378737238198224849
863465033159410054974700593138339226497249461751545728366702369745461014655997933798537483143 786841806593422227898388722980000748404719 (a 543-digit number).

Last night, after two days of intense competition, the Olympic Heptathlon finally finished. Team GB’s Jess Ennis secured gold despite the scoring system that is weighted against her, as we showed a few weeks ago.  In the wake of this, we’re going to have a look at what would have happened using the scoring system that we proposed as an alternative at this Games. Would the results have been the same? Would it have been fairer? 

Firstly, the official results for the Heptathlon are here for those who want to view them as we go along. Secondly, I recommend you read the first post before we continue; this post may not make much sense without that background.

Hey! Before I start this post, I better introduce myself, I’m Theo, and I’m a new writer on this blog along with Ned. Just a word of warning, this post does get quite technical later, but I do think you guys will like it, so please persevere, and enjoy!

I start with a problem for you guys…

Is there such thing as something with no end?

If I’m honest, I’ve never really thought about this very much. I know what you are thinking: “ What is this guy on about? He thinks he can write about maths but he hasn’t even considered what is one of the biggest, literally, subjects in maths?” and for you, I have no answer, It just never seemed me to be that interesting, nothing to prove wrong and calculations to make, infinity is infinity, a never ending number, and as brilliant that is, it has never had a real meaning to me…until yesterday: