Number Theory

    

Definitive Proof of Brocard's Conjecture

Authors: Kenneth A. Watanabe

This paper presents a formal proof of Brocard’s Conjecture, which posits that there are at least four prime numbers between the squares of any two consecutive primes pi2 and p_i+12 for i>1. By defining the function π*(n) that approximates the prime counting function π(n), we establish a lower bound for the number of primes in these intervals. Using mathematical induction, we demonstrate that the minimum number of primes in the interval, Δπ*(pi), is consistently greater than or equal to 4 for all pi ≥ 3. The proof is further supported by a rigorous error analysis, bounding the maximum possible deviation between the estimated prime count π*(n) and the actual prime counting function π(n).

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[v1] 2026-04-11 21:48:39

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