Wikipedia:Main Page history/2022 January 7b

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Visualization of 6 as a perfect number
Visualization of 6 as a perfect number

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. Perfect numbers are natural numbers that equal the sum of their positive proper divisors (all divisors excluding the number itself). There is a one-to-one correspondence between the Mersenne primes and the even perfect numbers. This is due to the Euclid–Euler theorem, partially proved by Euclid and completed by Leonhard Euler: even numbers are perfect if and only if they can be expressed in the form 2p − 1 × (2p − 1), where 2p − 1 is a Mersenne prime. (Full list...)

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Flag and seal of Illinois

This historical coat of arms of Illinois is an illustration from State Arms of the Union by Henry Mitchell, published by Louis Prang in 1876. It depicts a bald eagle perched on a rock carrying a shield with the stars and stripes. In the eagle's beak there is a banner with the state motto, "State Sovereignty, National Union."

Illustration credit: Henry Mitchell; restored by Andrew Shiva

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