I think the two-square-sum problem has been done to death. If you can factor X, it's easy to determine whether it has such representations;
i've been looking at the three square case. i assume the squares have to be equal length (there are many more triples if you assume leading zeroes). aside from the obvious sets of squares of length 1, i've found the following. i especially like the last one. 400, 256, 121 625, 400, 196 676, 400, 256 784, 625, 256 841, 169, 100 5625, 4096, 2500 6400, 5776, 1156 6400, 5776, 4489 7056, 3721, 1444 7396, 3600, 1225 8281, 2916, 1024 8464, 2601, 1156 8464, 3844, 1024 8464, 4489, 2601 42025, 14641, 10000 43681, 36100, 31329 44100, 33124, 11664 44100, 34969, 32041 51076, 48400, 33856 55225, 34596, 32400 55225, 35721, 20164 55696, 40000, 37636 59049, 51984, 44521 60025, 39204, 11881 62001, 34225, 14884 65025, 25921, 20164 65536, 55696, 12100 68121, 42436, 11664 70225, 26244, 14641 75625, 55696, 35344 77284, 22801, 11025 77841, 66049, 11664 81225, 22500, 18496 82369, 16641, 12100 84100, 15129, 11881 84100, 33856, 15376 85264, 24964, 23104 91204, 29584, 12544 95481, 44944, 15129 97344, 44521, 13689 99225, 12996, 10000 301401, 242064, 123201 erich