x, x+1 - two consecutive natural numbers
[tex](x+1)^3-x^3=1657 \\
x^3+3x^2+3x+1-x^3=1657 \\
3x^2+3x+1=1657 \\
3x^2+3x-1656=0 \\
3x^2+72x-69x-1656=0 \\
3x(x+24)-69(x+24)=0 \\
(3x-69)(x+24)=0 \\
3x-69=0 \ \lor \ x+24=0 \\
3x=69 \ \lor \ x=-24 \\
x=23 \ \lor \ x=-24[/tex]
x must be a natural number, so x=23.
[tex]x=23 \\
x+1=23+1=24[/tex]
The numbers are 23 and 24.
