# Are there "Petrov primes" for a > 5000 ?

Discussion in 'Number Theory' started by olgaSamara, Aug 19, 2021.

1. ### olgaSamara

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I apologize for my English!!

The other day, the Russian futurologist and mathematical engineer I. B. Petrov (Ivan Borisovich, he has a namesake - Igor Borisovich) published an article "[Petrov I. B. "Quasi-exponential primes", SI, 2021] (only in Russian, for now), where he proposed to consider a sequence of numbers of the form a^a-a-1, where a > 2, a is any natural number, to search for large primes.

Petrov himself defined prime numbers for the exponent a < 5000. Moreover, the last number found by him for which a^a-a-1 is a prime a = 1379. But then it's interesting. There were people who ran Petrov's formula to a = 10,000 (or 100,000 - I don't remember) and didn't find a single prime number. They used the Miller-Rabin algorithm for only a few rounds (which seems to me undoubtedly small for numbers with an index a > 1000, the probability is not large), but the question arose, are there any prime numbers after a = 1379 ?

In general, I don't see any reason why there can't be such numbers. Although the topic has been raised on the web - I honestly think it's stupid to discuss the finiteness of prime numbers for any sequence of natural numbers. The practical side of the issue is another matter. Why did Petrov even suggest such a strange sequence? And he didn't explain it anywhere...

I forgot how such numbers are called (by type a^a), but somewhere it was written about them. Among the features they have a sharp increase in the bit depth of the final number. And perhaps that's all! Unless only to not be petty and immediately break the records of Mersenne and GIMPS numbers. By the way, if we convert the Petrov numbers, we get: a (a^{a-1}-1)-1.

It is easier then to consider: a^{a-1}-1. Also an interesting sequence...

olgaSamara, Aug 19, 2021

2. ### MathLover1

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I dealing with math for last 50 years, but I have never heard about Igor Borisovich Petrov. I would like to do some research on this topic before answering your question. By the way, your English is good. What is your native language?

MathLover1, Aug 19, 2021
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3. ### olgaSamara

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My native language is Russian. But I communicate a little in English.

But I just wrote that the author is "Quasi-exponential primes", SI, 2021 Ivan (non Igor) Borisovich Petrov. He is not a scientist - just an amateur, but just a fairly well-known online futurist, science fiction writer and author of unusual ideas (from outright nonsense, to very interesting ideas, for example, a multidimensional numeral system). Unfortunately, there are several authors with the initials "Petrov I. B." and all of them are related to mathematics in one way or another.))

Last edited: Aug 19, 2021
olgaSamara, Aug 19, 2021
4. ### olgaSamara

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Just in case, I attach the author's publication (is allowed free distribution). But unfortunately it is only in Russian. I wrote the author a proposal to translate it.

#### Attached Files:

• ###### petrov-ivan-kvazieksponencialnye-prostye-chisla-si-358249.pdf
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olgaSamara, Aug 19, 2021
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5. ### MathLover1

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nice to meet you
thank you for the link, I did learn Russian and it will not be a problem to understand

MathLover1, Aug 19, 2021
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6. ### nycmathguy

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nycmathguy, Aug 19, 2021
7. ### MathLover1

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yes, I learn it back in my country

MathLover1, Aug 19, 2021
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8. ### nycmathguy

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I can read, write and speak Spanish. I was born in the Dominican Republic but raised in NYC.

nycmathguy, Aug 20, 2021
9. ### MathLover1

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my granddaughter learning Spanish
my native language is Serbo/Croatian ( Serbian and Croatian)
I also speak German

MathLover1, Aug 20, 2021
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10. ### nycmathguy

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Do you know female guitarist Ana Vidovic? She is from the Croatian part of the world.

nycmathguy, Aug 20, 2021
11. ### MathLover1

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No, she is born 8 November 1980 (age 40).
I know her father Ljubomir Vidovic, a musician from Karlovac, Croatia and the father of top and world-renowned musicians Viktor, Ana and Silvije Vidovic.

MathLover1, Aug 20, 2021
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12. ### nycmathguy

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Thanks for the video clips.

nycmathguy, Aug 20, 2021
13. ### syndixxx

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Hello! It was I who first drew attention to Petrov's article on the Web. https://oeis.org/A065798. It turns out that this is a well-known sequence. Only there the check was up to a = 3000, apparently. Petrov finished up to 5000, I finished up to 10000. something like that))

syndixxx, Aug 20, 2021

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