Let a positive integer be x.
Then the second integer = x + 1
Sum of the squares of the above integers = x2 + (x+1)2
= x2 + x2 + 2x + 1
= 2x2 + 2x + 1
By problem 2x2 + 2x + 1 = 613
2x2 + 2x - 612 = 0
x2 + x - 306 = 0
x(x + 18) - (x + 18) = 0
(x - )(x + 18) = 0
x - = 0 (or) x + 18 = 0
x = or -18,
we do not consider -18
Then the numbers are (, + 1)
, are the required two consecutive positive integers.