solving diophantine equations with modulo

jmedsy · #1 $Old$ November 3rd, 2009, 12:13 AM

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

How can I use modulo to solve this?

alexmahone · #2 $Old$ November 3rd, 2009, 12:54 AM

Quote:

Originally Posted by jmedsy $View Post$

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

How can I use modulo to solve this?

Consider modulo 5.
$x^4=0,\ 1\ (mod\ 5)$
$2009=4\ (mod\ 5)$

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

jmedsy · #3 $Old$ November 3rd, 2009, 01:20 AM

Quote:

Originally Posted by alexmahone $View Post$

Consider modulo 5.
$x^4=0,\ 1\ (mod\ 5)$
$2009=4\ (mod\ 5)$

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

alexmahone · #4 $Old$ November 3rd, 2009, 01:23 AM

Quote:

Originally Posted by jmedsy $View Post$

That makes sense. Did you choose (mod 5) just because one of the terms including a variable was a multiple of 5?

Yes.