Mersenne primes and perfect numbers are two deeply interlinked types of natural numbers in number theory. Mersenne primes, named after the friar Marin Mersenne, are prime numbers that can be expressed as 2p − 1 for some positive integer p. For example, 3 is a Mersenne prime as it is a prime number and is expressible as 22 − 1. The answer is 8. This is because Fermat's little theorem tells us that 1911 ≡ 19 (mod 11). We can further calculate that 19 = 1 × 11 + 8, which means that 19 (mod 11) ≡ 8, as claimed. When using Fermat's little theorem, always remember to verify the assumptions! We could use this theorem here since 11 is a prime number. The first known attempt at establishing an offworld presence occurred twelve million years ago, following the unification of the Thirteen Primes. Given the violent period of war that had preceded their reign, "Onyx Prime" proposed turning the innate fury of all Cybertronians outward in a galactic expansion, the Pax Cybertronia.

The paper by Kaoru Motose starts as follows: "Let q be a prime divisor of a Mersenne number 2^p-1 where p is prime. Then p is the order of 2 (mod q). Thus p is a divisor of q - 1 and q > p. This shows that there exist infinitely many prime numbers." - Pieter Moree, Oct 14 2004

Step 1: First create a list of numbers from 2 to 100 as shown above. We leave the number 1 because all prime numbers are more than 1. Step 2: We start from the first number 2 in the list. We cross out every number which is a multiple of 2 except 2. For example, we cross 4, 6, 8, 10, 12, 14, 16, and so on up to 100.

Q.1: From the list of prime numbers 1 to 1000 given above, find if 825 is a prime number or not? Solution: The list of prime numbers from 1 to 1000 does not include 825 as a prime number. It is a composite number since it has more than two factors. We can confirm this by prime factorisation of 825 also. Prime Factorization of 825 = 3 1 × 5 2

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what are the 13 primes