# Thread: solving diophantine equations with modulo

1. ## solving diophantine equations with modulo

For the Diophantine equation $\displaystyle x^4+5y^3=2009$, either find all integer solutions, or show that there are no integer solutions.

How can I use modulo to solve this?

2. Originally Posted by jmedsy
For the Diophantine equation $\displaystyle x^4+5y^3=2009$, either find all integer solutions, or show that there are no integer solutions.

How can I use modulo to solve this?
Consider modulo 5.
$\displaystyle x^4=0,\ 1\ (mod\ 5)$
$\displaystyle 2009=4\ (mod\ 5)$

$\displaystyle 2009-x^4=4,\ 3\ (mod\ 5)$
$\displaystyle 2009-x^4=5y^3$ has no integer solutions.
$\displaystyle x^4+5y^3=2009$ has no integer solutions.

3. Originally Posted by alexmahone
Consider modulo 5.
$\displaystyle x^4=0,\ 1\ (mod\ 5)$
$\displaystyle 2009=4\ (mod\ 5)$

$\displaystyle 2009-x^4=4,\ 3\ (mod\ 5)$
$\displaystyle 2009-x^4=5y^3$ has no integer solutions.
$\displaystyle x^4+5y^3=2009$ has no integer solutions.
That makes sense. Did you choose (mod 5) just because one of the terms including a variable was a multiple of 5?

4. Originally Posted by jmedsy
That makes sense. Did you choose (mod 5) just because one of the terms including a variable was a multiple of 5?
Yes.