## Three Smallest Integers Greater Than 1000 Which Cannot Be Written as the Sum of Three Squares

\[4^n(8k+7)\]

cannot be written as the sum of three squares (No Number of Form 4^n(8m+7) Can Be Written as a Sum of Three Squares). Any number congruent to 7 modulo 8 is of this form, and the other possibilies are multiples of such numbers by a power of 4.\[1004=4 \times 251\]

and \[251 \equiv 3 \; (mod \; 8)\]

.\[1007 \equiv 7 \; (mod \; 8)\]

\[1008=4^2 \times 63\]

and \[63 \equiv 7 \; (mod \; 8)\]

\[1012=4 \times 253\]

and \[253 \equiv 5 \; (mod \; 8)\]

\[1015 \equiv 7 \; (mod \; 8)\]

Hence 1007, 1008 and 1015 cannot be written as the sum of three squares.