Bruno Glecer

In 2017 my friend Juli asked himself this question: Is it possible to generate the prime numbers by starting with a constant (\(f_1=2. \cdots \)) that when rounded down gives the first prime (\( \lfloor f_1 \rfloor\ = p_1 = 2 \)), and then by multiplying this prime by some other real number encoded in \(f_1\), we can reach a value \(f_2=3. \cdots \) that rounds to the next prime (3), and so on. It turns out that it's possible, using this recursive formula:

\[ \large f_{n+1} = \lfloor f_n \rfloor \left( 1 + \{f_n\} \right) ~~~~~~~~~~ \lfloor f_n \rfloor = p_n \]

Where \( \{f_n\} \) is the mantisa of \( f_n \)

The starting value to generate all primes, \( f_n \) , also refered to as \( \lambda \) can be calculatedas:

\[ \large \lambda = \sum_{k=1}^{\infty} \frac{p_k -1}{\prod_{i=1}^{k-1} p_i} = \frac{2 - 1}{1} + \frac{3 - 1}{2} + \frac{5 - 1}{2 \cdot 3} + \frac{7 - 1}{2 \cdot 3 \cdot 5} + \dots \]

This value is approximatley equal to 2.920050977316134712. You can download 1 million digits here.

Together with Juli, Dylan and Massi, we where able to prove some properties of this recursive formula and associated constant, including that it's irrational. Mathematician and math communicator James Grime helped us publish this result in the American Mathematical Monthly. The preprint is available on arxiv.

The YouTube channel Numberfile for which James makes regular appearences made a video on the topic!

And a follow-up video: