\section*{Exercise 2 - 091022}

We are looking for four natural numbers \(a_{1}<a_{2}<a_{3}<a_{4}\) such that each of the numbers

\[
d_{1}=a_{4}-a_{3} ; \quad d_{2}=a_{3}-a_{2} ; \quad d_{3}=a_{2}-a_{1} ; d_{4}=a_{4}-a_{2} ; \quad d_{5}=a_{3}-a_{1} ; d_{6}=a_{4}-a_{1}
\]

is a prime number, where the same prime numbers may also occur.
}

Assume \(a_{1}, \ldots, a_{4}\) are the four numbers of the desired type. Then, for the numbers \(d_{1}, \ldots, d_{6}\) formed according to the problem statement, the equations

\[
d_{4}=d_{1}+d_{2}, \quad d_{5}=d_{2}+d_{3}, \quad d_{6}=d_{1}+d_{2}+d_{3}
\]

hold. Now, only the following cases are possible:
I. \(d_{1}, d_{2}, d_{3}\) are odd prime numbers. This leads to the contradiction that \(d_{4}\) and \(d_{5}\) are even and greater than 2, and thus not prime numbers. Therefore, this case is excluded.

II. \(d_{1}, d_{2}, d_{3}\) are even prime numbers (i.e., each is equal to 2). This leads to the same contradiction.

III. Exactly one of the numbers \(d_{1}, d_{2}, d_{3}\) is even (i.e., equal to 2). This leads to the contradiction that \(d_{6}\) is even and greater than 2.

IV. Exactly two of the numbers \(d_{1}, d_{2}, d_{3}\) are even (i.e., each is equal to 2).

a) \(d_{2}\) is one of them. Then it follows that either \(d_{4}\) or \(d_{5}\) is even and greater than 2.
b) \(d_{1}=d_{3}=2 ; d_{2}\) is an odd prime number. Then it follows that \(d_{4}=d_{5}=d_{2}+2\) and \(d_{6}=d_{2}+4\). Among the three integers of the form \(d_{2}, d_{2}+2, d_{2}+4\), one is always divisible by 3.

The only prime number divisible by 3 is 3. Since \(d_{2}>1\), \(d_{2}+2\) and \(d_{2}+4\) are greater than 3, the only possibility is \(d_{2}=3\).

From this, it follows further that

\[
a_{2}=a_{1}+d_{3}=a_{1}+2, \quad a_{3}=a_{2}+d_{2}=a_{1}+5, \quad a_{4}=a_{3}+d_{1}=a_{1}+7
\]

Therefore, \(a_{1}, \ldots, a_{4}\) can only satisfy the conditions of the problem if they are of the form

\[
a_{1}=0, \quad a_{2}=n+2, \quad a_{3}=n+5, \quad a_{4}=n+7
\]

with a natural number \(n\).

Conversely, any four numbers of the form \((*)\) indeed satisfy the conditions of the problem; for them, each of the numbers is a prime number

\[
\begin{array}{lll}
d_{1}=a_{4}-a_{3}=2, & d_{2}=a_{3}-a_{2}=3, & d_{3}=a_{2}-a_{1}=2 \\
d_{4}=a_{4}-a_{2}=5, & d_{5}=a_{3}-a_{1}=5, & d_{6}=a_{4}-a_{1}=7
\end{array}
\]