Find two consecutive numbers whose cubes differ by 127

Question

2 consecutive numbers whose cubes differ by 127

Answer 1

If the numbers are x and x+1, then:

(x+1)³=x³+3x²+3x+1.

so (x+1)³-x³=3x²+3x+1=127.

3x²+3x-126=0,

x²+x-42=0=(x+7)(x-6).

Therefore x=6 and x+1=7.

7³-6³=343-216=127.

The two numbers could also be -7 and -6, because:

(-6)³-(-7)³=-216+343=127.