Question

Find a counter example to the statement that every positive integer can be written as the sum of the squares of three integers.

Short answer

The counter example of the integer that cannot be written as addition of squares of three integers is 7.

Step-by-step solution

Introduction

The purpose is to get a counter example of the statement that every positive integral value can be written as the addition of squares of three integral values.

Examples for the form of \({x^2}\)where\(x \in Z\)

For some examples, here the addition of square of 3 integers means that integers can be written in the form of \({x^2}\)where\(x \in Z\).

1=0+0+1

2=0+1+1

3=1+1+1

4=0+0+2²

Here it can be easily observed that 1 and 0 are squares of itself. Similarly following up this, a counter example can be obtained.

Counter example

So, the counter example is 7 which cannot be written as addition of 3 squares which can be observed below;

\(7 = \_ + \_ + 4\)

Since the only integers that are less than 7 and that are squares, are 0, 1, 4. So, at least one of the addends is to be 4. On the other hand, only one addend could be 4, since\(4 + 4 > 7\). So, need the other two addends to add up to 3. But 0 and 1 by themselves can never make 7. So, any combination of these 3 integers never adds up with the result 7.

Hence, the integer cannot be written as addition of squares of three integers is 7.