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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.

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I suspect this is beyond reach. If we could prove there are infinitely many primes $p$ such that $10p+3$ is prime, we'd have it, but this is beyond what's currently known. –  Gerry Myerson Feb 15 '13 at 2:18
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The first thing I thought was to attack this problem (mow an open one) by contradiction: suppose that there are finite brothers numbers. Let $A$ be such set. Now, there is a maximal length of brothers number in $A$, suppose that $\text{length $\{p_i \unrhd p_j\}$}=k$. Now - a problem. Sorry, if it sounds stupid. :) –  Salech Alhasov Feb 15 '13 at 2:53
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@GerryMyerson I agree this seems just out of reach, but we can prove that for infinitely many prime pairs $(p,q)$, the numbers $10p+q$, $100p+q$, $1000p+q$, $10000p+q$ are simultaneously prime. The catch, of course, is that we don't know how to prove this for bounded $q$! –  Erick Wong Feb 15 '13 at 4:32
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1 Answer

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.

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It would be most interesting to find a pair of primes that are brothers in one base but not in another. Among $2$, $3$, $5$, and $7$, I think the brothers in base ten are also brothers in binary, although not always in the same order. –  Todd Wilcox Feb 15 '13 at 4:00
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Well, not all questions dependent on a particular base lie outside of the mainstream. There is the recent breakthrough of Mauduit and Rivat on the distribution of the sums of digits of primes (answering a question posed by no less than Gelfond). So most of the time these seem like useless questions, but once in a while the number theoretic machinery becomes powerful enough to overcome their apparent arbitrariness, and it's truly a sight to behold. –  Erick Wong Feb 15 '13 at 4:12
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