I am interested in finding a method to determine all four-digit primes (notation $ p = xyzw $) such that (its mirror) $ q = wzyx$ is also a prime in other words, to solve the system with digits $x,y,z,w; \space xw\ne 0$.
$$\begin{cases} 1000x+100y+10z+w = prime\\1000w+100z+10y+x= prime\end{cases}$$
Some examples of solutions are $xyzw=1009, 1031, 1033, 1061, 1091, 1103, 1151, 9041,
9967$.
For two-digit primes there are the four solutions: $$13,17,37\space \text{and}\space79$$ and for three digit-primes (discarded the fifteen trivial solutions $101,131,151, 181, 191, 313, 353, 373, 383, 727, 757,787, 797, 919$ and $929$) there are the fourteen ones:
$$107,113,149,157,167,179,199,337,347,359,389,709, 739\space \text{and}\space769$$
The "method" used to determine this is easy to guess (brute force some people say). I wonder about a method without quotation marks.
We note first that, unlike the case of three-digit primes, in the case we try of four- digit primes there is no trivial solution, because all number of the form $ abba $ is divisible by $11$.
I have these restrictions so far:
$$\begin{cases}1)\space1009\le1000x+100y+10z+w\le9967\\2)\space{\{x,w}\}\subset{\{1,3,7,9}\} \\3)\space x+y+z+w=3m\pm1\space\text{(i.e. not multiple of 3)}\\4)\space (x+z)-(y+w)\ne 11k \space\text{(i.e. not multiple of 11)}\\5) \text{ other criteria for divisibility}\end{cases}$$
Constraint $1)$ comes from the "method" used for two-digit and three-digit primes (i.e. $1009$ is the smallest solution and $9067$ is the greatest one).
Constraint $2)$ allows us to transform the system $$\begin{cases} 1000x+100y+10z+w = prime\\1000w+100z+10y+x= prime\end{cases}$$ of four unknowns $x,y,z,w$ in sixteen systems $$\begin{cases} 1000a+100y+10z+b = prime\\1000b+100z+10y+a= prime\end{cases}$$ of two unknowns $y,z$ with $a,b= 1,3,7,9$
Apply divisibility criteria(constraint $3)$, $4)$ and $5)$ could also perhaps help.
What's needed to have a solution method without quotation marks, I mean without comparing all of the four-digit primes in a table? Is it possible?