<ul><li>The largest known prime numbers have a specific form of 2^n - 1, known as Mersenne numbers, making them easier to test for primality.</li><li>Lehmer's theorem establishes conditions under which a Mersenne number is prime, providing a practical primality test for Mersenne primes.</li><li>The Lucas-Lehmer test simplifies primality testing by calculating values modulo Mp, significantly reducing computational complexity.</li><li>A comparison between naive and optimized versions of the Lucas-Lehmer test shows significant performance differences, especially with large primes like M521.</li></ul>

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