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

Entry: A day with Fermat in Toulouse [1640] - A33894002
Author: NotaFBene - U11273605

------- A day with Fermat in Toulouse -------
---- Or : On Fermat and the cubic roots of unity (mod p^k)
---- 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 (see next) 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.

----------(begin NB)----------

---- The Pascal triangle:
(a+b) = 1.a + 1.b
(a+b)^2 = a^2 + 2ba + b^2
(a+b)^3 = a^3 + 3a^2.b + 3a.b^2 + b^3
(a+b)^4 = a^4 + 4a^3.b + 6a^2.b^2 + 4a.b^3 + b^4 (etc)

The pattern of coefficients can be written more compactly as:
. . . .1 1
. . . 1 2 1
. . .1 3 3 1
. . 1 4 6 4 1 (etc: the sum of two neighbours yields a next-row entry)

---- Residue arithmetic, and the carry:
Number notation in base p notation uses n = c.p + r, with carry 'c' and unique residue r from 0 to p-1. For residue arithmetic reset carry c=0.

---- Fermat's Small Theorem (FST): take any odd prime p (having no proper divisor, like 3; 5, 7; 11, 13; 17, 19; 23; 29, 31; etc.
Notice all p >3 are in set 6m +/- 1, and p >5 in 2.3.5.m + {1,7,11,13,17,19,23,29} = 30m +/- {1,7,11,13} ) and do residue arithmetic mod p.

Fermat considered p-th powers g^p mod p, trying to find a generator g < p such that all its powers g = g^1, g^2, g^3, ..., g^(p-1) would be different. For instance mod p=7, then g=2 yields g^2= 4 and g^3= 8 = 1 mod 7 will not do since only half of the residues r < 7 are generated. But 3 does work : 3, 3^2 = 9 (base 10) = 1.7 + 2 = 2 mod 7, 3^3 = 3.3^2 = 3.2 = 6 = -1 mod 7. We can stop at -1 mod p , because the remaining h=(p-1)/2 powers mod p are the complement -g^i of those g^i (i=1..h) already found. Now 3 is called a primitive root of unity 1 (mod 7), since it generates all p-1 non-zero residues mod p. In short the generated p-1 cycle is: 3, 2, -1, -3, -2, 1 (mod 7).

Now FST states that for every prime there is such primitive root g, generating a full p-1 cycle. It suffices (usually) to look for a primitive root among the divisors of p-1 (not proven), for instance p=191 has p-1 =2.5.19 of which 19 is a primitive root (but not 2 or 5), its powers mod 191 generate a cycle of length 190.

Notice that in any such full cycle we have g^(p-1) = 1 mod p, the end of the cycle before it repeats itself, and g^h = -1 halfway. In fact it is easy to see that for any non-negative residue r < p holds r^(p-1) = 1 mod p, or : r^p = r (mod p).

---- Extension to higher precisions mod p^k (k digit base p arithmetic) also holds :
For odd prime p and any precision k > 1 there is a primitive root g < p that generates a cycle of length (p-1).p^{k-1} of residues mod p^k, denoted extension FST_k. The reason is that the powers of p+1 generate mod p^k a residue cycle of length p^{k-1}. For instance the cycle mod 7^2 is 6.7 = 42 long (D.Adam's magic number), generated for instance by 03, given in 7 columns of 6 residues with two 7-ary digits each:

+03 +43 +13 +53 -44 -04 -34
+12 +62 +42 +22 -65 -15 -35
+36 +46 +56 +66 -61 -51 -41
+44 +04 +34 -03 -43 -13 -53
+65 +15 +35 -12 -62 -42 -22
+61 +51 +41 -36 -46 -56 -66 = +01

Notice (mod 7^2) : -56 = 11 = p+1 = [3^{p-1}]^{p-1} , and (11)^7 = 01 : a length p cycle.
This cycle of 42 residues sums to 0, as do all subcycles (dividing 42),
like 7-cycle {61,51,41,31,21,11,01} = (11)^* and 6-cycle {43,42,-1,-43,-42,01}.
So the cubic roots a^3=1 (3-cycle) have sum=0: a +a^2 +1=0, here: 42 + 24 + 01 = 00 (mod 7^2),
. . . and they are p-th powers : 42 = 3^14 = (3^2)^7, 24 = 3^28 = (3^4)^7 (mod 7^2), 01 = 3^42 = (3^6)^7, satisfying FLT mod 7^2.
----------(end NB)----------

And indeed, after some more trials, at p=7 there was something interesting mod 7^2 (using base 7 number code): 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. This p-1 cycle in arithmetic mod p^k is called the 'core' of the full (p-1).p^{k-1} cycle that produces unity 1. It provides a solution to his modulo k-digits 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, which is called :
----> the cubic root solution of his equation mod p^k (prime p=6m+1).
Notice prime p = 1 (mod 6) for a cubic root of 1 to exist, since 3 must divide p-1, and p is an odd prime, so even p-1 has already a factor 2.

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 ... the cubic-roots [2] were the only possible type of 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 ?

----------
NB: For intro's see on author's homepage (N.F.Benschop = NotaFBene) :
. . . http://home.iae.nl/users/benschop/nf-abstr.htm
and online the full published direct FLT proof (Nov.2005) :
. . . http://pc2.iam.fmph.uniba.sk/amuc/_vol74n2.html (pp 169-184)

Hello NotAF,

Thanks for posting your Entry to Peer Review - as it was posted more than once before (and, as a result, these were subsequently removed) the discussion is spread out a bit, so we're going to post in those threads to tell those who gave feedback that the discussion is occurring in this thread.

A first point: It isn't necessary to put the full text of the Entry in your posts, as we can already see it in the Entry itself. In Peer Review we can look at smaller sections of the text.

I think the main point from previous discussions was that in order to fit in with the Edited Guide, this would have to be rewritten in order to be accessible to a lay audience. Anyway, the feedback in these two Peer Review threads still applies, so if you can address those that would be a great start:

F9700211?thread=5256836
F9686445?thread=5250339

(We'll drop a note in the other discussions to direct those giving feedback to this thread).

Just a note for anyone new to this discussion: although this exists on the web, it is the work of the original author.

Anyway - all the best with this ,

h2g2 Editors

Popular science and math for a lay audience (and that is where the major part of the publication market is, after all) works best if the author presupposes no specialized knowledge on the part of the audience. Humanizing the the proponents ina historical treat ment also enhances lay audience appeal.

Popular science and math for a lay audience (and that is where the major part of the publication market is, after all) works best if the author presupposes no specialized knowledge on the part of the audience. Humanizing the the proponents in a historical treatment also enhances lay audience appeal.

Indeed, I tried my best on reducing presupposed special knowledge by adding the required arithmetic, without which a sensible explanation of FLT is impossible, I'm afraid: _why_ would anyone who does not know simple arithmetic math be at all interested in Fermat in the first place?

Personalizing the actor(s) at that historical time was my first concern in this text, which was born out of frustration regarding the lackluster attitude to my approach for a direct FLT proof of 15 pgs - eventually after 10 years published by Acta Mathematica at U-Bratislava in Nov.2005 (vs. 150+20 pgs by Wiles/Taylor in 1995). Showing what Fermat could have known re a solution I found quite exciting. Namely his just discovered Small Thm, and its extension to k-digit p-ary code, with the cubic roots of 1 mod p^k as residue solution, easily found by hand, occurring first at mod 7^2, which can't be extended to integers because the exponent distributes over addition (a+b)^p = a+b = a^p + b^p (mod p^k) due to FST_k.

Because they want to learn.

Because it helps them to identify with figures of the story on an emotional and emotive basis.

Precisely. That is also why a direct short proof of FLT is so appealing (versus the extremely difficult to follow indirect proof of Wlles, who actually proved quite something else: the Tanyama-Shimura conjectture on the equivalence of elliptic curves and modular forms). FLT shows the impossibility of extending Pythagoras' statement on quadratics (relating integer arithmetic to geometry: the right angle) to higher powers.

Most people get Pythagoras at school, and thus may be interested in FLT, _if_ there is a direct approach that they somehow can follow, and what I suggest Fermat could have found (at least the first half of the proof: the cubic roots of 1 mod p^k for prime p=1 mod 6, via FST which he just discovered - related to the Pascal triangle which he and Pascal were working on at that time!) -- NFB

Its usually safe to assume a good working knowledge of basic arithmetic on the part of a lay audience, but assuming a ready facility even with linear algebra is stretching it a little.

I gave up thinking in Machist symbolic logic strings back in my Junior year in high school. At first that seemed like the fast track to rapid development of new insights. Then I found that if I'd hit on some novel insight I wanted to communicate to others, it would take longer to translate from the symbolic logic into the plain language than it would to work up the argument in the clear in the first place.

Dear ITIWBS,
I don't know what 'Machist symbolic logic' means. But if your point is that I use an unorthodox method in my arithmetic which few people can follow: the contrary is true. I use a straightforward residue-and-carry method (the standard way of representing numbers base p) that connects directly with Fermat's Small Thm which he discovered at about the same time. While the opposite holds for Wiles' method of Dirichlet L-spectra and much more esoteric stuff. Frankly, this first half of my FLT proof (using the cubic roots of 1 mod p^k) is at the undergraduate level, or even lower.
Including the worked examples, I don't see how I can make it more simple, resp. more interesting qua characters and historic details involved. I do NOT claim to be able to reach *everyone*, just those who are interested in basic arithmetic maths, and some of its history. -- NFB

A basic format on Machist* symbolic logic:

{[...A...] + [...B...] yields [...C...] + [...A'...] yields ... ... nth }

Its presupposed that the reader will do the logical reasoning with respect to the internal intrinsic logic of each of the boxes and their concatentions.

For example, diagramming your intro for A333894002 - A Day With Fermat in Toulouse [1640]

(Title) { subset [A(t)] of [B(t')]}

(line 1) {[A(t)][B]}

Or:

(line 2) { [A] + [...{unity(mod p^k)} exp -3...]}

(line 3) { imagine + picture [B][C(t')]}

(...for the duration of this treatment...)

A is always Fermat,
(t) is a day,
B is always Toulouse,
(t') is a date: [1640],
[...{unity(mod p^k)} exp -3...] is the cubic roots of unity (mod p^k)


*(After Ernst Mach who made extensive use of this in his pioneering articles on physics, cosmology and psychological biophysics. Helps if one reads some of his material, but he is infrequently anthologized precisely on account of the difficulty his symbolic logic has for most people.)

...more later, i need to log out...

It seems to me there's no advantage in this Mach logic. It is just a 1-1 translation into symbols. Associative string logic (semigroups) is bracket-free algebra. But this example, being natural language, is not associative, so placing brackets does count. Is _that_ the point of this Mach logic: bringing (a hierarchical) structure into a sequence?

Yep, useful for outlining and organizing, for example, personalities, ideas, interactions. Its in the expansion into low level language that sense and emotion emerge with increased detail and depth. Each of the interactions described in your article provides a basis for highly interesting story in its own right.

Correction: Post 12, "low level language" should read "high level language".

Maybe you were correct after all: developing "high level language" into a network of "low level language items"

I won't even pretend to have a grasp of anything discussed herein, but do the other reviewers who 'do' think there is potential here for an EG Entry, or should this be moved on to the Flea Market or back to Entry in time?

Is it time now?

It would appear that you've taken great pains to illuminate a bit of the history of mathematical discovery during an age ripe with insights.

It occurs to me that the story of the work is far more interesting, useful, and worthwhile than trying to create a text entry of the logic and math where the font and structure of the site won't support the finer intracies of your entry.

It also appears, at one point, that you use 7^2=42, which would be wrong if interpreted as 'seven to the second power' as many notational forms would have it. 7^2 would, of course, equal 49.

I'm beginning to think that you've selected the wrong site on which to post this. It appears, from your home personal space, that this is the only entry which you have created and then submitted.

As much as I appreciate new researchers, if this is the only reason you've stopped by, it seems to be the wrong stop. If, however, this was just to be the first of many, I want to welcome you!

Either way, I'd suggest simplifying, simplifying, and simplifying some more. Or, take it in sections and amplify, amplify, amplify, so that a 10-year-old can understand it (we have some, you know).

There's no gain to anyone to post an entry that has been removed, believing somehow that we have a 'right' to have our stuff in the E.G. This is not a democraticly run website, and we agree when we join that the powers-that-be have the right to determine what stays and what goes.

I'm looking forward to more entries from you!

It occurred to me that an example may help to illustrate my point.

I hate to have to resort to using one of my own entries, for I'm neither the most learned or prolific researcher here, but I wrote one on "The Amazing 42-Minute Gravity Sled" A2960633
http://www.bbc.co.uk/dna/h2g2/alabaster/A2960633

If you read it, you will no doubt notice that I've completely covered the facts and history, and given the reader everything actually useable on the subject (hopefully in an engaging and entertaining style), but resisted pressures to add the actual math. Once an entry has gone through our somewhat rigorous Peer Review, the reader can pretty much assum that the math works without having to get out pencil and paper.

Gee, on your entry, my would have run out of ink!

author gone since Apr 22, 2008 Scout proposes move to fleamarket

Could just be a semester of school, or a reallly long holiday.
Or, the researcher could just be out researching (or lurking); but, it wouldn't hurt to move it out of PR, again.