I thought about the following problem, probably it already appears in mathematical literature.
Operator $\unrhd$, is binary operation, defined for natural numbers as follows: To every $n,m$ naturals, $n\unrhd m$ is first comes $n$ and then comes $m$.
Example: if $n=56$, $m=67$, then $n\unrhd m = 5667$
Primes $p_k$ and $p_s$ will called brothers, if $p_k \unrhd p_s$ or $p_s \unrhd p_k$ are primes.
Example: $p_1=2$ and $p_2=3$ are brothers, as $2 \unrhd 3=23$ is prime.
Prove that there is infinite brothers numbers.