Fermat’s last theorem, proved by Andrew Wiles, states that there are no positive whole number solutions for
to the equality
for 
Yet in an episode of The Simpson’s it was noted that
, apparently contradicting Fermat’s last theorem.

Of course this is not a contradiction because
is not actually equal to
.
So why did The Simpson’s episode say these two numbers were equal?
Well,

while

So, starting from the left we see that the
digit in
is a 9, while for
it is an 8.
David Radcliffe (@daveinstpaul ) tweeted:
“What is the best way to measure the “closeness” of a near-solution to a^p+b^p=c^p? E.g. how “close” is 3987^12 + 4365^12 = 4472^12 really?”
x
The left most digits of these large numbers represent very large quantities.
x
Maybe one way to compare these two numbers is to look at their ratio

.
x
If we have a calculator that’s accurate to 10 decimal places we get the result
x
x
So to 10 decimal places the ratio

is 1.
x
It’s not until we get a more accurate calculator that we see, in fact,
x
x
Another way to think about the relationship between

and

is to take the twelfth root of

.
x
If we really had equality between

and

then

would be

.
x
If we have a calculator that’s accurate to 12 digits, we get

x
It’s only with a more accurate calculator that we see

x
This raises a sort of interesting question:
x
Given a “small” potential error

and an integer

, what solutions are there to
x
x
for positive integers

?