Is Petrov’s “metaremultion” finite?

[syndixxx]

I am sorry for my English.

This is a repost from one forum member from another mathematical forum. I'll try to translate it into English (as best I can). [this is a text from another person on the forum]:

Since Petrov is being quoted here, he also has a much more interesting article devoted to prime numbers: “Petrov I.B. METAREMULTION (general superficial numerical study of an interesting prime number)” Author’s article, self-publishing, 2023, 5 pp. (attached to topic). [I read it, but, alas, it is only in Russian (you can translate it through a translator)].

Actually, the author there cites the so-called metaremultion or multi re-near-repdigits:

2777277772777777277777777777777777772777777777777777777777777777777
7777777777777777777777777777777777777777777777777777777777777777777
7777777777777777777777777777777777777777777777777777777777277777777
7777777777777777777777777777777777777777777777777777777777777777777
7777777777777777777777777777777777777777777777777777777777777777777
7777777777777777777777777777777777777777777777777777777777777777777
77777777777777777777777777777777777777777777777777777

This is a number with a cyclic digital recording of blocks 277...7, that is, the number is formed by adding block 277...7 to the original (2777), and the number that appears when adding such a block is also simple. For the given number it is:

• 2777
• 277727777
• 2777277772777777
• 277727777277777727777777777777777777
• 27772777727777772777777777777777777727777777777777777777777777777777777777777
77777777777777777777777777777777777777777777777777777777777777777777777777777
77777777777777777777777777777777777777
• 27772777727777772777777777777777777727777777777777777777777777777777777777777
77777777777777777777777777777777777777777777777777777777777777777777777777777
77777777777777777777777777777777777777277777777777777777777777777777777777777
77777777777777777777777777777777777777777777777777777777777777777777777777777
77777777777777777777777777777777777777777777777777777777777777777777777777777
7777777777777777777777777777777777777777777777777777777777777777777777

Each one is simple. This is actually very interesting!! The author of the article provides a small study and for these numbers displays a sequence in increasing order of digit capacity (you can see the formula in the article). Yes, of course, for the next number this sequence is not correct, since its subsequent term is equal to ~196, which is less than the digit capacity of the last number found by the author. But, if there are no such metaremultion numbers at all, then their entire sequence fits into a certain formula! This suggests that at least some of the prime numbers have an absolutely definite (not random) distribution.Unfortunately, I do not have sufficient programming skills to test such “metaremultions” beyond the number that the author stopped at. And checking them using standard mathematical utilities is already quite difficult due to the large digit of numbers. We need an algorithm to quickly check for simplicity. But for some reason it seems to me that this is a finite sequence of prime numbers and the author has actually already found them all. It would be great because there is a sequence of bits.

Yes, Petrov gives the following hypothesis:

Hypothesis: There is such a large prime numbermetaremultion, which contains infinitethe number of metaremultions of lower orders, with the last blockrepeating number 7, each subsequent such number (from the smallestto more) will be significantly larger than a similar blockprevious one.

But I don't think it's true. In fact, all series of prime numbers driven into the framework of complex structures (such as metaremultion) are finite. But I could be wrong, in general the topic is interesting.

[I am writing from myself]:
The topic seemed quite interesting to me. But the problem is that it is not so easy to factor large metaremultions. I'm trying to write a program to test a number for primality by adding "7" to the last one from the article and checking. So far I have added/checked about ~ 5000 “7” added to the last number - there are no simple ones. But the accuracy of the program algorithm is very low.

 

[syndixxx]

P.S.: by the way, I searched among the near-repdigits sequences, but I didn’t find anything in this interpretation (that is, exactly this formation of the number). And at the same time, the numbers are actually interesting.

 

[conway]

Forgive me...you will not like my interpretation.

You cannot name a finite number that is NOT composed of infinite/infinitesimal numbers. The number one is finite, but it is composed of infinitesimals. "You are what you are composed of". This is a matter of perspective. That which is finite is only so, from a given perspective. So to that which is infinite. So then it is that which is finite that is not real, only a perspective. It is that which is infinite that is real. All things are infinite, from the right perspective.

To be fair you might ask...is this or that finite, from my perspective. But then again consider the number one.

 

[syndixxx]

conway, Thank you for your reply! I agree with you.

I think that the question was initially posed incorrectly (I’m not the author of the topic, one might say). Petrov's metaremultion is essentially a sequence of prime numbers with a special structure. The question is rather about whether such a sequence of prime numbers can be infinite? That is, it is possible that it will not be possible to find the next metaremultion (that is, the next such prime number) by adding 2 and a set of repeating 7s at the end of the last number.

But in fact, I found, I still see no reason to think that there are no such prime numbers larger than Petrov. The sequence of metaremultions is infinite, why should there be a finite number of prime numbers in it?

In theory it should be a prime number:

27772777727777772777777777777777777727777777777777777777777777777777777777777
77777777777777777777777777777777777777777777777777777777777777777777777777777
77777777777777777777777777777777777777277777777777777777777777777777777777777
77777777777777777777777777777777777777777777777777777777777777777777777777777
77777777777777777777777777777777777777777777777777777777777777777777777777777
7777777777777777777777777777777777777777777777777777777777777777777777277...7

And then the next number:

27772777727777772777777777777777777727777777777777777777777777777777777777777
77777777777777777777777777777777777777777777777777777777777777777777777777777
77777777777777777777777777777777777777277777777777777777777777777777777777777
77777777777777777777777777777777777777777777777777777777777777777777777777777
77777777777777777777777777777777777777777777777777777777777777777777777777777
7777777777777777777777777777777777777777777777777777777777777777777777277...7277...7

And so on...

 

[syndixxx]

The only thing Petrov says is that the number of 7 in the space of blocks 277...7 will increase (I don’t understand why he wrote that??! What is the reason that 7 will be greater?). That is, {the largest metaremultion-number}277...(a lot of 7)...7277...(very a lot of 7)...7

 

[conway]

I am the resident (new) philosopher on this forum. As such my replies often fall short. But to give you a purely mathematical response I would say this. It is inherent in the game you, or he "set" up. Or in the question that you ask. That is, "how many 7's are next", and "based on what rules". Consider the nature of exponents. It is a bit like that, in my opinion. So say you change the rules...say only "one" more seven is added to each repeating block, and there is only the original set (27777...),....yet still his hypothesis remains true. Given a large enough "block", and "time"... 7 will be infinite. As again...infinite is all that is real...it is finite that is "fake" or an illusion based on our perspective of the given "block" of numbers.

This is a case of exponential's. Or rather the self inherent (axiomatic) characteristic found in exponents...is also found here. Though they are not exponents.

 

[syndixxx]

Thank you for your reply! I agree with you on this too. I wrote a message a little higher, but for now it is being moderated (for some reason).

 

[conway]

Thank you for your reply! I agree with you on this too. I wrote a message a little higher, but for now it is being moderated (for some reason).

It is a great blessing to be of service to others!