\section*{Exercise 2 - 081012}

Determine all prime numbers \(p\) with the following property!

If you add the number 50 to \(p\) and subtract the number 50 from \(p\), you get two prime numbers.
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Assume a prime number \(p\) has the mentioned property. Then the numbers \(p-50\) and \(p+50\) are also prime numbers.

Now, \(p+50=p+48+2\) leaves the same remainder as \(p+2\) when divided by 3, and \(p-50=p-51+1\) leaves the same remainder as \(p+1\) when divided by 3. Among the three consecutive natural numbers \(p, p+1, p+2\), one must be divisible by 3, and thus one of the numbers \(p, p-50\), and \(p+50\) must also be divisible by 3.

Since there is exactly one prime number divisible by 3, which is 3 itself, it must appear among these three numbers, and it must be the smallest of them, otherwise \(p-50\) would not be a natural number, and thus not a prime number. Therefore, \(p-50=3\); hence, only \(p=53\) can have the property mentioned in the problem. Conversely, this is indeed the case; because \(p=53\), \(p+50=103\), and \(p-50=3\) are all prime numbers.

Adapted from \([5]\)