The number $137$ is a prime number.One of the permutation of $137$ is $173$, which is a prime number. The summation of $137$ digits is $11$ which is again a prime number!
My question: Is there a name for this type of numbers?
There are some other Properties for $137$ in Wikipedia. I found another Properties such that $137=2^7+2^3+1$ or the only way to write the number $137$ as a summation of two square numbers is $137=4^2+11^2$.
thanks for your advice and suggestions.
Edit:After reading comments and answers, I want to suggest a definition for such numbers like $137$. At first, I want to excuse me for this Venture that I want to make a definition. In the following definition, we suppose that the number one is a prime number.
Definition: We call a prime number like $p$, $\color{red}{\text{Sum and Permutation Prime}}$ of order $i\geq 1$ or in abbreviation SPP of order $i\geq 1$ if and only if
The summation of digits of $p$ be prime.
There is a number like $i\geq 1$, where all permutation of the $p$ of length $i$ be prime numbers.
If the last condition holds for $1\leq i \leq L(p)$, where $L(p)$ is the length of prime number $p$, we call the prime number $p$, SPP with full order. The last condition is because of one of the answers that said all permutation of $137$ of length two are prime numbers.
The number $113$ is a SPP numbers with full order but $137$ is not SPP with full order because $371$,$713$ and $731$ are not prime numbers. In fact $137$ is a SPP number of orders $1$ and $2$.
The SPP numbers of order ($\geq2$), do not have prime numbers $2$ and $5$ in their digits and just are consist of numbers $1$, $3$, $7$ and $9$.
Thanks again for all nice comments and beautiful answers.
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$\begingroup$There is no name for that property. If Sloane's OEIS listed these numbers, it would probably just call them "Prime numbers such that the digits can be permuted to form another prime number and the sum of digits is also prime."
The most relevant search result I could find is "Numbers $n$ such that $n$ and the digital sums of $n$ and $n$-th prime are primes." But it doesn't say anything about permutations of digits.
The OEIS does list numbers called "absolute primes" or "permutable primes," but to be considered one of those, every permutation of the base $10$ digits must give another prime. $137$ fails this test because $371 = 7 \times 53$.
I suggest you figure out what the first hundred or so prime numbers of this sort are and ask the Editors of Sloane's OEIS to add a listing for it.
$\endgroup$ 1 $\begingroup$Another property: Any combination of 2 digits from 137 is also prime (13,17,31,37,71,73).
$\endgroup$ 1 $\begingroup$As I know there is no name for a such property.
For the first sight it may be very interesting how special is the number 137. But there a many other numbers with many other interesting properties, so from general point of view the number 137 is only a drop in the ocean.
Such numbers are chiefly for non-mathematician who think about some mystical message related to them.
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