The fact that each prime number (greater than $9$) ends with one of the four digits $1,3,7,9$, allows us to classify the tens in which the primes are found according to which of these four digits, added to the tens, yields to a prime number.
For example, for the first ten we have $1 \rightarrow \{1,3,7,9\}$. In fact, $10+1$, $10+3$, $10+7$ and $10+9$ are all primes. Conversely, for the twentieth ten the association reads $20 \rightarrow \{\}$, since there are no primes between $200$ and $209$.
It is easy to see that each ten is associated to one (and only one) group of symbols, chosen among the following $16$ distinct alternatives: $\{\}$, $\{1\}$, $\{3\}$, $\{7\}$, $\{9\}$, $\{1,3\}$, $\{1,7\}$, $\{1,9\}$, $\{3,7\}$, $\{3,9\}$, $\{7,9\}$, $\{1,3,7\}$, $\{1,3,9\}$, $\{1,7,9\}$, $\{3,7,9\}$, $\{1,3,7,9\}$.
For the sake of simplicity, we can identify each of these $16$ distinct groups of symbols with a single symbol, or with a single color, as illustrated below:
Arranging the tens in a Pascal's triangle, we find (*) the following structure (omitting the first ten on the edges of the triangle):
(*) I hope that my code is correct! It would be great if someone, more skilled than me, could confirm the emergence of such structure. In case you are interested in double-checking, please have a look to this post for details.
However, assuming that I did not mess up too much with the code, my conjecture is that
For very big tens, there cannot be colored squares other than on the outer diagonal of the triangle.
In other words, big primes $p$ must be in the form $p=10^{\binom{n}{k}}+1$, or $p=10^{\binom{n}{k}}+3$, or $p=10^{\binom{n}{k}}+7$, or $p=10^{\binom{n}{k}}+9$, and $k=1$. Clearly, a weaker version of such conjecture is that $k$ can oscillate among some little integer $2,3,4,5\ldots$ (which ones?).
This is probably an obvious result for the experts (I apologize, in case), nevertheless I would be glad to understand the connections between this approach and others, and also to know if there is some technique to attack such problem.
Sorry for possible naivety, and thank you very much for your comments and suggestions!
EDIT: The conjecture was based on the assumption that the plot of the triangle was correct. But, as Ross has shown, that was not the case. An improved version of the code producing that plot, in fact, results in this picture:
Best Answer
The large white area in the middle claims that if $N$ is any number in that area of the triangle then none of $10N+1, 10N+3, 10N+7, 10N+9$ are prime. The claim is false as ${37 \choose 13}=3562467300$ and $35624673001$ is prime.