Integer-integer everywhere

Number Theory Level 3

Find out the number of integeral solutions of $4x^3 -7y^3 =2003$

4 0 6 none of these 3 2

1 solution

Andy Hayes
Oct 27, 2015

Rearranging the equation: $7y^3=4x^3-2003$

This means that $4x^3-2003\equiv0\pmod{7}$

Using modular arithmetic, we obtain $x^3\equiv2\pmod{7}$ .

However, $2$ is not a cubic residue $\pmod{7}$ . Therefore, there are no integer solutions.

nice solution:)

Atul Shivam - 5 years, 7 months ago