# General solution to diophantine equations

• mr_garlic
In summary, There is no general method for solving a diophantine equation, and the problem was posed by David Hilbert at the beginning of the 20th century. Martin Davis and Julia Robinson proved that it is impossible to have a general solution. However, there is a specific solution for linear diophantine equations that involves modifying a brute force algorithm to find the smallest solution. This can be applied to a problem of finding the steps to reach a certain value using n jugs of arbitrary sizes.

#### mr_garlic

Hello, I'm writing an application for a java class that solves the problem where you are given n jugs of arbitrary sizes and have to come up with the steps to reach a certain value.

I have figured out(read: did research) how to do this in a different way than the original, but it requires math that I don't know how to solve.

My question is, what is the general method for solving a diophantine equation of the form asub1x+asub2y+...asubnz = something or if you could point me to a paper or article on the subject

Sorry to disappoint you, but there is no general way to solve a diophantine equation. David Hilbert posed finding a general solution to diophantine equations as his tenth problem at the onset of the 20th century. Martin Davis and Julia Robinson proved that no general solution exists in the middle of the 20th century.

Gah, I meant linear diophantine equations, sorry for not specifying.

Ignoring a general mathematical solution, I realized that I can modify my brute force algorithm to find the smallest solution(closest to 0) to linear diophantine equations. An interesting application to an otherwise useless problem. I'll link to the library in a few minutes.