Let 1 number be `x` .  Since the numbers are consecutive, the other number would be `x + 1.`

Square both and add together to equal 313.

`x^2 + (x+1)^2 = 313`

`x^2 + x^2 + 2x + 1 = 313`

`2x^2 + 2x - 312 = 0`

`x^2 + x - 156 = 0`

`(x - 12)(x + 13) = 0`

`x = 12, x = -13`

Since the number are defined to be natural, x = -13 is not a solution.

Therefore, `x = 12` , and `x+1 = 13` .

The two numbers are 12 and 13.

`12^2 + 13^2 = 313`