On the Eight Levels theorem and applications towards Lucas-Lehmer primality test for Mersenne primes, I

Lucas-Lehmer test is the current standard algorithm used for testing the primality of Mersenne numbers, but it may have limitations in terms of its efficiency and accuracy. Developing new algorithms or improving upon existing ones could potentially improve the search for Mersenne primes and the understanding of the distribution of Mersenne primes and composites. The development of new versions of the primality test for Mersenne numbers could help to speed up the search for new Mersenne primes by improving the efficiency of the algorithm. This could potentially lead to the discovery of new Mersenne primes that were previously beyond the reach of current computational resources. The current paper proves what the author called the Eight Levels Theorem and then highlights and proves three new different versions for Lucas-Lehmer primality test for Mersenne primes and also gives a new criterion for Mersenne compositeness.
33 pages, 4 tables, 3 new versions for the Lucas-Lehmer primality test, 4 new combinatorial identities. arXiv admin note: substantial text overlap with arXiv:2108.13792