Published in 1997 as a follow up to a BBC documentary about the discovery of a proof of Fermat’s last theorem in 1994/5 this 362 page book takes a deep dive into the history of the theorem and the various attempts at a solution over the 358 years that it remained a mathematical puzzle. The joy of Pierre de Fermat’s last theorem is that it is very simple to understand but turned out to be incredibly difficult to prove. Anyone who has had Pythagoras’s theorem relating to the sides of a right angled triangle drummed into them at school will understand the basic concept. That theorem states that the square on the hypotenuse equals the sum of the squares on the other two sides or put simply in a diagram with the best known whole number solution as an example:

Fermat stated that although this clearly works for squared numbers, and indeed there are infinitely more combinations of whole number solutions (such as x = 5, y=12 and z=13 as 25 + 144 = 169) there are no whole number solutions if the power that you raise x, y and z to is greater than 2. Fermat stated that he had a proof, although he wrote this in the margin of his copy of the ancient Greek mathematician Diophantus’s Arithmetica and stated that ‘the proof was too large to fit in the margin’. Fermat was a mathematical genius but also extremely annoying as he would often taunt fellow mathematicians by writing to them that he had discovered a proof to some mathematical conjecture and challenge them to also find the solution and would rarely write down his own proofs in a rigorous manner. Certainly no example of Fermat proving his last theorem has ever been found. Fermat of course didn’t refer to it as his last theorem, it gained the name as slowly all his other conjectures were proved correct leaving just this one which would become notorious and also the driver of other mathematical insights as people tried to prove, or disprove, it over more than three and half centuries.

Let’s come back to that date of 1994/5 for the final proof. English mathematician Andrew Wiles had worked for many years on attempting a proof, but without admitting to his fellow mathematicians that he was working on it as it was seen as a waste of time and as a professor at Princeton University, New Jersey, USA it wouldn’t be appropriate to be seen to have an interest in the subject. However all that changed in the mid 1980’s when it was shown that Fermat’s last theorem would be effectively proved if there was a proof discovered to the seemingly unrelated Taniyama-Shimura conjecture. This conjecture deals with two extremely complex areas of mathematics and indicated that they were inter-related and indeed one could be used to solve problems in the other. These two concepts were elliptic curves, which were Wiles’s Phd speciality and modular forms, a four dimensional topological ‘structure’. Now I sort of understand the basics of elliptic curves but the use of modular forms is beyond me even with the basic description provided by Simon Singh in this book. Wiles saw this as a legitimate use of his time and would give him a proof of Fermat, which had fascinated him since he was ten years old, whilst working on a ‘genuine’ mathematical problem, the proof of Taniyama-Shimura. The problem was that this, like Fermat’s last theorem, was considered impossible to prove. He still decided to work in secret though and for many years came up against brick walls preventing his proof from working until in 1994 he took three lecture slots at a convention in Cambridge, England and under the deliberately opaque title of “Modular Forms, Elliptic Curves and Galois Representations” endeavoured to present his proof. The mathematical world was astounded and Wiles was hailed for his outstanding achievement, problems however were found during the rigorous checking before the proof could be published and it took several more months before Wiles finally fixed the error in his proof hence 1994/5 being given as the date of the solution. The 1994 proof was so close to being correct, but relied on another conjecture which it turned out wasn’t proved so proving this other theorem was what took the extra time.

Now it may well be, if you are still reading this blog, that you are thinking no way am I going to read this book it sounds far to complex but you would be wrong. Singh has done a remarkable job in not only summarising Andrew Wiles’s work and still making it approachable, but the history of the various attempts to solve Fermat is fascinating. I first read this book back in 1997 when it came out and have picked it off the shelves two or three times in the intervening decades and each time I love descriptions of the failed attempts and the progress, or otherwise, that they led to, along with the various other puzzles included which help to get your brain engaged in the problem. Each time I get that little bit further in understanding just what Wiles actually proved with the specific part of the Taniyama-Shimura conjecture (named after the two Japanese mathematicians who came up with it in 1957). Taniyama-Shimura would finally be proved for all variants in 2001 by four of Andrew Wiles’s former students and renamed The Modularity Theorem. A note on terminology conjectures are unproved but seem to work, theorems are fully proved

Give your brain a workout, I definitely recommend giving it a go.

As a child I was fascinated by mathematics, but especially by tricks and shortcuts that could be done. I started reading Martin Gardner’s section of Scientific American when I was eleven or twelve years old, I don’t claim to have understood all of it but each month my knowledge of recreational mathematics was stretched just that little bit more. I’ll cover one or more of his books in a later blog. However in 1977, when I was fifteen, this book was published and it was written by somebody who, at least partly, earned her living from amazing feats of mental arithmetic, I had to get a copy, and this book is still on my shelves today. Some of it I already knew but there were whole sections where she explained how to do tricks that I had seen done but which had baffled me such as calculating the day of the week for any date given to you or working out square and cube roots in your head. I remember practising these tricks for hours until I could do them too.

The book starts of simply by looking at each of the digits 0 to 9 in detail, explaining what is special about each of them and giving tips around multiplying and dividing by them, patterns in their multiplication tables etc. She then moves on to chapters about multiplication, addition, division and a very short chapter on subtraction. These chapters not only suggest shortcuts, which I still use today, to perform such calculations but ways to quickly check if the answer you get makes sense such as casting out nines. The book really caught my attention however when we reach calculating squares, cubes, square roots and cube roots. Amazingly cube roots which non mathematicians would assume to be much more difficult then square roots are actually very simple and fifth roots are even easier, square roots proved to be quite tricky. But just to see how easy extracting a cube root lets look at all you need to know, worryingly forty five years later I can still remember this:

• 1 cubed = 1
• 2 cubed = 8
• 3 cubed = 27
• 4 cubed = 64
• 5 cubed = 125
• 6 cubed = 216
• 7 cubed = 343
• 8 cubed = 512
• 9 cubed = 729

Assuming that we are starting with 474,552 (which is 78 x 78 x 78) how do you get the right answer? Well first of all look at the thousands i.e. 474, this comes between 343 and 516 so the first digit is the cube root of the lower number which is 7. Next you will notice that all the cubes in the list above end with a different number and you just need to find the one that ends with the same digit as the number you are trying to extract the root of which in this case is 2 which matches 512 or 8 cubed and there we have the answer, the 7 from the thousands value along with the 8 from the final digit gives the required answer of 78. Notice that it was simply a case of knowing the first nine cubes and no actual calculation was performed on 474,552 in order to get the right answer.

Calculating the day of the week is a bit more tricky as you need to memorise four tables, admittedly the first of which is simply the first four values from the seven times table so this barely counts as a table and the working out is also more involved. I can’t do this in my head anymore and frankly with the all pervading computers or mobile phones with calendars on them what was once a occasionally handy ability is now of no use whatsoever as you are rarely that far from a device where you can look up the day for a specific date if you need it. When I was a teenager however this was quite impressive at least amongst the other maths fans at school and I got to be pretty quick at it.

The book finishes with chapters on special numbers and finally tricks and puzzles most of which, even then, I had already encountered but this book stretched still further my mathematical skills and I loved it. It has been great fun reading it again and finding out what I remembered and what I had forgotten. Shakuntala Devi died at the age of 83 in 2013 and wrote several books on mathematics along with astrology and oddly ‘The World of Homosexuals’ which she claimed was inspired by her marriage to a homosexual man but Figuring: The Joy of Numbers is probably her best known work, at least outside India although sadly it appears to now be out of print. If you know a child interested in mathematics I suggest trying to get a copy for them, it really is a joy.