§ 1. Some properties of Mersenne numbers § 2. Theorem of E. Lucas and D. H. Lehmer § 3. How the greatest of the known prime numbers have been found § 4. Prime divisors of Fermat numbers § 5. A necessary and sufficient condition for a Fermat number to be a prime. § 6. How the fact that number $2^(2^{1945}) + 1$ is divisible by $5*2^{1947}+1$ was discovered