Solving Third-Order Diophantine Equations: Resources and Assistance

[Oxymoron]

Does anyone know of any resources on the web (or if you prefer, provide me directly with assistance) which will help me understand how to solve equations of the form

$$ax^3+by^3=c$$

I believe they are third-order Diophantine equations.

 

[Oxymoron]

Perhaps I should be a little more specific.

How would you go about proving that a third order Diophantine equation has no solutions?

 

[Oxymoron]

I decided to have a go and here is how I went. Let me know if I made any mistakes

Consider the Diophantine equation

$$x^3+117y^3=5$$

Choose mod 5.

The equation tells us that $5|x^3 \Rightarrow 5^3|x^3 \Rightarrow x = 5X$. Therefore

$$125\cdot X^3 +117y^3 = 5$$

By the same procedure as above, this equation tells us that $5|y^3 \Rightarrow y = 5Y$. Therefore

$$125\cdot X^3 + 117\cdot 5Y^3 = 5$$

Divide through by 5 and we have

$$25\cdot X^3 + 117\cdot Y^3 = 1$$

This equation now tells me that

$$25X^3 \equiv 1(\mod 5)$$

But $25\equiv 0 (\mod 5)$. Hence there is no such $0,1,2,3,4$ such that $X^3\equiv 1(\mod 5)$.

However, if we had checked Y first we would have found

$$117Y^3 \equiv 1(\mod 5)$$

which implies that

$$Y^3 \equiv 2(\mod 5)$$

since $117\equiv 2 (\mod 5)$. And since if we let $Y = 3$ then

$$Y^3 = 3^3 = 27 \equiv 2(\mod 5)$$

then this tells us that there is a solution. But since it failed for X there is no solution.