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So, 6 is a perfect number because the proper divisors of 6 are 1, 2, and 3, and 1 + 2 + 3 = 6. [2] [4] Euclid proved c. 300 BCE that every prime expressed as M p = 2 p − 1 has a corresponding perfect number M p × (M p +1)/2 = 2 p − 1 × (2 p − 1). For example, the Mersenne prime 2 2 − 1 = 3 leads to the corresponding perfect number 2 2 ...
This works with all perfect numbers () with odd prime p and, in fact, with all numbers of the form () for odd integer (not necessarily prime) m. Owing to their form, 2 p − 1 ( 2 p − 1 ) , {\displaystyle 2^{p-1}(2^{p}-1),} every even perfect number is represented in binary form as p ones followed by p − 1 zeros; for example:
Pages in category "Perfect numbers" The following 11 pages are in this category, out of 11 total. This list may not reflect recent changes. ...
Toggle List of Mersenne primes and perfect numbers subsection. 1.1 Comments by RunningTiger123. 1.2 Accessibility review (MOS:DTAB) 1.3 Image review — Pass.
This is a list of articles about prime numbers. A prime number (or prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. By Euclid's theorem, there are an infinite number of prime numbers. Subsets of the prime numbers may be generated with various formulas for primes.
Such a number is algebraic and can be expressed as the sum of a rational number and the square root of a rational number. Constructible number: A number representing a length that can be constructed using a compass and straightedge. Constructible numbers form a subfield of the field of algebraic numbers, and include the quadratic surds.
A list of articles about numbers (not about numerals). Topics include powers of ten, notable integers, prime and cardinal numbers, and the myriad system.
The following table lists the progression of the largest known prime number in ascending order. [4] Here M p = 2 p − 1 is the Mersenne number with exponent p, where p is a prime number. The longest record-holder known was M 19 = 524,287, which was the largest known prime for 144 years. No records are known prior to 1456. [citation needed]