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

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.

Answer 2

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 =
`√(2), √(3)`,... cannot be integer

∴
`y^(2)=4-x^(2)`

`y^(2)=4-1`

`y = √(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.