## Some Things are Just Hard to Explain: The Taniyama-Shimura Conjecture

As a technical person you are often in a position of having to explain intricate technical issues to folks who aren't deeply technical or knowledgeable in a particular field. Perhaps the listeners are your superiors, your investors, your work colleagues, or your customers. Whoever they are you need to be able to explain seemingly complex technical issues in a language they understand. This isn't always easy as you may not know the level of interest or understanding the listener has of the particular topic of conversation. My advice is to initially offer the "simplified, but not completely dumbed-down executive summary" response, and only offer up further technical details if requested. That way you are not insulting them by a gratuitous generalisation but you are giving them the choice of going deeper if they request it, rather than forcing an in-depth discussion on them and suffering the indignity of being cut-off before you've made your point. The Board of Directors doesn't want to hear a techo-babble sermon.

Clearly, this is just sensible communication skills, and is something that can easily be examined during interviews. For instance, a potential Product Manager might be asked:

*Can you explain a how a database works to a 7-year old?*

Or a maths geek with post-graduate education applying for a programming gig might be thrown a curve ball out of left field with something like this (yes I've actually seen that before, believe it or not)...

*How would you explain to high school students the Taniyama-Shimura Conjecture as it applies to Fermat's Last Theorem?
*

So WTF is that you're asking? Well it's a particularly complex mathematical theorem that even Harvard professors have trouble explaining, and therein lies the problem. You might be clever enough to have an understanding of the subject matter but conveying it to others is particularly challenging.

But first a slight digression to get everyone on the same page...

## Background Info

Perhaps one of the most famous theorems in mathematics is **Fermat's Last Theorem**. This was a conjecture proposed by Fermat in the 17-th century. It states that no three positive integers a, b, and c can satisfy the equation a^{n} + b^{n} = c^{n} (the Diophantine equations) for any integer value of *n *greater than two. Famously, Fermat made a margin note in one of his manuscripts saying he had a proof for it but didn't include it due to space restrictions. Without Fermat committing his proof to paper the problem stood as an unproven conjecture for some 300 years. Because of the importance of the conjecture, the simplicity of the equation, the number of famous mathematicians who had failed to find a proof for it, and the theatrical story around the "missing proof", Fermat's Last Theorem, which was actually still a conjecture at the time, became very famous in the mathematics world.

Fermat's Last conjecture was a conjecture for over 300 years, meaning it had never been proven, but the *conjecture *became a *theorem *when Mathematics Professor, Andrew Wiles, proved it in the mid 90s. But a key part of the breakthrough proof for the conjecture was a lesser known but equally important conjecture known as the **Taniyama-Shimura conjecture** for elliptic curves. This conjecture, which had been around since the 50s, stated that every elliptic curve was a *modular form*. It was later proved and became known as the **Modularity Theorem**.

So how exactly is it related to Fermat's theorem? Well it provided the key link that bridged 2 disparate fields of mathematics: *modular forms* and *elliptic curves*, and in doing so brought a proof for Fermat's theorem much closer than it had been before. The Taniyama-Shimura conjecture had as one of its consequences, Fermat's Last Theorem, meaning that formally proving Taniyama-Shimura would result in a proof for Fermat's theorem. Basically the plan of attack was as follows:

1. Assume that Fermat was wrong and there is in fact a solution to the equation a^{n} + b^{n} = c^{n} , *n*>2.

2. With the assumed solution we can generate an elliptic curve with some rather weird properties - the curve seems not to be modular, but the Taniyama-Shimura conjecture stated that all elliptic curves are modular - a clear contradiction.

3. This gave incentive for researchers to find a proof for Taniyama-Shimura because the discovery of such a proof would be the critical piece in proving Fermat's Last Theorem.

In essence, this is *proof by contradiction* (a simple example of this can be found here).

## Explanation By Way of an Analogy

So how do you summarise the relationship between these two conjectures so that high school students - who have never studied any number theory - can understand it? Use a simplified analogy! Here's one I came up with that I hope shows how these 2 pieces are related:

*Imagine you have to figure out if there exists a special one-of-a-kind metallic needle located somewhere within a massive haystack the size of Texas. We don't know for sure if the needle is in the haystack or not. Some people say that the special needle is not in the haystack at all and we'd like to confirm this before we waste time searching. We know that it's not really feasible to try and search through the haystack because it's just too damn big so we need another way to confirm that the needle does or doesn't exist. Coincidently an eccentric Japanese scientist has recently claimed that he has built the worlds' first super-range needle magnet that, he claims, can suck in a metallic needle even if it is as far away as the moon is from the earth. If we use his super-range needle magnet and we don't find anything it doesn't really tell us that the needle doesn't exist - it could mean that, or it could mean that his magnet is not as good as he claims. Given that it is relatively easy to try the whiz-bang new magnet - and most likely not find anything as most people suspect - we only need to prove that the super-range needle magnet can detect a needle at a distance of "the size of Texas" to be completely confident that the needle doesn't exist. This seems like an easier problem to tackle so you decide to focus on it.*

In this analogy, finding the needle in the haystack is analogous to proving Fermat's Last Theorem, and checking the range of the new magnet is analogous to proving Taniyama-Shimura. High school students would at least comprehend the analogy.

*12 Jul 2009*
*Damien Wintour*

on 23 Jul 2009 at 8:51 am 1Adam BreenJesus would be proud

on 03 Mar 2010 at 4:04 pm 2Ace AventuraNo doubt one of the Taniyama guys commited suicide

on 11 Mar 2011 at 8:04 am 3paulowow. How do magnets work?

on 21 Mar 2011 at 8:20 pm 4DAMichaudAre all needles magnetic?

on 21 Mar 2011 at 8:57 pm 5Damien WintourDan, as long as it is not stainless-steel it would be magnetic. But good point!