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Perfect number. 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.
For example, 3 is a Mersenne prime as it is a prime number and is expressible as 22 − 1. [ 1 ][ 2 ] The numbers p corresponding to Mersenne primes must themselves be prime, although not all primes p lead to Mersenne primes—for example, 211 − 1 = 2047 = 23 × 89. [ 3 ] Meanwhile, perfect numbers are natural numbers that equal the sum of ...
Demonstration, with Cuisenaire rods, of the 2-perfection of the number 6. In mathematics, a multiply perfect number (also called multiperfect number or pluperfect number) is a generalization of a perfect number. For a given natural number k, a number n is called k-perfect (or k-fold perfect) if the sum of all positive divisors of n (the divisor ...
For example, the interval from C to G is a perfect fifth, as the note G lies seven semitones above C. The perfect fifth may be derived from the harmonic series as the interval between the second and third harmonics. In a diatonic scale, the dominant note is a perfect fifth above the tonic note.
Square number. Square number 16 as sum of gnomons. In mathematics, a square number or perfect square is an integer that is the square of an integer; [1] in other words, it is the product of some integer with itself. For example, 9 is a square number, since it equals 32 and can be written as 3 × 3.
The Euclid–Euler theorem states that an even natural number is perfect if and only if it has the form 2 p−1 M p, where M p is a Mersenne prime. [1] The perfect number 6 comes from p = 2 in this way, as 2 2−1 M 2 = 2 × 3 = 6, and the Mersenne prime 7 corresponds in the same way to the perfect number 28.
In 1652 the Polish polymath Jan Brożek noted that there was no perfect number between 10 4 and 10 7. [9] Despite over 2,000 years of study, it still is not known whether there exist infinitely many perfect numbers; or whether there are any odd ones. [9] Today the term "perfect number" is merely historic in nature, used for the sake of tradition.
Superperfect number. In number theory, a superperfect number is a positive integer n that satisfies. where σ is the divisor summatory function. Superperfect numbers are not a generalization of perfect numbers but have a common generalization. The term was coined by D. Suryanarayana (1969).