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

Integers of the form
$4^n(8k+7)$
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.