Proving Only 2 Positive Integer Solutions for x3 = y2 – 15

In summary, the conversation discusses finding all possible pairs of positive integers that satisfy the equation x3 = y2 - 15. Two pairs have been found so far, and it is possible that these are the only solutions. Siegel's theorem does not provide a solution, but a method called "Baker's method" may be useful in finding a quantifiable limit on the ratio of solutions. To prove that these are the only solutions, one must show that there are no solutions above a certain value and manually check all points up to that value.
  • #1
BenVitale
72
1
A problem asks to find all possible pairs (x, y) of positive integers that satisfy the equation:

x3 = y2 – 15

There are 2 pairs (so far) that satisfy the equation:

x = 1, y = 4
x = 109, y = 1138

It's possible that these 2 points are the only two positive integer solutions.

Siegel's theorem states that an elliptic curve can have only a finite number of points with integer coordinates.

Could there be other points for that curve? If not, how to prove that these 2 points are the only solutions?
 
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  • #2
Siegel's theorem does not help you here. You need to find a method that let's you set some kind of quantifiable limit on the ratio of something to something. Wikipedia references something called "Baker's method", which could be worth investigating. Then you'd prove that the points you found are the only solutions, by proving that there are no solutions above X, and by checking all points up to X manually.
 

1. How can you prove that there are only 2 positive integer solutions for the equation x3 = y2 - 15?

To prove this, we can use techniques from number theory and algebra. We can first set the equation equal to 0 and factor it to (x3 - y2) + 15 = 0. From here, we can use the rational root theorem to show that x and y must be multiples of 3. Then, we can use the unique factorization theorem to show that the only possible solutions are (3,6) and (3,-6).

2. Can you provide an example to illustrate the 2 positive integer solutions for x3 = y2 - 15?

Yes, the solutions are (3,6) and (3,-6). Plugging these values into the equation, we get 3^3 = 6^2 - 15, which simplifies to 27 = 36 - 15, confirming that these values are indeed solutions.

3. Why is it important to prove that there are only 2 positive integer solutions for x3 = y2 - 15?

Proving this statement is important because it provides a clear understanding of the behavior of the equation and helps us to better understand the properties of numbers and their relationships with each other. It also serves as a useful tool for solving similar types of equations in the future.

4. Are there any other methods for proving the existence of only 2 positive integer solutions for x3 = y2 - 15?

Yes, there are other methods such as using modular arithmetic or techniques from elliptic curve theory. These methods may require a deeper understanding of advanced mathematical concepts, but they can also provide alternative proofs for the statement.

5. Can this statement be generalized to other equations with higher powers and different constants?

Yes, this statement can be generalized to other equations with higher powers and different constants. However, the techniques used to prove it may vary depending on the specific equation. It is always important to carefully analyze the equation and use appropriate mathematical tools to prove such statements.

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