Answer
The positive integer 7 cannot be written as the sum of the square of three integers.
Work Step by Step
The claim is false and we are looking for a counterexample.
The squares of integers are:
0^{1}, 1^{2}, 2^{3}, 3^{2}, etc or 0,1,4,9,etc.
For example, 7 is a positive integer. If 7 is the sum of squares of three integers, then the squares cannot be larger than 7. Let us determine the sum of any squares of three integers with the squares smaller than or equal to 7.
0+0+0=0
0+0+1=1
0+1+1=2
1+1+1=3
0+0+4=4
0+1+4=5
1+1+4=6
0+4+4=8
1+4+4=9
4+4+4=12
We have noted that none of the sums of squares of three integers(with squares less than or equal to 7) is equal to 7 and thus 7 cannot be written as the sum of squares of three integers.
