Let the two numbers be x and y, in which x is larger than y. From these variables, we can formulate the following equations:
x = y + 1 --> eqn 1
x³ - y³ = 631 --> eqn 2
Since there are two unknowns and two independent equations, this system is solvable. Substitute equation 1 to equation 2, then solve for y.
(y + 1)³ - y³ = 631
Solving for y,
y = 14
x = y + 1 = 14 + 1 = 15
Therefore, the two consecutive numbers are 14 and 15.