Find all integers 1<=k<=169 for which 169 is not the sum of k nonzero squares.

The squares are not necessarily unique. For example k=5: 169=1+4+4+16+144.

This should be done without a brute force program.

k=169  169*1

k=166  165*1 + 1*4

k=163  161*1 + 2*4

k=161  160*1 + 1*9

k=160  157*1 + 3*4

k=158  156*1 + 1*4 + 1*9

k=157  153*1 + 4*4

k=155  152*1 + 2*4 + 1*9

k=154  149*1 + 5*4   OR  153*1 + 1*16

k=152  148*1 + 3*4 + 1*9

k=151  145*1 + 6*4   OR  149*1 + 1*4 + 1*16

k=149  144*1 + 4*4 + 1*9

k=148  141*1 + 7*4   OR  145*1 + 2*4 + 1*16

k=147  143*1 + 2*4 + 2*9

k=146  140*1 + 5*4 + 1*9

k=145  137*1 + 8*4

k=144  139*1 + 3*4 + 2*9

Patterns seem to repeat every k-3 so I think all values of k in the upper range other than those missed above have a solution.

I'll start on the lower values of k.

Posted by Nosher on 2004-11-04 22:38:15