
Fermat's Last Theorem
Martin Carrudus on the Proof of 'Fermat's Last Theorem'
Fermat was a 17th Century mathematician, who made many 'conjectures', but rarely provided actual proofs of them. One such conjecture was that whole numbers of the form 2^{2^n}+1 i.e. 2 raised to the power 2 raised to the power n, with a 1 added to the whole lot, were always prime for all whole numbers (integers), n. This was eventually disproved by an 18th Century mathematician, Euler, for the case n=5, i.e. 2^{2^5}+1 can be factorised, and it is now thought that, to the contrary, that these socalled 'Fermat Numbers' are, in fact, all factorisable for n greater than 5.
Note that a 'prime' number is a whole number that has no 'factors' other than one, and itself. For example, the first prime number is 3, since it can only be divided by 1, or by itself. The opposite of a prime is a 'composite'. This is a number that is divisible by some number less than itself and not one. For example the number 4 can be divided by itself, by 1 and also by 2. The 'Fundamental Theorem of Arithmetic', states that all whole numbers are either prime or expressible as unique product (multiplied together) of primes less than it, some of which may be raised to certain powers. This may seem obvious, (really? ED) but there are certain domains of 'number like' objects that do not have this property i.e. you can get the same result by 'multiplying together' two entirely different pairs of 'numbers'.
'Fermat's Last Theorem' was a conjecture that Fermat made, that was not the last he made, it is just the last that remained either to be proved, or disproved. These 'conjectures' were in the field of 'Number Theory', where we are dealing with the properties of the whole numbers, 1, 2, 3, 4, ...
For these purposes, zero is not considered to be a true 'whole' number.
Equations that are to be solved with all whole numbers in them are called 'Diophantine' Equations, after Diophantus, a Greek mathematician, of the 3rd Century A.D., who first wrote down results concerning such equations.
Around 1637, Fermat was studying a Latin translation of Arithmetica, by Diophantus, where Diophantus was trying to separate the square of a whole number into the sum of the squares of two other whole numbers. In Latin in the margin of the book Fermat wrote:
'On the other hand, it is impossible to separate a cube into two cubes, or a biquadrate into two biquadrates, or generally any power except a square into two powers with the same exponent. I have discovered a truly marvellous proof of this, which, however, the margin is not large enough to contain.'.
What Fermat was saying was that you can't find whole numbers, x, y and z, such that:.
xn + yn = zn, for any n greater than 2.
This, so simply stated, is 'Fermat's Last Theorem', and its proof has baffled some of the best mathematical brains for over 400 years!
The equation above IS soluble in the trivial case where the x or y is zero, and, also when n=2, it amounts (remember Pythagorus, from your school days!) to finding a right angled triangle with all whole number sides. You are also probably familiar with the 3, 4, 5 triangle, because 32+42=52, but, in fact, there are a countably infinite number of solutions. The 'trick' is to choose any two (whole) numbers, m and n, then, if you let:.
x = m^{2}  n^{2}
y = 2.m.n
and
z = m^{2} + n^{2}
THEN
x^{2} + y^{2} = z^{2} ALWAYS!
E.g. Letting m=3 and n=2, leads to x=5, y=12 and z=13, and 5^{2} + 12^{2} = 13^{2}.
This is not the only result to do with the sum of powers of whole numbers e.g.
3^{3} + 4^{3} + 5^{3} = 6^{3} and 18^{2} + 26^{2} = 1000 = 10^{3} and many other amazing results!
As you probably guessed, Fermat's proof of the Theorem was never found.
One infuriated mathematician even sent someone to search Fermat's house, after Fermat's death, in order to find the proof! In the 1840's a German mathematician, called Kummer did a lot of work on the Theorem and established an overriding theory, called the 'Theory of Ideals'. Certain numbers, in his theory, are 'ideal'.
Up until 1988, the Theorem had been proved for all exponents up to 125,000, and, if there was indeed a solution to Fermat's equation, it would involve numbers in the excess of 125,000^{125,000}, or, with over a million digits in them!
Perhaps the greatest mathematician of them all, Gauss, refused to work on it, saying that it would drop out as a sideresult of a much larger theorem. He proved to be right.
The theorem involved is the socalled 'TaniyamaShimura' or 'Shimura Taniyama' Conjecture. This concerns 'elliptic curves', which are curves in 2D in x and y coordinates, which obey the equation y^{2}=a.x^{3}+b.x^{2}+c.x+d, where the a, b, c and d are rational constants i.e. they are fractions formed by the division of one whole number into another. The circle is a special form of elliptic curve. There are also functions (calculations with numbers that result in another number) called 'modular forms' that have special properties.
The 'modular form' associated with the circle are the trigonometric functions sine and cosine.
The 'ShimuraTaniyama' Conjecture, was that EVERY elliptic curve has an associated 'modular form'.
In 1984, a mathematician called 'Gerhard Frey', at a conference in the Black Forest, made a 'joke' assertion that, if there were a solution to Fermat's equation, it would lead to an elliptic curve that could not possibly have an associated modular form. But another mathematician, Ken Ribet, managed, amazingly, to show that Frey's 'joke', was, in fact true!
A Cambridge Mathematician, Andrew Wiles, who had dreamed of solving Fermat's Last Theorem, saw his chance. Prove the 'TaniyamaShimura' Conjecture, and there could be no solution to Fermat's equation, since this would lead to an elliptic curve without a modular form, which was impossible.
Wiles' work was simplified by the fact that the class of elliptic curves involved, was a subset of all the curves, called semistable.
Wiles started with a technique called 'Horizontal Iwasawa' Theory, but found it only applied in a limited set of circumstances. He returned to the mathematical community for ideas.
Then Wiles heard that a young student mathematician, Matthias Flach, had found a way of numbering the elliptic curves, using some earlier work by a Russian mathematician, Kolyvagin. The socalled FlachKolyvagin numbering. Wiles now saw the method, which is called induction. Show that if an elliptic curve numbered 'n' has an associated modular form, then it follows that the elliptic curve numbered 'n+1' ALSO has an associated modular form. Then prove it for n=1, and you have proved it for all n. Like knocking over an infinite line of balanced dominoes by knocking the first.
Wiles spent the next seven years secretly working on the problem. Whenever he got stuck, he quietly approached friends for help. He likened it to entering a dark room, bumping over the furniture in the dark, until he found the light switch, then entering another dark room, and doing the same thing, until, finally, the whole house was lit.
What Wiles was doing, in effect, was running through all the contingencies that could arise, and eliminating them one by one.
In May 1993, Wiles overcame one final hurdle, and he had a proof. All 200 pages of it. He presented it in the June at a Conference on Number Theory at Cambridge. The leading experts in Number Theory studied the proof  and found a flaw in it!
Wiles went to ground again for a whole year. It was like he was trying to cover a floor with a carpet with a hole in it. No matter how he shifted the carpet around, he couldn't get rid of the hole!
He was just giving up, when, on Monday, September 19th 1994, he thought he would write down the exact conditions under which his proof was failing. With a blinding flash, he saw these conditions were EXACTLY the ones under which Horizontal Iwasawa Theory applied. It was the extra piece of carpet he needed! He couldn't believe what he'd written down. He had to go away for an hour, and then study it again to see if it was true.
And it was just such a neat result, it was so beautiful that it made him weep.
Wiles' work was built on the backs of centuries of work by other mathematicians. It was a triumph of 20th Century Mathematics. Many of the techniques involved just didn't exist in other centuries before. We now believe that Fermat thought up a proof that he thought he could generalise to higher powers, without realising that, in these number systems, you lost the 'unique factorisation' property. Probably one of the greatest Mathematical 'jokes' of all time, the way it perplexed generations of Mathematicians to come!
Postscript...
What are the practical results of the proof of Fermat's Last Theorem? It is only of interest to Mathematicians! There is one 'interesting' result. The symbol x or, the modulus of x, is x with the sign taken out  e.g.
2 = 2.
If we consider curves in x and y of the form: x^{n} + y^{n} = 1, for n greater than 2, the socalled called 'hyper ellipses', then they pass through no point where both x and y (the coordinates) are rational, except where x=0 or y=0. The method of proof, as used in the proof of Fermat's Last Theorem, is to assume the opposite of what you are trying to prove, and show this leads to an absurdity. Socalled reductio ad absurdum proof.
Suppose, then, that we can have both (positive) rational x and y in x^{n} + y^{n} = 1.
We can always reduce two rationals to a 'common denominator', by multiplying them on the top and bottom by appropriate numbers.
Suppose this is 'd' and the rationals are a/d and b/d, where a, b and d are whole numbers. Then (a/d)^{n} + (b/d)^{n} = 1, or, multiplying on both sides by d^{n}, we get a^{n} + b^{n} = d^{n}, which is impossible for whole numbers a, b and d, and n>2 according to Fermat's Last Theorem. This is an absurdity, which has arisen by assuming we can have both coordinates rational. We conclude that we can't have both coordinates (the x and y) being rational.
Reductio ad Absurdum!
Martin Carrudus
