Solving the Equation 2n x2 7 for Natural Numbers x

In this article, we explore the solution to the equation (2^n x^2 7), where (x) is a natural number. This problem can be approached systematically using a series of mathematical steps and properties of natural numbers.

Initial Analysis and Constraints

Given that (x) is a natural number, the smallest possible value for (x^2) is 1 (since (x 1)), making (x^2 7 geq 8). Consequently, (2^n geq 8), implying (n geq 3). This sets the lower limit for (n).

Odd and Even Cases for (x)

Let's first consider the scenario where (x) is an even number. We express (x) as:

[x 2y - 1, quad y in mathbb{N}]

The equation becomes: [2^n 4y^2 - y - 2]

Define (m n - 2 geq 1); the equation transforms into:

[2^m - 2 y(y - 1)]

This can be further analyzed in two cases: when (m 1) and when (m geq 2).

Case 1: (m 1)

When (m 1), we have:

[2^1 - 2 y(y - 1) implies 0 y(y - 1)]

The solutions are (y 0) or (y 1). Since (y) must be a natural number, (y 0) is not valid, leaving us with (y 1). This implies (n 3) and (x 1).

Case 2: (m geq 2)

For (m geq 2), the possibilities for (y) and (y - 1) must be considered:

If (y) and (y - 1) are both even, (y 2u) and the equation simplifies to: [2^{m - 1} - 1 u2u - 1 implies 2^{m - 2} u]

When (m 2), the equation becomes:

[4 - 1 u2 - 1 implies 3 2u - 1 implies u 2]

Hence, (y 4 - 1 3), giving (n 4) and (x 3).

If (y) and (y - 1) are both odd, (y 2v - 1) and the equation further simplifies to:

[4v - 5v 2^{m - 3} - 2 implies -v 2^{m - 3} - 2]

For (m 4):

[8 - 11 -1 implies -3 2^1 - 2 implies v 1]

This results in (y 3) and (n 7), (x 11).

Odd Value of (x)

Alternatively, if (x) is odd, express (x) as:

[x 2y - 1, quad y in mathbb{N}]

The equation becomes:

[2^{m - 1} - 1 2y - 1(y - 1) implies 2^{m - 2} y - 1]

For (m 2):

[4 - 3 1 implies y - 1 1 implies y 2, , x 3]

For (m geq 3), the analysis is similar, but for larger (m), no solutions are found.

Solutions

The solutions to the equation (2^n x^2 7) where (x) is a natural number are:

(n 3, x 1) (n 4, x 3) (n 5, x 5) (n 7, x 11) (n 15, x 181)

This equation is a known form, referred to as the Ramanujan-Nagell equation. For a detailed proof and further exploration, refer to the provided literature.