Why 42 ?

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" >http://pc2.iam.fmph.uniba.sk/amuc/_vol74n2.html (p 169-184)


BTW: my name NotaFBene comes from the Latin "Nota Bene" (Notice Well), and my initials in real life are NB (Nico Benschop, The Netherlands). The middle F is from my middle name Frits, and also from one of my main interests: 'Function' - which I see roughly as a black box with inputs, outputs and internal states. Function composition then is: making a network of boxes coupled by arrows that represent the flow of information. I spent my working life of 32 years in (industrial) research mainly on function composition (Digital Network synthesis, VLSI Very Large Scale Integrated circuits) and function decomposition methods (algebraic and otherwise). So when 'NotaBene' was already taken, I made it into NotaFBene ;-)
For my cv, hobbies, and other stuff see:

http://home.iae.nl/users/benschop/mycv.htm" >http://home.iae.nl/users/benschop/mycv.htm />http://home.iae.nl/users/benschop/work.htm" >http://home.iae.nl/users/benschop/work.htm />http://home.iae.nl/users/benschop/play.htm" >http://home.iae.nl/users/benschop/play.htm />http://home.iae.nl/users/benschop/ferm.htm" >http://home.iae.nl/users/benschop/ferm.htm : "A day with Fermat in Toulouse (1640)"
http://home.iae.nl/users/benschop/carry.htm" >http://home.iae.nl/users/benschop/carry.htm : Revival of the lost carry, or: how to break the Hensel lift for FLT.
http://home.iae.nl/users/benschop/nf-abstr.htm" >http://home.iae.nl/users/benschop/nf-abstr.htm : On Fermat & the cubic roots of unity (mod p^k).
http://home.iae.nl/users/benschop/ng-abstr.htm" >http://home.iae.nl/users/benschop/ng-abstr.htm : On Goldbach & the lattice of idempotents (mod \prod first k primes).
http://home.iae.nl/users/benschop/sgrp-flt.htm" >http://home.iae.nl/users/benschop/sgrp-flt.htm : Using semigroups (= function composition) for a direct FLT proof.

---------------- One is always halfway anyway --------- 1 = AHA -----------

___________ Math = the Art of separating Necessity from Coïncidence.

___________ Life = making the Best of Necessity, using Coïncidence...

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