n n n
Over naturals , a + b = c ===> n < 3 | abc = 0 .
Or equivalently , a power can be split into 2 like powers only in a
trivial case or if the power is at most a square .
The Fermat-Wiles Theorem was formerly known as Fermat's Last Theorem , and is distinguished from Fermat's Theorem , formerly known as Fermat's Little Theorem .
The Fermat-Wiles Theorem was the long-sought goal of an enterprise which
turned into one of the most productive in the history of mathematics .
If Fermat had a proof of the Fermat-Wiles Theorem , it was lost .
In 1994 , Andrew Wiles proved it after seven years of brilliant work ,
constructing the last part of an incredibly elaborate edifice
incorporating centuries of work by other mathematicians .
Ironically , Fermat's Theorem is of far more day-to-day use in research
and computation in number theory than the Fermat-Wiles Theorem .
Wiles' proof culminates a search for answers to some very natural
questions , questions which often occur to very young children ,
including Wiles , who became interested when 10 years old .
If you have a line of 23 marbles , can you split them into 2 lines ?
Can you split a first power into 2 first powers ?
Of course .
If you have 23 coins , you can divide them into a pile of 12 coins and a
pile of 11 coins .
Such a split can easily be done , typically in many ways .
If you have a square of toy soldiers ,
can you split them into 2 squares ?
Only sometimes .
What about a cube into 2 cubes ?
A fourth power into 2 fourth powers ?
Wiles proved that no power higher than a square can be split into 2 like
powers .
Walter Nissen
2008-01-28