A prime number "Game of Life": can floor($y \cdot p\#$) be prime for all $p$>=2?

TL;DR

The paper investigates whether a floor transformation of the primorial, $\lfloor y\cdot p\#\rfloor$, can be prime for all primes $p\ge2$ by constructing a slowly growing, Mills-like prime-generation family $\{q\}$ tied to $p\#$ via a fixed $y$. It develops a probabilistic, stage-by-stage framework linking prime distribution in primorial-sized intervals to Chebyshev bias, $\theta(p)$, and (conditionally) RH, introducing quantities like $\omega(s)$ and $\psi(p)$ to predict survival and growth. Computational results up to stage 331 show high survival probabilities for multiple branches and reveal an extremely narrow admissible $y$-range suggesting a near-unique infinite run, while Appendix material analyzes variations, descendants, and real-world feasibility of the construction. The work broadens the landscape of prime-generating sequences, connects to classical prime-bias results, and raises open questions about the existence and uniqueness of a universal $y$ yielding infinite primes, as well as potential semi-sequences and broader applications.

Abstract

A new sequence in the spirit of the Mills primes is presented and its properties are investigated.

Figures (6)

• Figure 1: Comparison. With the prime number output becoming more generous, the forecast is increasingly accurate.
• Figure 2: Histogram of the standard deviations from the predicted number of primes at each stage $s = 99…251$. 70.6% of the values are within 1$\sigma$, 92.2% within 2$\sigma$, and all within 3$\sigma$. There are 78 negative deviations vs. 75 positive ones.
• Figure 3: Number of possible values for $y$ (for all we know). From $s = 95$ onward, all primes $q$ emerge from one common ancestor at $s = 43$, namely 1292942159746921794791923187781692727375711831825607985864285936838920812561. Naturally, the branches in the picture are not equally strong. For $s = 331$ ($p = 2221$), the following number of probable primes divide up on the nine given groups: 11477 + 9305 + 64706 + 123386 + 137475 + 101078 + 13648 + 77384 + 271928.
• Figure 4: Alleged number of possibilities up to stage 100.
• Figure 5: The value is growing in the process (e.g. about 52% for $s = 252$), so we can soon expect another threefold split. In fact, the next one appears to be in the upper region of $p = 613$, $s = 112$.