We are tasked with solving the equation $x^3 + x^2 + x + 1 = p^n$, where $x$ and $n$ are natural numbers, and $p$ is a prime number. Our goal is to find all possible combinations of $x$, $n$, and $p$ that satisfy the equation.

Step 1: Factor the Left-Hand Side

The left-hand side of the equation, $x^3 + x^2 + x + 1$, can be factored:

$x^3 + x^2 + x + 1 = \frac{x^4 - 1}{x - 1} \quad \text{for} \quad x \neq 1$

So, the equation becomes:

$\frac{x^4 - 1}{x - 1} = p^n$

Step 2: Consider Small Values of $x$

Case 1: $x = 1$

Substituting $x = 1$ into the equation:

$1^3 + 1^2 + 1 + 1 = 4$

This gives $p^n = 4$, but $p$ must be a prime number. Since $4$ cannot be expressed as a power of any prime, there is no solution in this case.

Case 2: $x = 2$

Substituting $x = 2$ into the equation:

$2^3 + 2^2 + 2 + 1 = 8 + 4 + 2 + 1 = 15$

Here we have $p^n = 15$. Since 15 is not a power of any prime, there is no solution in this case either.

Case 3: $x = 3$

Substituting $x = 3$ into the equation:

$3^3 + 3^2 + 3 + 1 = 27 + 9 + 3 + 1 = 40$

Again, $p^n = 40$, and since 40 is not a power of a prime, there is no solution here.

Case 4: $x = 5$

Substituting $x = 5$ into the equation:

$5^3 + 5^2 + 5 + 1 = 125 + 25 + 5 + 1 = 156$

Once again, $p^n = 156$, which is not a power of a prime, so there is no solution here.

Step 3: Analyze Larger Values of $x$

For values of $x > 5$, the left-hand side grows very rapidly, making it increasingly difficult for it to be a power of a prime number. From this trend and prior analysis, we observe that the equation cannot yield powers of primes for larger $x$.

Conclusion:

After testing small values of $x$ and factoring the equation, there are no natural number solutions where $x^3 + x^2 + x + 1 = p^n$, where $p$ is a prime and $n$ is a natural number.

Would you like a deeper exploration of other factoring methods or specific prime-based approaches?

Here are 5 related questions to consider:

1. Can you express a similar factorization for $x^k + x^{k-1} + \dots + 1$ in general?
2. How would the equation change if we allowed $x$ or $n$ to be non-natural numbers?
3. Could other polynomial forms, such as $x^3 + x^2 + x - 1 = p^n$, have solutions?
4. How does the behavior of the equation change for non-prime powers?
5. What insights can number theory provide for similar exponential equations?

Tip: In Diophantine equations, testing small values and analyzing the growth of terms is often a key step in finding potential solutions or ruling out possibilities.