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Perfect numbers are natural numbers that equal the sum of their positive proper divisors, which are divisors excluding the number itself. So, 6 is a perfect number because the proper divisors of 6 are 1, 2, and 3, and 1 + 2 + 3 = 6. [2] [4]
The first known European mention of the fifth perfect number is a manuscript written between 1456 and 1461 by an unknown mathematician. [10] In 1588, the Italian mathematician Pietro Cataldi identified the sixth (8,589,869,056) and the seventh (137,438,691,328) perfect numbers, and also proved that every perfect number obtained from Euclid's ...
An equally tempered perfect fifth, defined as 700 cents, is about two cents narrower than a just perfect fifth, which is approximately 701.955 cents. Kepler explored musical tuning in terms of integer ratios, and defined a "lower imperfect fifth" as a 40:27 pitch ratio, and a "greater imperfect fifth" as a 243:160 pitch ratio. [ 13 ]
We know that the sequence of perfect numbers starts 6, 28, 496, 8128, 33550336. But some sources say 2096128 rather than 33550336 is the fifth perfect number.
The fifth, M 13 = 8191, ... a Mersenne number cannot be a perfect power. That is, ... A Mersenne–Fermat number is defined as ...
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No odd perfect numbers are known; hence, all known perfect numbers are triangular. For example, the third triangular number is (3 × 2 =) 6, the seventh is (7 × 4 =) 28, the 31st is (31 × 16 =) 496, and the 127th is (127 × 64 =) 8128. The final digit of a triangular number is 0, 1, 3, 5, 6, or 8, and thus such numbers never end in 2, 4, 7, or 9.
A manuscript of 1456 gave the fifth perfect number: 33,550,336. Gradually mathematicians found further perfect numbers (which are very rare). In 1652 the Polish polymath Jan Brożek noted that there was no perfect number between 10 4 and 10 7 .