No Number of Form 4^n(8m+7) Can Be Written as a Sum of Three Squares
A positive integer
\[n\]cannot be expressed as the sum of three squares if it is of the form
The squares modulo 8 are 0,1, 4 so a sum of three squares can be congruent modulo 8 to any of the numbers 0, 1, 2, 3, 4, 5, 6 but not 7. Hence no number of the form
\[8m+7\]can be a sum of three squares.
Suppose for some
\[n \gt 1, \; m \ge 0\]we have
The left hand side of the last expression is congruent modulo 4 to 0 and as squares modulo 4 are either 0 or 1, it has to be the case that
\[x, \; y, \; z\]are all even. Put
\[x=2x_1, \; y=2y_1. \; z=2z_1\]we get
\[n-1 \gt 1\]then
\[x_1, \; y_1, \; z_1\]are all even and we can write
We can continue in this way, eventually expressing
\[8m+7\]as a sum of three square - a contradiction so the assumption that
\[4^n(8m+7)\]can be written as a sum of three squares is false.