Determine the Value of X

Answer

To solve the equation $16p + 1 = x^3$ where $p$ is a prime number, we need to find the value of $x$. Let's break down the solution step-by-step.

Step 1: Understand the Equation

The given equation is:

$16p + 1 = x^3$

Rearranging it, we get:

$16p = x^3 - 1$

The expression $x^3 - 1$ can be factored using the identity for the difference of cubes:

$x^3 - 1 = (x - 1)(x^2 + x + 1)$

Thus, the equation becomes:

$16p = (x - 1)(x^2 + x + 1)$

Step 2: Analyze the Factors

We know that $p$ is a prime number, which implies that the product $(x - 1)(x^2 + x + 1)$ divided by 16 should result in a prime number. Let's analyze the factors:

• $x - 1$ must be a multiple of 16 for $p$ to remain a prime number.
• $x^2 + x + 1$ must be an odd number because the sum of an even number $x(x+1)$ and 1 is always odd.

Step 3: Solve for $x$

Given that $x - 1$ must be a multiple of 16, let's assume:

$x - 1 = 16k$

where $k$ is a positive integer. Therefore, $x = 16k + 1$.

Substituting back into the equation:

$16p = (16k)(x^2 + x + 1)$

Since $p$ is a prime number, $k$ must be 1 to keep $p$ prime. Therefore:

$x - 1 = 16$ $x = 16 + 1 = 17$

Step 4: Verify the Solution

Substituting $x = 17$ back into the original equation:

$16p + 1 = 17^3$ $16p + 1 = 4913$ $16p = 4912$ $p = \frac{4912}{16} = 307$

Check if 307 is a prime number:

• 307 is not divisible by any prime number up to its square root (approximately 17.5), confirming that 307 is a prime number.

Thus, the solution is consistent, and the value of $x$ is:

$\boxed{17}$