Question

Prove or disprove:

a) Any positive integer can be written as the sum of the squares of two integers

b) For every integer n, the number 3(n2 + 2n + 3) ? 2n2 is a perfect square

Answer 1

Step-by-step explanation:

a). Let $x^{2}+y^{2}=4$

     $y^{2}=4-x^{2}$

Now since $y^{2}$ is a square of an integer, its value is $\geq$ 0.

Therefore, $4-x^{2}$ ≥ 0, since x is an integer

where $4-x^{2}$ = 1,2,3,...

and x = 1

But as x = $\sqrt{2}, \sqrt{3}$,... cannot be integer

∴ $y^{2}=4-x^{2}$

   $y^{2}=4-1$

   $y = \sqrt{3}$

              = 1.732      which is not an integer

Thus, any positive integer cannot be written as the sum of the squares of the two integers.

b). Let n be an integer

∴ $3(n^{2}+2n+3)-2n^{2}$

On solving we get,

 $3n^{2}+6n+9-2n^{2}$

 $n^{2}+6n+9$

 $n^{2}+3n+3n+9$

 $n(n+3)+3(n+3)$

  $(n+3)(n+3)$

  $(n+3)^{2}$

which is a perfect square

Hence proved.

Answer 2

Answer:

Step-by-step explanation:

a). Let

   

Now since  is a square of an integer, its value is  0.

Therefore,  ≥ 0, since x is an integer

where  = 1,2,3,...

and x = 1

But as x = ,... cannot be integer

∴

 

 

             = 1.732      which is not an integer

Thus, any positive integer cannot be written as the sum of the squares of the two integers.

b). Let n be an integer

∴

On solving we get,

 

 

which is a perfect square

Hence proved.