To know whether everyone knows about primes

Number Theory Level 4

Find the smallest prime number $n$ , such that for all prime numbers $p \geq n$ , $p^2 + 2$ is always composite.

The answer is 5.

1 solution

Kazem Sepehrinia
Aug 9, 2015

For $p\ge 5$ we have $p^2 \equiv 1$ mod $3$ and therefore $p^2+2 \equiv 1+2 \equiv 0$ mod $3$ and $p^2+2$ is always composite. So $n_{min}=5$ . Note that for $p=3$ , $p^2+2=11$ is a prime.

Even though I guessed between 4 and 5, I think the answer should be $4$ since it doesn't specify $n$ has to be a prime and $p\ge 4$ makes sense.

Xuming Liang - 5 years, 10 months ago

But $p$ is a prime and cannot take the value of $4$ . So it's not necessary to say $n$ has to be a prime.

Kazem Sepehrinia - 5 years, 10 months ago

$4$ is the smaller value of $n$ than $5$ and it still works (so the answer would be $4$ - the problem asked for the smallest value of $n$ , and $4$ works), because the set of all primes $\ge 4$ is equal to the set of all primes $\ge 5$ . But now the problem statement has been edited and it says that $n$ is prime, so the answer $5$ is now correct.

mathh mathh - 5 years, 10 months ago

@Mathh Mathh – I still have doubt on this one, I don't know. Maybe one of challenge masters will clarify this for us?

Kazem Sepehrinia - 5 years, 10 months ago

@Kazem Sepehrinia – Thanks. Those who answered 4 have been marked correct.

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Calvin Lin Staff - 5 years, 10 months ago

To know whether everyone knows about primes

The answer is 5.

1 solution

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