Find two consecutive numbers whose cubes differ by 631

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.