Conference rusure::math

    The problem with this approach is that the size of N for which it
    can be proved that no solutions can exist is prohibitively large.
    From Beiler's "Recreations in the Theory of Numbers", p. 288:
    
    	"...in 1941 it was shown that if x^p+y^p+z^p = 0 and no one
    of x, y, or z is a multiple of p, the p must be not less than
    253747889."
    
    Since the multiplicity condition in this statement is fairly easy
    to prove, we are already at the limits of the storage capacity of
    our largest computers just to store the coefficients of the polynomial
    which is to be factored.

Andrew Granville and Michael Monagan (Univ. of Saskatchewan) recently
extended the exponent in the previous note to 714,591,416,091,389. They 
used MAPLE extensively in the proof, including factoring some integers of 
153 digits, and evaluating the determinants of matrices of order 43 whose 
elements are univariate polynomials of order up to 50. Not the sort of 
thing you do on the back of an envelope!

> 153 digits, and evaluating the determinants of matrices of order 43 whose 
> elements are univariate polynomials of order up to 50. Not the sort of 
> thing you do on the back of an envelope!

    ... or the margin of a book

From: [email protected] (Andrew P. Mullhaupt)
Newsgroups: sci.math
Subject: Re: number theory proofs?
Organization: Morgan Stanley & Co. NY, NY
 
In article <[email protected]>, [email protected] (H. de Bruijn) writes:
> When I was young, NOBODY could present a proof of Fermat's Last Theorem:
>  |
>  | There exist NO non-zero integers  x, y, z  such that:
>  |
>  |   x^3 + y^3 = z^3
>  |
> 
> Or am I not well informed anymore ?!?
Logically speaking, this depends on when you were young. You had to either
have been young a very long time ago, (Ahh, I'm not exactly sure but I think
certainly you would be a contemporary of Euler - in which case - you have
my sincere congratulations!), or the latter.
 
Later,
Andrew Mullhaupt

>Andrew Granville and Michael Monagan (Univ. of Saskatchewan) recently
>extended the exponent in the previous note to 714,591,416,091,389. 

... and this figure has recently been extended to 7.56E17, by Coppersmith
["Fermat's Last Theorem (Case 1) and the Wieferich Criterion", Mathematics 
of Computation Vol 54 pp.743-750 (1989)]. Reported in the MAPLE Technical 
Newsletter, Fall 1991.

Wow!  Math moves by leaps and bounds.  The 1991 result by Buhler, Crandall,
and Sompolski was (only) that FLT is true for n <= 10^6.  Now it's 7.56�10^17.

Using Inkeri's 1953 result: if (x,y,z) is a primitive counterexample to FLT
with exponent p (p prime) then x > [(2p�+p)/log10(3p)]^p.  When combined with
the result n <= 7.56�10^17, this means x must have at least 3.982�10^19 digits!


Also, in 1983 Faltings proved that for every n >= 3, fermat's equation has only
finitely many primitive solutions (unlike the generators of Pythagorean triples).

        re .7,
        
        From .1 and "(Case 1)" in .5, it appears the the bound
        given for p is only for the case were p does not divide
        xyz, i.e., p does not divide any of x, y, or z.
        
        Dan

re .7
    
>Also, in 1983 Faltings proved that for every n >= 3, fermat's equation has only
>finitely many primitive solutions (unlike the generators of Pythagorean
>triples).
    
    I happened to be reading about this the other day. Here are some more
    details that I couldn't find already mentioned in this conference.
    
    Suppose a^n + b^n = c^n then we can write this as x^n + y^n = 1 where
    x = a/c and y = b/c.
    
    Since for n=2 it is well known that there are an infinite number of
    primitive Pythagorean triples, the unit circle passes through
    infinitely many rational points.
    
    Mordell's Conjecture was that:
    
    No curve whose genus exceeds one can pass through more than a finite
    number of rational points.
    
    This was proved by the (then West) German mathematician Gerd Faltings
    for which he won a Fields Medal.
    
    Hence the result in .7 that for any given n there are no more than a
    finite number of primitive counter-examples to FLT. To my knowledge
    there is no known *upper* bound on "n" for counter-examples so
    therefore it is an open question as to whether the total number of
    primitive counter-examples to FLT is finite or infinite.

    (Implicit in the above is that the "supercircles" for n>2 are of genus
     greater than one. EFTR.)
    
    P.S. I am close to completing a proof of FLT but am temporarily out of
    1" by 11" paper. If nothing else I should be able to make a fairly
    large M�bius strip. :-)