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.