This project will advance fundamental mathematical knowledge by developing new tools to understand the deep connections between addition and multiplication within number systems. It bridges two major areas of pure mathematics: additive combinatorics, which studies patterns formed by adding numbers, and multiplicative number theory, which focuses on properties related to multiplying numbers, especially prime numbers. The PI will create novel techniques to solve complex problems where these additive and multiplicative structures interact. Findings of the project will benefit fields beyond mathematics, particularly cybersecurity, where improved understanding of number-theoretic functions could lead to stronger encryption methods protecting digital information. The combinatorial methods developed will also have the potential to enhance the reliability of data transmission and storage systems. Additionally, the project will train undergraduate and graduate students and contribute to educational outreach through math competitions and K-12 programs to inspire future scientists. The project will tackle challenging open questions involving prime numbers, smooth numbers, and general multiplicative functions. More specifically, the PI will develop new methods in additive combinatorics that can be applied to questions involving primes, such as finding narrow progressions in primes and finding solutions to linear equations in subsets of primes. The PI will analyze the behavior of mult