His numbers do not add up to the same thing. In other words, 4709^20 != (4110^20 + 4693^20). (The difference is ~10^61 or so, whereas the numbers are ~10^73. In other words, they diverge at ~ the 12th digit, whereas many calculators only display 10.)
Google verifies the answer is 434437^15 + 588129^15 = 588544^15: https://www.google.com/search?q=434437%5E15+%2B+588129%5E15+...
> >>> 434437* * 15 + 588129* * 15 - 588544 * *15
> -604550152144288043930860169354171954730939671404246170822386878582482
(Edit: how do I display two asterisks in a row? It's supposed to be number(asterisk)(asterisk)number.)
This is using Python, which does arbitrary-precision integer arithmetic.
Google's calculator probably uses floating-point numbers internally, and hence starts losing precision.
In case anyone is curious, the above "solutions" are called near-misses, since they're almost correct. A clever person came up with an algorithm to generate interesting near-misses for low exponents. See the table on page 15: http://arxiv.org/pdf/math/0005139v1.pdf
Related: http://math.stackexchange.com/questions/526330/fermats-last-...
Have fun!