Search for prime numbers - "metaremultions" (programs and algorithms)?

I was interested in the topic of searching for the largest prime numbers, the so-called “metaremultions” (or multi re-near-repdigits). As I understand it, such a sequence of prime numbers has not yet been registered. That is, it has not been studied.

I first learned about them from the publication of Petrov I.B.: “Petrov I.B. METAREMULTION (general superficial numerical study of an interesting prime number)” Author’s article, self-publishing, 2023, 5 pp Here and further I will use quotes from this article by Petrov for the sake of convenience. We are talking about prime numbers of the form:



The peculiarity of these numbers is that if you discard the blocks “277...7” from behind one after another, then the remaining numbers will also be prime. I wrote it crookedly, but it will be clear like this:



In other words, if we take the largest known prime number, put 2 at the end and add n-number of digits 7, then sooner or later we will find a prime number. And then repeat...

The beauty may be that there are an infinite number of such numbers possible. The author himself states:



Here it is very crookedly formulated; by “metaremultion” one should apparently mean prime numbers. And then the meaning comes down to the phrase I wrote above about the infinity of the series of such prime numbers. I didn’t understand about the increase in the number of digits 7 in distant blocks, but on one forum they told me that this is most likely the case (why??). The author, as usual, is silent (this is generally his “trick”, I’m not surprised).

Now the actual question. What is the best way to check the simplicity of these numbers using any programs or algorithms? The numbers are not simple, they are difficult to write down in a simple formula (like the Mersenne numbers), but I tried it a little later.

I don’t really want to write my own utility (in terms of the algorithm). There are LLR and Prime95, but I understand that they only test certain prime numbers. Prime95 - Mersenne numbers, did not understand LLR. You can, of course, create some simple algorithm in C, but its error will be very large, that is, it is unreliable. What do you advise?

Yes, of course, I understand that for large prime numbers, roughly speaking, data centers are needed, but for now we are talking about checking the available numbers. I ran these numbers through some home-made utility to check numbers of approximately 5000 characters, after which the utility said “glitch” and ordered not to continue the search. But I am not sure of the correctness of her results.