Solving the Integer Equation x2 2n 153 Using Advanced Number Theory Techniques

The given equation, (x^2 2^n 153), is an interesting problem that requires a deep understanding of number theory. Let's explore the solution step-by-step.

Initial Solution and Verification

The initial solution given is when (n 4), leading to the equation:

(x^2 2^4 153)

This simplifies to:

(x^2 153 - 16 137)

Solving for (x), we get:

(x 13)

Thus, the pair that satisfies the equation is ((x, n) (13, 4)).

Exploring Non-Standard Methods

To further explore the solution, we can factorize 153 in different ways and see if we can find additional solutions. Notably, we know that:

(153 9 times 17 3 times 51)

Testing smaller numbers such as:

(9^2 - 4^2 153)

Gives us:

(x 13) and (n 4)

Further exploration reveals that:

(27^2 - 24^2 153)

Which simplifies to:

(27^2 - 2^{log_{2} (576)/(log_2 (2))} 153)

Since (log_{2} (576)/(log_2 (2)) 9.169925001), it does not provide an integer solution for (n).

Advanced Techniques for Solutions

We can rewrite the equation as:

(x^2 - 153 2^n)

Considering the problem in the ring of integers of (Q[sqrt{17}]), where:

(x^2 - 153 x - 3sqrt{17}x 3sqrt{17})

This can be factored as:

(2 (frac{3}{2} - frac{1}{2}sqrt{17})(-frac{3}{2} - frac{1}{2}sqrt{17}))

Thus, we can write:

((frac{3}{2} - frac{1}{2}sqrt{17})^n (frac{3}{2} frac{1}{2}sqrt{17})^m x - 3sqrt{17}x 3sqrt{17})

The greatest common divisor (GCD) of the terms must be considered. Simplifying, we find that:

(gcd(x - 3sqrt{17}, x 3sqrt{17}) gcd(-6sqrt{17}, x 3sqrt{17}) g)

The possible values of (g) are 1 and (3sqrt{17}), but since (x 0) is not a solution, we discard these.

Given the form of the equation, we can explore different values for (m) and (n). For example, when (m n 1), we get:

(x 13, n 4)

For (m 2, n 1), we get:

(6 - 2sqrt{17})

Confirming that the term on (sqrt{17}) strictly increases in magnitude as (m) increases.

Using SageMath, we find:

((frac{3}{2} - frac{1}{2}sqrt{17})^4 (frac{3}{2} frac{1}{2}sqrt{17})^4 20)

((frac{3}{2} - frac{1}{2}sqrt{17})^5 (frac{3}{2} frac{1}{2}sqrt{17})^4 -24 - 8sqrt{17})

Thus, the only integer solution for this problem is ((x, n) (13, 4)).