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

Entry: Why 42 ? - A33637557
Author: NotaFBene - U11273605

Created: 19th March 2008
Why 42 ?


When Douglas Adams was asked why he took the number 42 as the answer to all questions about the Universe, he replied that it just came to his mind. You might think this was 'at random'. But, as you know, nothing happens at random. In this case the reason for 42 is a mathematical one, namely the clue to Fermat's Last Theorem (FLT) which Fermat posed around 1640 in the margin of his book 'Aritmetica' (by Diophantes) when trying to generalize Pythagoras' equation a^2 + b^2 = c^2. We learn at school there are some special solutions for integers, the smallest one being 3^2 + 4^2 = 5^2 (re the sides 3,4,5 of a rectangular triangle).

His proposed generalization a^n + b^n = c^n for integer exponents n>2 has no solution (he claimed to have a marvelous proof of this, which however was too large for the margin This kept many mathematicians busy till Andrew Wiles published in 1995 a very long and indirect proof (some 150 pgs, using the result of another paper of 20 pgs by his student dr. Taylor). Clearly not the kind of proof Fermat had in mind...

Pythagoras' eqn is the essence of right angles, and of the 3 dimensional (3D) space we live in, and no extension of this is possible by FLT (at least not for integer space For FLT it suffices to consider odd prime exponents p>2. Now the clue of 42 in this story, namely: there _are_ solutions to FLT in residues mod p^k (odd prime p, and any precision k>0; for k=1 Fermat's Small Thm holds FST: a^p = a mod p for any prime p and all 'a' coprime to p, which he just discovered around 1640). The solutions for k=2 are crucial (can be extended to any k>2). In fact for any prime of form p=6m+1 (that is: p=1 mod 6), to solve a^p + b^p = c^p take for a, b, c the three cubic roots of 1 mod p^k, which sum to 0 (mod p^k). Then a^p + (a^2)^p = -1 (mod p^k). Moreover holds: a^p + b^p = (a+b)^p = a + b, since a^p=a, b^p=b and c^p=c for any such cubic-root solution in residues (re an extension of FST to mod p^k for k>1). So exponent p distributes over a sum (EDS) for such solutions, hence no extension to integers is possible, since the EDS property does not hold for integers.

The smallest case is p=7, with the crucial solution a=24, b=42, c=66 (in 7-ary code), all three in the length (p-1).p = 42 (decimal) cycle of units (mod 7^2). There you are: 42 occurs as solution to the residue generalization of Pythagoras which is the clue to our 3D space, both as cycle-length (decimal) and as cubic root of unity (base 7). If that's not a clue to the Universe, then I don't know

If you want to read more about it, see for a direct FLT proof (16 pgs):
"Additive structure of the group of units mod p^k, with core and carry concepts for extension to integers",
online at http://pc2.iam.fmph.uniba.sk/amuc/_vol74n2.html (p 169-184)

Traveller in Time on top
"Hi there and < <./>Welcome</.> > to HooToo


Think it is the wrong question 'Why 42 ?' does not give the answer 42. "

Traveller in Time previewing the forum
"Any time now an ACE will give you a streaming Welcome on your Personal Space

This will probably also give you an idea what we are looknig for on this Review Forum "

You're probably going to be summarily dismissed as a crank for posting something into Peer Review that most ordinary people will see as arcane gobbledegook. I'm aware of the mathematics you mention - I think it's covered in de Sautoy's "Music of the Primes" - but you will have a job on your hands writing it in terms that the layman can understand.

This would be an interesting thing to discuss over at the h2g2 Maths Lab (A895205) though, so I suggest you start the conversation there.

It's not currently an entry which meets our Writing-Guidelines, so I suggest you remove it from Peer Review and work on making it something far more accessible and clear. Take a look at other Maths-related entries in the Edited Guide to give you an idea of what you should be aiming for: C63

Icy

Wow, that was a cold shower from Icy North (true to your name )
Just watch those Northpole glaciers dripping away, Icy!
I have often been called a crank, so my elephant skin is used to that by now: publishing the FLT stuff in an Acta Mathematica is my revenge.
This place is bigger than I thought - I was already wondering what Brunel in the URL had to do with BBC/h2g2, but I've an inkling now...

I would love to remove my 'Why 42?' entry from Peer Review, but how to do that?
BTW, I applied to the math department. Am curious what comes of it...

Well, I didn't mean to offend, so apologies for that. I hope the advice I've given is useful - I've written a few of those entries in the Maths category myself, so I have some idea what it will take to get one into the Edited Guide.

People at the Maths Lab will help out - there are some doctorate-level mathematicians around here from time to time - don't expact an instant response, though - they'll turn up eventually.

To remove the entry, click on PeerReview then scroll down to the list of entries, and click on the link next to yours to remove it.

Oh, and welcome to h2g2! They're not all as icy as me around here - some of them can be quite friendly on occasion (grumble).

Icy

Hi Icy, I just removed my entry from the Peer Review list.

The reason for me to try h2g2 (which I found by chance via a Google listing of someone visiting a homepage of mine with a counter) is to get some response to the direct FLT proof I published in the Bratislava Acta Mathematica (nov 2005). It seems to be ignored despite frequent discussions, for instance on the sci.math newsgroup, about what kind of approach Fermat might have had to prove his claim (clearly not Wiles' way).

Since he did not elaborate later in his correspondences, that idea he must have found insufficient for a complete proof. If it was what I think: start expanding FST to FLT mod p^k (k-digit arithmetic base p) and notice the cubic roots of 1 (e.g. mod 7^2 already: easy to chack by hand calculations) - he had no proof that these solutions for p=1 mod 6 (since 3 must divide order p-1 of the cyclic group of units, which he discovered with his FST) are the only ones, which in fact they are NOT: there are special cubic-root extension types of solutions that start with p=59, which you can hardly find otherwise that by computer search.

There seems to be no interest in my residue-and-carry approach which starts with an extension of FST (his Small Thm that he discovered about the same time as he made his 'marginal note') using base p number notation (prime exponent p in his FLT claim), followed by induction on the precision k. Notice that the Hensel lift can be broken due to the special 'triplet' structure of the solutions of FLT mod p^k, for which the exponent distributes over a sum.

I did write a story about this suggestion, more for interested laymen,
-- A day with Fermat in Toulouse --
see http://home.iae.nl/users/benschop/ferm.htm
Would that be appropriate for a Guide Entry text?

Interesting, and you should ask others' opinions too. My view is that it is still too advanced. I'd start from the basis of explaining your concept without any equations whatsoever (like Hawking's "Brief History of Time", for example) and only then add those equations which are absolutely essential. Remember that many people will immediately switch off when they see a mathematical symbol. In two of my entries I ask people to skip sections if they don't want to see the maths.

As to material which has already published on other sites, you may have a problem unless you rewrite it (which I believe you'd have to do anyway with this one). The BBC won't publish anything where the copyright is held elsewhere. More info at <./>Writing-Copyright</.>

I'll add a link to the conversation you started in the Maths Lab, to point others towards this.

Icy

This is very good, in fact I'm gobsmacked. I managed to forget all about number theory and was feeling very well and at peace with myself when it comes and smacks you in the face and says I'm still here
This is by far the best reason for 42, beats my observation by a mile and should be published.

Hi Icy; thanks for your useful comments.
Regarding people who switch off at the first equation: this story is clearly not for them (they *should* switch off and not waste their precious time...) It is meant for *interested* laypersons, meaning that they are interested in basic math and its development/history. If necessary, I'll include a paragraph on residue arithmetic (like watching the time on a clock mod 12 , and on the three cubic roots of unity as three points in a circle at 0, 120, 240 degrees of the unit circle [mod 7^2] of length (p-1)p = 42).

PS: see the next enthusiatic reply (from Alex the Gray) to my story "A day with Fermat in Toulouse (1640)" , which also gives plenty historical pointers of how it was in those times (I guess...) -- NFB (also Gray: 65+ )