Question involving prime numbers, *brothers* numbers. I thought about the following problem, probably it already appears in mathematical literature.
Definition 1:
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 $n\unrhd m = 5667$ 
Definition 2:
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.
Question:
Prove that there is infinite brothers numbers. 
 A: Just to slightly elaborate on Gerry's answer given in the comment, notice that your question relies heavily on the way numbers are presented (in particular you seem to assume numbers are written in decimal notation and your binary operation is not on numbers but rather on numbers expressed in binary form). In stark contrast is the property Gerry mentions is beyond current reach does not depend on any particular notation for expressing numbers. In this case the former subsumes the latter. It is important to realize when a conjecture, or any statement, about numbers relies on the way numbers are written. It is somewhat unnatural to wonder about properties of numbers that are dependent on the way numbers are written, since then it is not so much a property of numbers but rather a combination of a number theoretic property and a typographical property. But aside from sporadic results here and there, who cares about the particularities of how numbers are written? For computations it could be relevant but not really for deep number theoretic results. So not only is your question probably not in the mainstream literature, it is also not likely to get there. 
