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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 ...
In number theory, a perfect number is a positive integer that is equal to the sum of its positive proper divisors, that is, divisors excluding the number itself. For instance, 6 has proper divisors 1, 2 and 3, and 1 + 2 + 3 = 6, so 6 is a perfect number. The next perfect number is 28, since 1 + 2 + 4 + 7 + 14 = 28.
Euclid gave a formula for (even) "perfect" numbers: N p = 2 p−1 (2 p − 1) where p and 2 p − 1 are prime numbers. [8] Euclid had listed the first four perfect numbers: 6; 28; 496; and 8128. A manuscript of 1456 gave the fifth perfect number: 33,550,336. Gradually mathematicians found further perfect numbers (which are very rare).
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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 perfect number is a natural number that equals the sum of its proper divisors, the numbers that are less than it and divide it evenly (with remainder zero). For instance, the proper divisors of 6 are 1, 2, and 3, which sum to 6, so 6 is perfect. A Mersenne prime is a prime number of the form M p = 2 p − 1, one less than a power of two.
Notably, absent consensus, please do not add articles about individual perfect numbers themselves (such as 6). Pages in category "Perfect numbers" The following 11 pages are in this category, out of 11 total.
Hyphenate all numbers under 100 that need more than one word. For example, $73 is written as “seventy-three,” and the words for $43.50 are “Forty-three and 50/100.”