Prime Successor Irreducibility: New Conjectures on Computational Limits
A new preprint on arXiv (2605.12504) introduces conjectures and theorems on the computational irreducibility of prime successor generation. The authors propose that no general algorithm can compute the next prime after a given prime p substantially faster than sequential trial division, except on sparse inputs. Three formal frameworks are developed: PSI-T (Turing machine complexity model) asserts lower bounds on running time; PSI-K (Kolmogorov complexity formulation) proves unconditional incompressibility of typical prime gaps for fixed c<1 using sieve bounds; PSI-W (weakness-based sparse-set anti-concentration) shows no small menu of algorithms can efficiently cover all primes. The work bridges number theory and computational complexity.
Key facts
- Preprint arXiv:2605.12504 released on arXiv
- Announce type: cross
- Develops Prime Successor Irreducibility (PSI) conjectures
- PSI-T: Turing machine complexity model for lower bounds
- PSI-K: Kolmogorov complexity formulation proved unconditionally for c<1
- PSI-W: weakness-based sparse-set anti-concentration
- Uses standard sieve bounds for PSI-K proof
- No general algorithm can compute next prime faster than sequential testing except on sparse sets
Entities
Institutions
- arXiv