Algebraic Characterization of Prime Numbers Through Binomial Coefficient Divisibility

This thesis presents a rigorous investigation into the characterization of prime numbers through the divisibility properties of binomial coefficients. It demonstrates that a positive integer is prime if and only if it divides all the interior binomial coefficients in its corresponding row of Pascal's Triangle. Using concepts from p-adic valuation theory and Legendre's method, the study provides complete and constructive proofs with clear step-by-step reasoning. The work introduces the original concept of primality signatures, which describe the unique pattern of prime numbers that appear as binomial coefficients in each row of Pascal's Triangle. It is shown that for every prime number greater than or equal to five, its primality signature contains only itself. Extensive computational tests up to ten thousand confirm the theoretical predictions with perfect accuracy, successfully identifying all prime and composite numbers, including every Carmichael number in that range. Finally, three algorithms are presented with detailed analyses of their computational complexity and real performance. The research also establishes strong links with several classical results in number theory, including the theorems of Lucas, Kummer, Wilson, and Wolstenholme, and provides a complete open-source implementation for verification and further exploration.