Abstract

This research investigates the upper bound on the probability of encountering prime numbers within any set of 100 consecutive natural numbers. Specifically, it is shown that this probability cannot exceed 0.25, implying that at most 25 primes exist in such an interval. Using a combination of analytical reasoning based on the prime number theorem and a mathematical induction framework, we rigorously establish this result. The study further examines the transition between overlap-ping consecutive intervals, accounting for the inclusion or exclusion of boundary numbers and confirms the invariance of the upper bound. This result provides a deeper understanding of the density of primes in finite intervals and offers insights into their distribution properties.

Keywords: Prime number theorem; Prime counting function; Mathematical induction; Probability; Natural numbers