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2. List of Mersenne primes and perfect numbers - Wikipedia

   en.wikipedia.org/wiki/List_of_Mersenne_primes...

   [7] [8] [9] It is widely believed, [10] but not proven, that no odd perfect numbers exist; numerous restrictive conditions have been proven, [10] including a lower bound of 10 1500. [11] The following is a list of all 52 currently known (as of January 2025) Mersenne primes and corresponding perfect numbers, along with their exponents p.

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   Sometimes there are 2 or 3 subjects which rotate between student bodies throughout the year. For example, the 8A students [10] might take Art in the first half of the year and Music in the second half. Off-timetable lessons: [11] sometimes an occasional lesson is scheduled "off the timetable" meaning before school, after school, or during lunch ...

4. Perfect number - Wikipedia

   en.wikipedia.org/wiki/Perfect_number

   In about 300 BC Euclid showed that if 2 p − 1 is prime then 2 p−1 (2 p − 1) is perfect. The first four perfect numbers were the only ones known to early Greek mathematics, and the mathematician Nicomachus noted 8128 as early as around AD 100. [3]

5. Euclid–Euler theorem - Wikipedia

   en.wikipedia.org/wiki/Euclid–Euler_theorem

   The Euclid–Euler theorem states that an even natural number is perfect if and only if it has the form 2 p−1 M p, where M p is a Mersenne prime. [1] The perfect number 6 comes from p = 2 in this way, as 2 2−1 M 2 = 2 × 3 = 6, and the Mersenne prime 7 corresponds in the same way to the perfect number 28.

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7. List of prime numbers - Wikipedia

   en.wikipedia.org/wiki/List_of_prime_numbers

   All prime numbers from 31 to 6,469,693,189 for free download. Lists of Primes at the Prime Pages. The Nth Prime Page Nth prime through n=10^12, pi(x) through x=3*10^13, Random primes in same range. Interface to a list of the first 98 million primes (primes less than 2,000,000,000) Weisstein, Eric W. "Prime Number Sequences". MathWorld.