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This year is year $2017$ and it is a prime number. Next year is year $2018=2\cdot 1009$. $1009$ is also a prime number.

In general, is there a law about prime numbers followed by $2\cdot (\text{another prime number})$?

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    This is somewhat related to "safe primes" where a prime is preceeded by two times another prime. (That turns out to be useful in cryptography.) See https://en.m.wikipedia.org/wiki/Safe_prime. – 2017-01-14
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    Related: http://math.stackexchange.com/questions/911690/are-there-infinite-many-primes-p-such-that-2p-1-is-also-prime – 2017-01-14
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    Your wording is confusing (at least for me, and for some answerers also). I would say "Primes $p=2q-1$ followed by 2 times a prime number $q$". – 2017-01-14
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    @Watson In fact I got this wrong as well! But the given example should have pushed me into the right direction ... – 2017-01-14
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    @Watson made the edit you suggest, I hope this makes it more clear for you. – 2017-01-15

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Probably there are infinite many primes $p$, such that $2p-1$ is prime as well. The Bunyakovsky-conjecture would imply this. There is a great statistical evidence for this claim, but as far as I know, no proof.

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    He's asking about primes $p$ where $2p - 1$ is also prime, not $2p + 1$. – 2017-01-14
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    I misread the question, but the situation is very similar – 2017-01-14