In a recent prestige edition of Time magazine about "Genius", Andrew Wiles appeared in an article that included a floating quotation that purported to state the gist of Fermat's "Last" Theorem, as follows:
The equation x^n + y^n = z^n is not solvable for integers n greater than 2.
But if this is the correct statement, I can find nine different examples that disprove it, and that's just for the case when n = 4:
0^4 + (-1)^4 = (-1)^4
0^4 + (-1)^4 = 1^4
0^4 + 0^4 = 0^4
0^4 + 1^4 = (-1)^4
0^4 + 1^4 = 1^4
(-1)^4 + 0^4 = (-1)^4
(-1)^4 + 0^4 = 1^4
1^4 + 0^4 = (-1)^4
1^4 + 0^4 = 1^4
Granted, these are uninteresting, obvious, even "trivial" counterexamples,
but the unfortunate wording of the statement means that they can be regarded
as correct solutions, because it was never stated that x, y
and z had to be (strictly) positive integers.
In fact, it was never even stated that x, y and z had to be integers at all; in that case, for any pair of nonnegative real numbers x and y, and any positive real number n, if we define the real number z by:
z = ( a^n + b^n ) ^ (1/n)
then we get a value that can be used to generate yet another counter-example:
a^n + b^n = z^n
So it should be clear that a proper statement of Fermat's conjecture would be something along the lines of:
When the integer exponent n is greater than 2, there is no solution of the equation x^n + y^n = z^n for which x, y and z are strictly positive integers.
No doubt the illustrator would persist in preferring the original statement. No doubt the readers of Time's "Genius" issue wouldn't care. But let's be clear that mathematicians care; in the same way that "Loose lips sink ships" we can state that "Mathematical goofs destroy proofs".