Proof that multiplication of two numbers plus $2$ is a prime number Let's say we have odd prime numbers $3,5,7,11,13, \dots $ in ascending order  $(p_1,p_2,p_3,\dots)$. Prove that this sentence is true or false : 
For every $i$,
$$p_i p_{i+1}+2$$
is a prime number.
Any ideas how can I prove this?
 A: I am essentially summarizing the comments - credit to the above users. 
Consider the counterexample $11(13)+2=143+2=145=29(5).$ We can see that these are consecutive primes in ascending order being multiplied, and $2$ is added. If the assertion holds, the result should be prime. But $145$ admits factorization $5*29$ and is as such not prime. This assertion is not true as stated.
A: It is false.
Suppose $p_i$ and $p_{i + 1}$ form a twin prime pair, that is, $p_{i + 1} = p_i + 2$, like 3 and 5 or 107 and 109. There is an even integer $n$ between them, such that $n = p_i + 1 = p_{i + 1} - 1$. Then $p_i p_{i + 1} + 2 = (n - 1)(n + 1) + 2 = n^2 - 1 + 2 = n^2 + 1$.
If $n \equiv 2 \pmod{10}$, then $n^2 \equiv 4 \pmod{10}$ and therefore $n^2 + 1 \equiv 5 \pmod{10}$, meaning that $n^2 + 1$ is an odd multiple of 5. The only positive odd multiple of 5 that is prime is 5 itself, but 1 is not a prime number, so $1 \times 3 + 2$ is useless here.
This means that any twin prime pair such that $p_i \equiv 1 \pmod{10}$ and $p_{i + 1} \equiv 3 \pmod{10}$ is a counterexample (an example that disproves) the assertion, e.g., 11 and 13, 101 and 103, 191 and 193, etc.; these correspond to 145, 10405, 36865, etc.

There are counterexamples besides the ones I've just described. Because the primes kind of thin out (without completely disappearing, of course), as $i$ gets larger it becomes increasingly unlikely that $p_i p_{i + 1} + 2$ is prime.
A: Building off of Robert Soupe's answer, thinking about modular arithmetic is a great way to come up with counterexamples.
The first counterexample I found was $31, 37$ - both equal $1$ (mod $3$), so when we multiply them and add $2$, we get a multiple of $3$. I'm slow at mental multiplication, so searching for a pair of consecutive primes which were both 1 or both 2 (mod 3) was easier for me than trying some small examples.
Similarly, $p_i, p_{i+1}$ form a counterexample if:


*

*$p_i=1$ (mod $5$), $p_{i+1}=3$ (mod $5$) - for example, $41$ and $43$.

*$p_i=4$ (mod $7$), $p_{i+1}=3$ (mod $7$) - for example, $53$ and $59$ ($57$ is not prime! :P).

*And so on.
