Matomäki's Short Intervals and the Prime Highway: Exponential Improvement
Abstract
We present an exponential improvement to Matomäki's "primes in short intervals" result using the Prime Highway. Matomäki (2024) proved, without L-functions, that primes exist in intervals (x - x^0.975, x] for all large x. The Highway's deterministic structure—where K consecutive primes modulo a primorial uniquely determine the next gap with zero collisions—yields intervals of length O(log²x) containing primes. Key results:- At x = 10^15, Highway intervals are 315 billion times shorter than Matomäki's- Verified to 10^9: 50.8 million primes, zero violations, zero collisions- Maximum gap ratio: 0.74 (well below Granville bound of 1.12)- Highway parameters: K=20, mod 210 The improvement is conditional on Highway determinism persisting to infinity. *v2 fixes typo: improvement factor is 315 billion, not trillion.
Fields of Study & Subjects
Analytic Number Theory Research
Limits and Structures in Graph Theory
Advanced Mathematical Theories and Applications
Publication details
- Journal / source
- Zenodo (CERN European Organization for Nuclear Research)
- Publication date
- 20 January 2026
- Document type
- Preprint