Problem 1: Let \(x\) be a positive integer and \(y = x^2+2\). Can \(x\) and \(y\) be both prime? The answer is yes, since for \(x=3\) we get \(y=11\), and both numbers are prime. Prove that this is the only value of \(x\) for which both \(x\) and \(y\) are prime.

Hint: Consider cases, depending on the remainder of \(x\) modulo \(3\).

Solution 1:
If \(x\neq3\), then \(x=1\) or \(x=2\) in \(x\) modulo \(3\).
\(x=1\) and \(x=2\) would be prime under the equation \(x^2 = 1\) (mod \(3\))
However, with the equation \(x^2+2\), both \(x=1\) and \(x=2\) result in 0 \(x^2+2=0\)(mod \(3\))
This means that \(3\) evenly divides into \(x^2+2\) for \(x=1\) and \(x=2\), meaning that \(y\) is not prime for both of these terms.
Additionally, \(y\neq3\) because the equation is only true for \(x=1\) and \(1\) is not a prime number.
Therefore, since for \(x=1\) and \(x=2\), \(y\) is divisible by \(3\) and \(y\neq3\), the only case where both \(x\) and \(y\) can be prime is \(x=3\)