Challenging equation

Number Theory Level 3

Find the number of integer solutions $(x,y)$ to the below equation.

$y^3=x^4-x^3+1$

1 solution

Akarsh Kumar Srit
Jan 9, 2016

(0,1) and (1,1) are d only solutions

How do you know?

Trung Đặng Đoàn Đức - 5 years, 5 months ago

As u see rhs has to be a perfect cube if we want lhs to be an integer so it is only possible for 2 values of x

Akarsh Kumar Srit - 5 years, 5 months ago

So, what does it implies? Why larger values of $x$ doesn't make the RHS a perfect cube?

Trung Đặng Đoàn Đức - 5 years, 5 months ago

@Trung Đặng Đoàn Đức – You can writ d expression s x^3(x-1) +1 as u put larger values it cant create a perfect cube

Akarsh Kumar Srit - 5 years, 5 months ago

@Akarsh Kumar Srit – Please write the solution carefully. Nobody will understand how did you do to the RHS if you are just saying like that.

Trung Đặng Đoàn Đức - 5 years, 5 months ago