Booth Id:
MATH018
Category:
Mathematics
Year:
2026
Finalist Names:
Al-Shafee, Hassan (School: Normandy High School)
Abstract:
This research establishes a transformative paradigm in analytic number theory by replacing stochastic prime gap analysis with a deterministic framework to enhance cryptographic efficiency. We demonstrate that the spacing between consecutive primes, gs = ps - ps-1, is strictly governed by the global analytic requirements of the Bernoulli-Zeta identity.
The procedure involved isolating a conserved Global Residual Constant K(1) through the Euler product and introducing the Pointwise Isolation Principle. This geometric anchor yields a deterministic bound of gs <= 0.1066 * sqrt(ps-1 * ln ps-1), which unconditionally improves the current p^0.525 record (BHP, 2001) and is tighter than what the Riemann Hypothesis suggests.
Numerical verification was conducted across a regime of 50,847,536 primes, confirming zero violations for s >= 3387 (the threshold) and a worst-case gap-to-bound ratio of 0.9755. This work bridges the global geometry of the Zeta function and local prime distribution to offer profound implications for cybersecurity:
REDUCES SEARCH LATENCY: Accelerates prime generation for RSA/ECC by replacing probabilistic search with a bounded window.
ENHANCES EFFICIENCY: Collapses 1024-bit search windows by a factor of 1.7908 x 10^7 and 2048-bit search windows by a factor of 6.4402 x 10^14 compared to traditional models.
STRUCTURAL PREDICTABILITY: Proves prime randomness is a structural illusion forced by the transcendental requirements of the Basel identity, Zeta(2) = pi^2/6.
Ultimately, this framework provides a new deterministic standard for the architecture of digital security.
Awards Won: