Sum of 5 distinct squares

[franktaw at netscape.net]

A004438 is the numbers that are not the sum of 5 distinct squares.  I 
think this sequence is finite.  It contains 76 values:

1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,2
8,29,31,32,33,34,35,36,37,38,40,41,42,43,44,45,47,48,49,52,53,56,58,59,60
,61,64,67,68,69,72,73,76,77,80,83,89,92,96,97,101,104,108,112,124,128,136
,188,224

and nothing else up to 5000.

If we look at non-sums of 5 distinct non-zero squares, there are 124 
values:

1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,2
8,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52
,53,54,56,57,58,59,60,61,62,63,64,65,67,68,69,70,71,72,73,74,76,77,78,80,
81,83,84,85,86,89,91,92,93,96,97,98,101,102,104,105,107,108,109,112,113,1
16,117,119,122,124,125,128,133,136,137,140,141,149,153,161,164,173,177,18
2,188,189,197,203,221,224,236,245

and again nothing else up to 5000.  (This sequence is not in the OEIS.)

I am confident, based on these results, that these sequences are in 
fact finite, and that the above lists are complete.  This isn't a 
proof, however.  Does anyone have any idea how to prove this?

Franklin T. Adams-Watters