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Many of the properties proved about odd perfect numbers also apply to Descartes numbers, and Pace Nielsen has suggested that sufficient study of those numbers may lead to a proof that no odd perfect numbers exist. All perfect numbers are also Ore's harmonic numbers, and it has been conjectured as well that there are no odd Ore's harmonic numbers other than 1. More recently, Carl Pomerance has presented a heuristic argument suggesting that indeed no odd perfect number should exist. there are any odd perfect numbers is a most difficult question". In 1496, Jacques Lefèvre stated that Euclid's rule gives all perfect numbers, thus implying that no odd perfect number exists. It is unknown whether any odd perfect numbers exist, though various results have been obtained. Thus every even perfect number is a pernicious number.Įvery even perfect number is also a practical number (cf. This works with all perfect numbers 2 p−1(2 p − 1) with odd prime p and, in fact, with all numbers of the form 2 m−1(2 m − 1) for odd integer (not necessarily prime) m. This can be reformulated as follows: adding the digits of any even perfect number (except 6), then adding the digits of the resulting number, and repeating this process until a single digit (called the digital root) is obtained, always produces the number 1. With each resulting triangular number T 7 = 28, T 31 = 496, T 127 = 8128 (after subtracting 1 from the perfect number and dividing the result by 9) ending in 3 or 5, the sequence starting with T 2 = 3, T 10 = 55, T 42 = 903, T 2730 = 3727815, . Equivalently, a perfect number is a number that is half the sum of all of its positive divisors including itself in symbols, σ 1 ( n ) = 2 n The sum of divisors of a number, excluding the number itself, is called its aliquot sum, so a perfect number is one that is equal to its aliquot sum. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number. In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. Illustration of the perfect number status of the number 6 For the 2012 film, see Perfect Number (film).