Formula for finding prime numbers f(a) = a * dr(a) + 1

Yesterday I saw on twirpx I saw an interesting article "Petrov I.B. Study of the prime number formula f(a) = a * dr(a) + 1".

The article discusses a new approach to the study of prime numbers, based on the numerical roots of numbers. the prime number formula f(a) = a * dr(a) + 1 is studied, where dr(a) is the numerical root of the number a. The study shows that this formula provides a competitive probability of finding prime numbers over a wide range of values of a, appearing to be a potentially important tool for numerical theory.

Essentially, the article is a description of the formula and small data on the statistics of the probability of prime numbers among the results of generating natural numbers by this formula. There is a comparison of this probability for the range up to a = 1,000,000 with known formulas for calculating prime numbers. (I won’t cite it here - everything is in the article). But the research itself is in fact absent there. This is simply a notation and description of the formula. Nevertheless, the topic is interesting.

The formula itself: f(a)=a∗dr(a)+1, where dr(a) is the numerical root of the number a. With a range from 1 to 1,000,000, the probability of prime numbers among the results is 15.79% (this is really true - I was not too lazy to make a program in C). The author gives the Euler polynomial for comparison: n2+n+41 for n from 1 to 1,000,000, the probability of prime numbers among the results is 26.11% (I’ve never heard that this formula is an “Euler polynomial”; in Soviet literature it goes without a name, or am I missing something? But oh well.. . there really is such a formula). There is also a comparison with the formulas of Riesel, Sophie Germain, and Fermat numbers. For a given range, they produce an even lower probability of prime numbers among the results (although for Riesel the author used a fixed n=1, which is a very strong limitation).

In general, Petrov’s formula is very interesting judging by the probability, but I ran it to a = 100,000,000 and there the probability drops to ~12%, which is logical (a larger range is involved in the sample). According to Petrov, the probability of primes among natural numbers is 11.11% - the “normal” distribution of primes among natural numbers. What is generally speculative is that the probability will vary depending on the limitations of the sampling range. If we take the number of prime numbers in the range covered by the formula with a given a, then in fairness the percentage of prime numbers there is less than 11.11%.

It didn’t work out to run the “Euler polynomial” even up to 10,000,000 - it took a very long time on my hardware, but I assume that it’s the same story as with Petrov’s formula.

Something else is interesting here. Essentially, if you make a sample based on odd natural numbers: 2∗a+1 - the probability of prime numbers should increase to Petrov’s formula (in fact, in Petrov’s formula there is no getting rid of even numbers, dr(a) will actually randomly (relative to the series of natural numbers up to 10) produce values from 1 to 9, and most importantly - the step of values in Petrov’s is larger ), but after checking, I saw that Petrov’s formula still gives a higher probability of primes than just the ancient Greek formula for an odd number! Why?

The topic is not so interesting for the formula itself (as it seems to me), but rather for the connection between the numerical root and the distribution of prime numbers. To be honest, I have never seen this approach before. The number root is usually perceived as pampering for entertaining problems... But this idea definitely has potential for study (I’m talking about the connection with primes).