Homer's Last Theorem

This project hopes to find Homer's Last Theorem, based on the subtle joke from the Simpson's. Brought to light by Simon Singh in his book: The Simpsons and Their Mathematical Secrets.
This particular secret involves Fermat's Last Theorem. A mathematical problem that has plagued mathematicians for over 300 years.

Fermat's Last Theorem states:
There should be no integers such that: aⁿ + bⁿ = cⁿ
for n > 2

But Homer has appeared to have found a solution:

And putting it in on the calculator seems to work! How is it that Homer Simpson, or more accurately the writer, David Cohen seems to have found a solution to the problem that took mathematicians centuries to solve?
The trick lies on the accuracy of the numbers used. The true solution is:

3,987¹² + 4,365¹² = 4,472.0000000070576171875¹²

Given that 4,472.0000000070576171875 is not an integer, this doesn't fit Fermat's Theorem. But why then does a calculator give an even 4,472 when giving the answer? The trick lies in the accuracy of the numbers used! Given that the usual scientific calculators only have a certain degree of accuracy. The 0.0000000070576171875 is lost due to the hardware not being able to store to that level of accuracy. Therefore the number appears to be 4,472 when in reality it is 0.000000002% larger!
This project aims to find more solutions like this that appear to find a solution to Fermat's Last Theorem to the unaware.
A useful trick to have to impress your friends!