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  2. Multiply perfect number - Wikipedia

    en.wikipedia.org/wiki/Multiply_perfect_number

    It can be proven that: . For a given prime number p, if n is p-perfect and p does not divide n, then pn is (p + 1)-perfect.This implies that an integer n is a 3-perfect number divisible by 2 but not by 4, if and only if n/2 is an odd perfect number, of which none are known.

  3. Perfect number - Wikipedia

    en.wikipedia.org/wiki/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.

  4. List of Mersenne primes and perfect numbers - Wikipedia

    en.wikipedia.org/wiki/List_of_Mersenne_primes...

    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 ...

  5. Sprained ankle - Wikipedia

    en.wikipedia.org/wiki/Sprained_ankle

    A grade 1 sprain is defined as mild damage to a ligament or ligaments without instability of the affected joint. A grade 2 sprain is considered a partial tear to the ligament, in which it is stretched to the point that it becomes loose. A grade 3 sprain is a complete tear of a ligament, causing instability in the affected joint. [3]

  6. Granville number - Wikipedia

    en.wikipedia.org/wiki/Granville_number

    The only known -perfect number with three distinct prime factors is 126 = 2 · 3 2 · 7. [2] Every number of form 2^(n - 1) * (2^n - 1) * (2^n)^m where m >= 0, where 2^n - 1 is Prime, are Granville Numbers. So, there are infinitely many Granville Numbers and the infinite family has 2 prime factors- 2 and a Mersenne Prime.

  7. Parity (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Parity_(mathematics)

    An integer is even if it is congruent to 0 modulo this ideal, in other words if it is congruent to 0 modulo 2, and odd if it is congruent to 1 modulo 2. All prime numbers are odd, with one exception: the prime number 2. [14] All known perfect numbers are even; it is unknown whether any odd perfect numbers exist. [15]

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