Is this Fermat’s Proof of the Last Theorem?
Dear Millenium Reader. Anyone with a passionate (or just passing) interest in discovering Fermat’s own proof to ‘Fermat’s Last Theorem’ may have been surprised that Steig Larsson included this subject in Part I Chapter 1 of Millennium II. What is more intriguing is that in Part IV Chapter 32, as Salander was approaching Zalachenko’s disued farm, and her possible death, she had a sudden incredible revelation of Fermat’s proof ? a revelation that appeared to make it simple and beautiful. (Was it on a par with Andrew Wiles’ 1994 revelation that helped him find a solution to the problem stopping him from proving FLT?). As a consequence I have looked through Millennium III to see if Salander had any recollection of this and found she didn’t. But for readers of the Millennium trilogy the following is what I believe is Fermat’s disarmingly simple proof.
Fermat’s Last Theorem
The questions that all mathematicians have asked is since his death is how did he start, and how did he prove it? Diophantus’Greek version of Arithmetica was published in 1621as the Diophanti Alexandrini Arithmeticorum Libri Sex et De Numeris Multanguls Liber Unus in both Greek and Latin by Claude-Gasper Bachet. In the early 1630’s Fermat became interested in the more subtle questions surrounding each of Diophantus’ questions, and around 1636/37 he thought that he could add something to the solution given to Question 8 of Bk II. This concerned the division of a given square into two squares, and it led him on to wonder if cubes and higher powers could be likewise divided. He played around with a few of the lower powers (as any modern mathematician would do) and found it was unlikely. Without any solution he eventually came to Q29 of Bk V which concerned the finding of three squares such that the sum of the squares is a square. Again looking to see what he could add, Fermat wrote in the margin that he could see a way of proving that a square could not be divided into two fourth powers, That is, in modern algebraic terminolgy x^4 y^4 ? z^2. His proof was never discovered, but an internet blogspot by Larry Freeman gives a conjectural reconstruction of it.
Fermat found that x^4 y^4 ? z^2 can be rewritten as x^4 y^4 ? z^4 because any z^4 can be written as z^2. Not only that, if x^4 y^4 ? z^4 then there are no solutions to any multiple (k) of the power 4. For instance, (x^4)^k (y^4)^k ? (z^4)^k is the convention for expressing a power raised to a power. Therefore, the multiples of power 4 (4k) that had no solutions are 4.1=4, 4.2=8, 4.3=12, 4.4=16, 4.5=20 …and so on. So how did Fermat go from here to proving his thoerem for all n > 2?
Although he may have searched for proofs to n =3, 5, 6, 7, etc, he simply went back to Q8 and asked himself the simple question: Why was it that squares be divided into two squares but not higher powers? He found that for any Pythagorean triple
z^2 – y^2 – x^2 always gave a zero difference. Applying this to higher powers he found that the nearest to a non-zero difference could only obtained with cubic triples (7,6,5), (9,8,6), (12,10,9), and the triple (3,2,1) for the fourth and higher powers. Therefore, there were no solutions for powers greater than squares. By applying the logic he used for power 4 he began to list all the multiples (nk) without solutions,
n (nk) multiples
3 3.1=3, 3.2=6, 3.3=9, 3.4=12, 3.5=15…
4 4.1=4, 4.2=8, 4.3=12, 4.4=16, 4.5=20 …
5 5.1=5, 5.2=10, 5.3=15, 5.4=20, 5.5=25…
6 6.1=6, 6.2=12, 6.3=18, 6.4=24, 6.5=30…etc..
But one thing stood out immediately. The fourth multiple of n = 3, 5, 6, etc is always numerically equal to a multiple of n = 4! Therefore, if the multiples of n = 4 had no solutions then the fourth multiples of any other power had no solutions. Using the logic of mathematical proof introduced by the ancient Greeks Fermat realized that he could now vigorously prove that all n > 2 had no solutions (indicated by sign ?);
Let A = (x^4)^1 (y^4)^1 ? (z^4)^1
Let B = (x^4)^k>2 (y^4)^k>2 ? (z^4)^k>2
Let C = (x^k>2)^4 (y^k>2)^4 ? (z^k>2)^4
Let D = (x^k>2)^1 (y^k>2)^1 ? (z^k>2)^1
As A is proved to have no solutions,
Then A implies B when k > 2, and B has no solutions.
As B is equal to C, then C has no solutions.
As D implies C, then D has no solutions.
Proof that no power greater than squares can be divided into two powers can be demonstrated by inserting any k in the equations. For instance let k > 2 = 3,5,6,7…etc. Fermat was obviously very pleased with this simple proof and he annotated the long-hand theorem in the margin of Q8 Bk II. The ‘marvellous’ aspect about it was that he didn’t have to prove the infinitude of powers individually.
With reference to Bachet’s question about the area of Pythagorean triangles not having square areas, this is conjecturally reconstructed in Larry Freeman’s blogspot on ‘Fermat’s One Proof’ and shows that P^4 – Q^4 ? z^2 can be expressed as x^4 y^4 ? z^4. This was achieved sometime in the mid to late 1650’s, when he began corresponding with Christiaan Huygens. Not only did he describe in general terms his proof of this, but also indicated that he had proved the case for n = 3. This proof has never been found, and I believe that it was just an intellectual exercise to show that he could. In these and all his proofs, it seems that the way he used his method of infinite descent depended on whether questions were ‘affirmative’ or ‘negative’.
Much has been made of Fermat’s reluctance to provide any solutions when he corresponded with other mathematicians. We must remember that in the period of Fermat’s mathematical education the Francois Viete algebraic notation (introduced between 1584 and 1589) was used in France, and he continued using it throughout his life. However, in 1637 Rene Descartes published a work which severed the dependence of all mathematics on geometry and became known as analytical geometry. This eventually superseded the Viete notation. It isn’t surprising that when Fermat did correspond with mathematicians from several European countries, his questions were posed in Latin, the language of the educated classes. Not only did he not want to give away the secrets of his method of infinite descent, but his solutions existed in the out-of-date Viete notation, and nobody would have understood them. This obviously made things difficult for his son Clement-Samuel when he came to republish the Arithmeticorum with his fathers notes and theorems.
So Millenium Reader, if you are interested, there several outstanding mysteries:
– After his death, what became of Fermat’s copy of the Arithmeticorum and the prolific amount of notes and jottings made during his life?
– Knowing that Fermat proved FLT for only one power, n = 4, why didn’t all