Diophantine equations of the form x^3-dy^3=1

[Mathguy15]

I was recently thinking about these diophantine equations. I'm looking for solutions (x,y) in integers with d as an integer also. I have found that whenever d is a perfect cube, the equation has finitely many solutions. However, I can't seem to figure anything else out regarding equations of this type. Any ideas?MathguyEDITh, and i don't know any advanced mathematics except for some linear algebra and bits and pieces of abstract algebra. So, please don't overwhelm this mere high schooler. Thanks

 

[Mathguy15]

Oh, and I've also figured out that if d is a cube-free integer such that the number of solutions to the equation is finite, then all integers of the equation x^3-d(n^3)(y^3)=1 also has a finite number of solutions.

 

[Feryll]

Well, we know it has no solutions x>1 (There's a special case x=0, d=1, and y=-1, though) when d is a cubed integer, as then d^3*y^3 would also be a cubed integer. And since there is no difference between two cubed integers that is only 1 (they grow further and further apart from each other), there is a finite number of solutions.

And there is an infinite number of solutions when y=1, as then x>1 would be also have d=x^3-1 to make the solution true. But on the specifics of x=/=1 and y=/=1, I do not know.

 

[Petek]

Look up Delaunay-Nagell theorem (or equation). However, this result uses advanced algebraic techniques, so you might not be ready to understand the proof (yet!).

 

[blue_raver22]

!Hello Mathguy,

Diophantine equations of the form x^3-dy^3=1 are known as "Cubic Thue-Morse equations" and they have been studied extensively in number theory. These types of equations are called Diophantine because they involve finding integer solutions (x,y) to an algebraic equation.

As you mentioned, when d is a perfect cube, the equation has finitely many solutions. In fact, it has been proven that for any given d, there are only finitely many solutions to this equation. This is known as the "Mordell Conjecture" and it was proven by the mathematician Louis Mordell in the early 20th century.

However, when d is not a perfect cube, the equation becomes much more difficult to solve. There are some techniques that can be used, such as the "Pell equation" method, but these require a deeper understanding of number theory.

One interesting fact about these equations is that they are closely related to the famous "Fermat's Last Theorem". In fact, the solutions to the equation x^3-dy^3=1 can be used to find solutions to Fermat's Last Theorem for certain exponents.

In summary, Diophantine equations of the form x^3-dy^3=1 are a fascinating area of study in number theory and have connections to many other important problems in mathematics. While they may seem daunting at first, with further study and a solid understanding of number theory, you may be able to make some interesting discoveries about these equations. Best of luck in your mathematical explorations!

Sincerely,