we interpret "the sum of their square is 313" to mean "the sum of their squares is 313"

let 'n' represent the first positive integer, then 'n+1' represents the second integer

the sum of their squares is 313:

n^2 + (n+1)^2 = 313

solving for 'n' we have a positive value of n=12

12+1 = 13

the numbers are 12 and 13