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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 Mersenne prime 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.
284 can be written as a sum of exactly 4 nonzero perfect squares. [3] 284 is a nontotient number which are numbers where phi(x) equaling that number has no solution. [4] 284 is a number that is the nth prime plus n. It is the 51st prime number plus 51. [5]
Furthermore, 114,055,847 is the largest exponent below which all other prime exponents have been tested at least once, so all Mersenne numbers below the 51st (M 82589933) have been tested. [34] ^ ‡ The number M 136279841 has 41,024,320 decimal digits. To help visualize the size of this number, if it were to be saved to disk, the resulting ...
With the exception of 1, a Mersenne number cannot be a perfect power. That is, and in accordance with Mihăilescu's theorem, the equation 2 m − 1 = n k has no solutions where m, n, and k are integers with m > 1 and k > 1. The Mersenne number sequence is a member of the family of Lucas sequences. It is U n (3, 2).
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The following table lists the progression of the largest known prime number in ascending order. [3] 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]
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