1. Find all prime numbers $p$ for which there exists a natural number $n$ such that $p^{n}+1$ is a cube of some natural number.
SOLUTION. Let's assume that for a natural number $a$ the following holds: $p^{n}+1=a^{3}$ (obviously $a \geq 2$). We will rearrange this equation to make it possible to factor one side:

$$
p^{n}=a^{3}-1=(a-1)\left(a^{2}+a+1\right) \text {. }
$$

From this factorization, it follows that if $a>2$, both numbers $a-1$ and $a^{2}+a+1$ are powers of the prime number $p$ (with positive integer exponents).

In the case $a>2$, we have $a-1=p^{k}$ or $a=p^{k}+1$ for a positive integer $k$, which, when substituted into the trinomial $a^{2}+a+1$, gives the value $p^{2 k}+3 p^{k}+3$. Since $a-1=p^{k}2$ no prime number $p$ satisfies the equation $p^{n}+1=a^{3}$, regardless of the choice of the exponent $n$.

For $a=2$, we get the equation $p^{n}=7$, so $p=7$ is the only prime number that satisfies the equation.

SUPPLEMENTARY AND ADDITIONAL PROBLEMS:

N1. Determine all triples $(a, b, c)$ of natural numbers for which the following holds:

$$
2^{a}+4^{b}=8^{c} \text {. }
$$

$[62-\mathrm{B}-\mathrm{I}-1]$

N2. Find all pairs of natural numbers $a, b$ such that $a b=a+b$. [We move all terms to the left side and factor it: $(a-1)(b-1)=1$. There is only one way to write the number 1 on the right side as a product of two non-negative integers, so $a=b=2$.]

N3. Find all prime numbers $p$ for which there exists a natural number $x$ such that $p^{5}+4=x^{2}$. [Clearly $x>2$ and from the rearranged equation $p^{5}=(x-2)(x+2)$, considering $x+2>x-2>0$, it follows that the pair $(x+2, x-2)$ is either $\left(p^{5}, 1\right),\left(p^{4}, p\right)$, or $\left(p^{3}, p^{2}\right)$. In the first case, $p^{5}-1=4$, in the second $p^{4}-p=4$, but neither of these equations has a prime solution. The remaining possibility is $p^{3}-p^{2}=4$, which gives $p=2$ and $x=6$.]

D1. Find all pairs of prime numbers $p, q$ for which there exists a natural number $a$ such that

$$
\frac{p q}{p+q}=\frac{a^{2}+1}{a+1}
$$

$[62-\mathrm{A}-\mathrm{I}-1]$

D2. Find all pairs of natural numbers $x, y$ such that

$$
\frac{x y^{2}}{x+y}
$$

is a prime number. [58-A-I-3]

D3. Prove that for no natural number $n$ is the number $27^{n}-n^{27}$ a prime number. [Slovak version $57-$ A-III-4, https://skmo.sk/dokument.php?id=215]

D4. Find all integers $n$ for which $n^{4}-3 n^{2}+9$ is a prime number. [61-A-III-1]