h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

Entry: A day with Fermat in Toulouse (1640) - A33637557
Author: NotaFBene - U11273605

Created: 19th March 2008
-------- A day with Fermat in Toulouse -- (or: Fermat and the cubic roots of unity) -----
-------- Just imagine . . . . picture Toulouse in southern France, 1640 . . .

Pierre de F. closes up at the end of a hectic day at the office, where he works as a magistrate at city hall. He gets a headache from those church prelates dominating everything, as if they had the truth in their pocket. After a sober but nourishing meal he retreats into his study and relaxes with his hobby: studying the old Greeks that were just recently translated into Latin, the international scientific language of the day. . . Especially the 'Arithmetica' of Diophantus is fascinating. He just discovered that Pythagoras' a^2+b^2=c^2 for integers, without common factor, occurs for primes c of form c=4n+1, starting with the well known 3^2+4^2=(4+1)^2, and 12^2+5^2=(3.4+1)^2.
Would it be possible to generalize this to higher powers? Could the sum of two cubes be a cube, etc.?

From his young friend Blaise Pascal in Paris he just got a letter, (say) about the multiplicative structure (using factorial n!) of the coefficients in the expansion of (a+b)^n arranged in a triangle, skipping the a's and b's (clever trick to dispose of cluttering detail) although the coefficients are generated purely by addition, like a two-dimensional FiBonacci array, summing two neighbours repeatedly. - In fact, he just found that factorial structure himself, and he answered Blaise: . . is'nt it marvelous that truth is the same in Toulouse as in Paris ?! Quite different from theology, where truth depended on the last priest or cardinal you talked to.

No, that would never happen to his dear Mathematics, brought forward as the new science by Galileo (from Italy, in his 7-th year of house arrest, and two years before his death, for criticising the geo-centric, if not ego-centric, worldview pushed by the church), and by Descartes (who fled to Holland to save his skin, and escape the sad lot of Geordano Bruno at the stake in 1600; Galileo's students prepared an escape for him to Holland, but he felt too weak and old for the trip). --- Authority had no place in Math: everybody with a clear and open mind, possibly helped by some calculation device, that he sometimes borrowed from Blaise who designed one (photo)* , could check it out and find new laws, and test for the truth himself.

Later, Leibniz in Germany, even made a multiplier from that device, which sold quite well. Of course it worked decimal, that beautiful code introduced from Arabia to Europe some 400 years ago by FiBonacci (the son of Bonacci). Leibniz did give a fleeting thought to binary code, which was just coming from China, where it was used for the philosophy of balance (yin/yang) and in a 3+3 bit code for the 8 x 8 = 64 hexagrams of the I-Ching (book of Changes) [ic-59] since 2000 BC. But the number wheels in the calculator would be too many for quick carry propagation which was the bottleneck for speed, as well as requiring too much of the crank turning the wheels.

So Pierre sharpened his feather to do some doodling by hand in his notebook, using p-ary code in order to find some structure, and possibly find solutions to [1] : a^p+b^p = c^p . . prime p>2 (composite powers he quickly found inessential).
From Pascals' triangle he saw that p divides each number in the p-th row, except the 1's at the two ends, proven easily by their newly discovered factorial coefficient nature. And slowly his Small Theorem (FST) took shape: a^p+b^p=(a+b)^p for the last p-ary digit, or as it is now known: n^p = n mod p for all n (using the modulo notation of Gauss who thus formulated arithmetic 'closure' for residues mod m, in 1801). This was exciting because the exponent distributes over addition, most unusual, and probably leading to inequality for integers, rather than equality. Could FST also be possible mod p^k for k>1, and some special a,b ? That would at least provide a solution to his root equation [1] mod p^k.

And indeed, after some more trials, at p=7 there was something interesting mod 7^2: 42+01=43 where a=42=a^7 and a+1=(a+1)^7, and another pair b, b+1 with similar properties: 24+01=25. Moreover b=1/a, and a+b= 66 = -1 (mod 7^2). So a^2+a+1=0, implying (a+1)^2=a and (a+1)^6=1 --> p-1 cycle mod p^2. He convinced himself that this holds for any prime p=6n+1, so 3 divides p-1, allowing a 3-cycle {a, a^2, a^3=1} in the p-1 cycle, to be called the 'core' of the cyclic group of roots of 1 mod p^k, for any k>=1 --- (now using the 'group' concepts of Abel and Galois, who factored Gauss' closure around 1830, into an uncoupled product resp. cascade- coupled product of cycles: solvable groups; the final chapter of what Nicolo Fontana, alias Tartaglia the stutterer from Brescia, started around 1540 when he found, just in time before a challenged competition, a formula for the roots of any cubic equation - subsequently stolen by Cardano and published in his "Ars Magna" in 1543, without reference to its source).

So this p-1 cycle would be his Medium Theorem (FMT) or 'Core theorem': n^p = n mod p^k (for odd prime p and any k>0, and p-1 special fixed-point core-residues coprime to p) as extension of FST to higher precisions k. It provides a solution to his k-digit Large Theorem (FLT/k): a^p+b^p= -1 mod p^k . . [2] for odd prime p, k>1 with {a,b} "in-core" where a^p=a, b^p=b and b=1/a, a^3=1 mod p^k, sothat a^2+a+1=0: the cubic root solution of his equation mod p^k (prime p=6n+1). Now the full FLT for positive integers that he was after, and which he suspected to yield inequality for all primes p>2, would follow if ... [2] were the only possible solution of FLT/k, apart from some scaling factor. Then the exponent distributes over a sum: a^p+b^p=(a+b)^p mod p^k, clearly impossible for integer p-th powers < p^{kp} since the 'mixed terms' with a^i.b^(p-i) in the expansion do not sum to zero (which they do mod p^k for the cubic root case).

In fact, notice the recursion (a+b)(a^n+b^n) = a^{n+1} + b^{n+1} + ab(a^{n-1} + b^{n-1}). Writing S(n) for a^n+b^n, and shifing n+1 --> n this becomes: S(n) = (a+b).S(n-1) - ab.S(n-2) . . [3]. Then it is not difficult to derive, given {a,b} in core, that: a+b = -1 --> ab=1, so actually a cubic root-pair is the only solution 'in-core' for k>2. In this case [3] becomes S(n)= -{S(n-1)+S(n-2)} which is like a negative FiBonacci sequence. Starting with -1, -1 it yields for increasing n the repeated sequence {-1,-1,+2}* with period 3 : precisely as expected for cubic roots!

But what about other possible solutions not in core ? . . Without a clue, and Pascal's Calculator (PC) too slow and not geared for p-ary code to do some more experiments, he quit. Putting his notebook away, he let the problem rest. . . If only he could have gone till p=59 (peanuts for our PC's) then he would have found a clue, and discovered a different kind of root which indeed is not 'in-core' with all three terms. With of course the next questions: if that would be the end of it - or maybe there would be still other types of FLT/k roots, and : would some EDS (Exponent Distributes over a Sum) type of argument hold for these new roots ?

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PS: For a published direct FLT proof (16 pgs) see:
http://pc2.iam.fmph.uniba.sk/amuc/_vol74n2.html (pp 169-184)
and intro's at http://home.iae.nl/users/benschop/nf-abstr.htm