What Tom does on a rainy Sunday afternoon

Should you ever have been curious about the integer points on the quartic surface x^4+y^4+z^4=N*t^4, the program here will find them with reasonable speed. Call it with 'four N limit'; if you have lots of processors to use, 'four (N) (limit) 0 (limit/4)' on the first one, 'four (N) (limit) (limit/4) (limit/2)' on the second and so on.

1949^4 + 4727^4 + 12389^4 = 2657^4 * 483

It takes a minute on a 2.4GHz Pentium 4 with a very old motherboard for 'four 163 10000' to find all the points on x^4+y^4+z^4=163t^4 with less than five digits in each term, or about twelvefifty¹ hours for 'four 1 500000' to find the first-discovered-in-1986 smallest point on x^4+y^4+z^4=t^4; the program uses about (limit/30000) megabytes of memory, and time a little over quadratic in (limit). I'd be interested to see how fast it runs on actually-fast modern computers, if any of my readers have one, and a C++ compiler, and the desire to run 'time four 163 10000' and wait a minute for the result.

Some of the points on these surfaces are known to be connected with points on elliptic curves lying on the surface (which is why a separate program of mine managed to find 7592431981391^4 + 22495595284040^4 + 27239791692640^4 = 29999857938609^4, which would be found by the simple search program only after some thousands of times the present age of the universe); the only problem is that I have no idea how to get from a known point to an elliptic curve that it happens to lie on. I obtained the big point by using a known elliptic curve and finding large points on that using standard software (part of which I developed for my PhD), but I haven't yet managed to find any elliptic curves for N not equal to 1.

On the other hand, it may well be that there are points on the surfaces which lie on no elliptic curve; nobody has a clue how to find those by any methods cleverer than the one implemented in the code above.

¹: it turned out that the run-time is a bit more over quadratic than I thought it was