Perfect number

In number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself (also known as its aliquot sum). Equivalently, a perfect number is a number that is half the sum of all of its positive divisors (including itself) i.e. σ₁(n) = 2n.

This definition is ancient, appearing as early as Euclid's Elements (VII.22) where it is called τέλειος ἀριθμός (perfect, ideal, or complete number). Euclid also proved a formation rule (IX.36) whereby is an even perfect number whenever is what is now called a Mersenne prime—a prime of the form for prime Much later, Euler proved that all even perfect numbers are of this form. This is known as the Euclid–Euler theorem.

It is not known whether there are any odd perfect numbers, nor whether infinitely many perfect numbers exist.

Examples

The first perfect number is 6, because 1, 2, and 3 are its proper positive divisors, and 1 + 2 + 3 = 6. Equivalently, the number 6 is equal to half the sum of all its positive divisors: ( 1 + 2 + 3 + 6 ) / 2 = 6. The next perfect number is 28 = 1 + 2 + 4 + 7 + 14. This is followed by the perfect numbers 496 and 8128 (sequence A000396 in OEIS).

Perfect Number (film)

Perfect Number (Hangul: 용의자X; RR: Yonguija X; lit. Suspect X) is a 2012 South Korean mystery-drama film directed by Bang Eun-jin. Adapted from Keigo Higashino's novel The Devotion of Suspect X, it centers around a mild-mannered mathematics professor (Ryoo Seung-bum) who plans the perfect alibi for the woman he secretly loves (Lee Yo-won) when she unexpectedly murders her abusive ex-husband.Cho Jin-woong received a Best Supporting Actor nomination at the 49th Baeksang Arts Awards in 2013.

Plot

Kim Seok-go showed a lot of promise as a brilliant mathematician when he was in school, resolutely focused on his studies rather than on friends throughout his childhood. Now in his 30s, he's an ordinary high school math teacher, a far cry from the promising future of his youth. Seok-go is solemn and introverted, and his morning exchanges with Baek Hwa-sun, the cafe employee he buys lunch from, is the brightest part of his day. When Hwa-sun's ex-husband mercilessly beats Hwa-sun and her niece, Hwa-sun kills him. Seok-go overhears the fight from his house next door and decides to cover up the killing and protect her from the police. He uses his genius in meticulously planning the perfect alibi for her, and thanks to his efforts, Hwa-sun is cleared in the case. However, the detective in charge, Jo Min-beom, believes that Hwa-sun is guilty and follows his intuition despite the lack of evidence. Min-beom also happens to have gone to the same high school as Seok-go, and when he finds out that his old school friend lives next door to the prime suspect, he starts digging deeper into Seok-go's life...