Fermat's Last Theorem (42 page)

Then add
a
²
– 2
ab
to both sides:

This can be simplified to

Finally, divide both sides by
a
²
–
ab
, and we get

The original statement appears to be, and is, completely harmless, but somewhere in the step-by-step manipulation of the equation there was a subtle but disastrous error which leads to the contradiction in the final statement.

In fact, the fatal mistake appears in the last step in which both sides are divided by
a
²
–
ab.
We know from the original statement that
a
=
b
, and therefore dividing by
a
²
–
ab
is equivalent to dividing by zero.

Dividing anything by zero is a risky step because zero will go into any finite quantity an infinite number of times. By creating infinity on both sides we have effectively torn apart both halves of the equation and allowed a contradiction to creep into the argument.

This subtle error is typical of the sort of blunder which caught out many of the entrants for the Wolfskehl Prize.

The following axioms are all that are required as the foundation for the elaborate structure of arithmetic:

1. For any numbers
m, n

2. For any numbers
m, n, k
,

3. For any numbers
m, n, k

4. There is a number 0 which has the property that, for any number
n
,

5. There is a number 1 which has the property that, for any number
n
,

6. For every number
n
, there is another number
k
such that

7. For any numbers
m, n, k
,

From these axioms other rules can be proved. For example, by rigorously applying the axioms and assuming nothing else, we can rigorously prove the apparently obvious rule that

To begin with we state that

Then by Axiom 6, let
l
be a number such that,
k
+
l
= 0, so