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...
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...
