A few years ago I read about an equation which Homer Simpson pondered. It was a Fermat near miss (FNM), an equation which almost satisfies Fermat's Last theorem. Here is an example I found a few days ago. It's a cubic equation which misses satisfying Fermat's Last Theorem by a very small error.
491933 + 509203 = 630863
The sum of the 2 cubes on the left is 251072400480057. The cube root of the sum = 63085.9999999994. The error is 0.0000000006
In more precise terms, an FNM is a pair of integers raised to an exponent n, and their sum is almost exactly the nth root of an integer. I wrote a perl script to find these Fermat misses. Some of them are quite interesting. A trivial miss is where a>>b so b^n ~ 0.00 and a = c. A prime miss is where all 4 numbers a,b,c, and n are prime. David X Cohen worked hard to find a FNM which does not violate parity.
My perl script uses double precision floating point numbers so the relative error is limited to 1e-16. Noam Elkies at Harvard has apparently written a more precise program because he finds FNM which have errors far below 1e-16.
While this is simply a mathematical curiosity I wonder at the implications that FNM can be found to any arbitrary precision. For example, suppose we wanted to find a FNM with an error of 1 over google, 1e-100. Is there any reason this is not possible? Just as we can prove that 0.9999..... = 1.000... I wonder about the implications of a FNM with an infinite number of digits. It's amazing to think that Fermat's Last Theorem can be violated to any arbitrary but finite precision with no harm or foul to mathematical rules.