The Baillie-PSW Primality Test - Finding Prime Numbers That Look Like Flags

Explore the fascinating Baillie-PSW primality test through the lens of the United States of America prime, a remarkable 5,382-digit prime number discovered by Vadim Ponomarenko that visually resembles the American flag. Learn how this powerful probabilistic primality test works and discover the mathematical concepts that make it so effective at identifying prime numbers. Master Fermat's Little Theorem and understand its role in primality testing, then delve into the base 2 strong probable prime test as the first component of the Baillie-PSW method. Examine the Legendre and Jacobi symbols, essential tools for determining quadratic residues, and explore how these symbols contribute to the test's reliability. Investigate Lucas sequences and their properties, understanding how they form the second crucial component of the Baillie-PSW test. Analyze why certain composite numbers like 119 can fool individual tests but struggle to deceive the combined Baillie-PSW approach. Gain practical knowledge of how to implement this test yourself to potentially discover prime numbers that visually represent your own country's flag, while appreciating the mathematical elegance behind one of the most reliable primality tests available.