solving diophantine equations with modulo

• Nov 2nd 2009, 10:13 PM
jmedsy
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?
• Nov 2nd 2009, 10:54 PM
alexmahone
Quote:

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.
• Nov 2nd 2009, 11:20 PM
jmedsy
Quote:

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?
• Nov 2nd 2009, 11:23 PM
alexmahone
Quote:

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.