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Perfect number

In number theory, a perfect number is a positive integer, equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3, 1 + 2 + 3 = 6, so 6 is a perfect number; the sum of divisors of a number, excluding the number itself, is called its aliquot sum, so a perfect number is one, equal to its aliquot sum. Equivalently, a perfect number is a number, half the sum of all of its positive divisors including itself. For instance, 28 is perfect as 1 + 2 + 4 + 7 + 14 + 28 = 56 = 2 × 28; this definition is ancient, appearing as early as Euclid's Elements where it is called τέλειος ἀριθμός. Euclid proved a formation rule whereby q / 2 is an perfect number whenever q is a prime of the form 2 p − 1 for prime p —what is now called a Mersenne prime. Two millennia Euler proved that all 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.

The first few perfect numbers are 6, 28, 496 and 8128. In about 300 BC Euclid showed that if 2p − 1 is prime 2p−1 is perfect; the first four perfect numbers were the only ones known to early Greek mathematics, the mathematician Nicomachus noted 8128 as early as around AD 100. Nicomachus states without proof that every perfect number is of the form 2 n − 1 where 2 n − 1 is prime, he seems to be unaware. He said that the perfect numbers end in 6 or 8 alternately. Philo of Alexandria in his first-century book "On the creation" mentions perfect numbers, claiming that the world was created in 6 days and the moon orbits in 28 days because 6 and 28 are perfect. Philo is followed by Origen, by Didymus the Blind, who adds the observation that there are only four perfect numbers that are less than 10,000.. St Augustine defines perfect numbers in City of God in the early 5th century AD, repeating the claim that God created the world in 6 days because 6 is the smallest perfect number; the Egyptian mathematician Ismail ibn Fallūs mentioned the next three perfect numbers and listed a few more which are now known to be incorrect.

The first known European mention of the fifth perfect number is a manuscript written between 1456 and 1461 by an unknown mathematician. In 1588, the Italian mathematician Pietro Cataldi identified the sixth and the seventh perfect numbers, proved that every perfect number obtained from Euclid's rule ends with a 6 or an 8. Euclid proved that 2p−1 is an perfect number whenever 2p − 1 is prime. For example, the first four perfect numbers are generated by the formula 2p−1, with p a prime number, as follows: for p = 2: 21 = 2 × 3 = 6 for p = 3: 22 = 4 × 7 = 28 for p = 5: 24 = 16 × 31 = 496 for p = 7: 26 = 64 × 127 = 8128. Prime numbers of the form 2p − 1 are known as Mersenne primes, after the seventeenth-century monk Marin Mersenne, who studied number theory and perfect numbers. For 2p − 1 to be prime, it is necessary. However, not all numbers of the form 2p −. In fact, Mersenne primes are rare—of the 2,610,944 prime numbers p up to 43,112,609, 2p − 1 is prime for only 47 of them. Although Nicomachus had stated that all perfect numbers were of the form 2 n − 1 where 2 n − 1 is prime, Ibn al-Haytham circa AD 1000 conjectured only that every perfect number is of that form.

It was not until the 18th century that Leonhard Euler proved that the formula 2p−1 will yield all the perfect numbers. Thus, there is a one-to-one correspondence between perfect numbers and Mersenne primes; this result is referred to as the Euclid–Euler theorem. An exhaustive search by the GIMPS distributed computing project has shown that the first 47 perfect numbers are 2p−1 for p = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801 and 43112609. Four higher perfect numbers have been discovered, namely those for which p = 57885161, 74207281, 77232917, 82589933, though there may be others within this range; as of December 2018, 51 Mersenne primes are known, therefore 51 perfect numbers. It is not known whether there are infi

Paddy Chayefsky

Sidney Aaron "Paddy" Chayefsky was an American playwright and novelist. He is the only person to have won three solo Academy Awards for writing both adapted and original screenplays, he was one of the most renowned dramatists of the Golden Age of Television. His intimate, realistic scripts provided a naturalistic style of television drama for the 1950s, dramatizing the lives of ordinary Americans. Martin Gottfried wrote in All His Jazz that Chayefsky was "the most successful graduate of television's slice of life school of naturalism."Following his critically acclaimed teleplays, Chayefsky became a noted playwright and novelist. As a screenwriter, he received three Academy Awards for The Hospital and Network; the movie Marty was based on his own television drama about two lonely people finding love. Network was a satire of the television industry and The Hospital was satiric. Film historian David Thomson called The Hospital "years ahead of its time. Few films capture the disaster of America's self-destructive idealism so well."

His screenplay for Network is regarded as his masterpiece, has been hailed as "the kind of literate, darkly funny and breathtakingly prescient material that prompts many to claim it as the greatest screenplay of the 20th century."Chayefsky's early stories were influenced by the author's childhood in The Bronx. Chayefsky was part of the inaugural class of inductees into the Academy of Television Arts & Sciences' Television Hall of Fame, he received this honor three years after his death, in 1984. Sidney Chayefsky was born in the Bronx, New York City to Russian-Jewish immigrants Harry and Gussie Stuchevsky Chayefsky. Harry Chayefsky's father served for twenty-five years in the Russian army so the family was allowed to live in Moscow, while Gussie Stuchevsky lived in a village near Odessa. Harry and Gussie emigrated to New York in 1909 respectively. Harry Chayefsky worked for a New Jersey milk distribution company in which he took a controlling interest and renamed Dellwood Dairies; the family lived in Perth Amboy, New Jersey and Mount Vernon, New York, moving temporarily to Bailey Avenue in the West Bronx at the time of Sidney Chayefsky's birth while a larger house in Mount Vernon was being completed.

He had two older brothers and Winn. As a toddler Chayefsky showed signs of being gifted, could "speak intelligently" at two and a half, his father suffered a financial reversal during the Wall Street Crash of 1929, the family moved back to the Bronx. Chayefsky attended a public elementary school; as a boy, Chayefsky was noted for his verbal ability. He attended DeWitt Clinton High School, where he served as editor of the school's literary magazine, "The Magpie." He graduated from Clinton in 1939 at age 16 and attended the City College of New York, graduating with a degree in social sciences in 1943. While at City College he played for the semi-professional football team Kingsbridge Trojans, he studied languages at Fordham University during his Army service. In 1943, two weeks before his graduation from City College, Chayefsky was drafted into the United States Army, served in combat in Europe. While in the Army he adopted the nickname "Paddy." The nickname was given spontaneously. Although Jewish, he asked to be excused to attend Mass.

"Sure you do, Paddy," said the officer, the name stuck. Chayefsky was wounded by a land mine while serving with the 104th Infantry Division in the European Theatre near Aachen, Germany, he was awarded the Purple Heart. The wound left contributing to his shyness around women. While recovering from his injuries in the Army Hospital near Cirencester, England, he wrote the book and lyrics to a musical comedy, No T. O. for Love. First produced in 1945 by the Special Services Unit, the show toured European Army bases for two years; the London opening of No T. O. for Love at the Scala Theatre in the West End was the beginning of Chayefsky's theatrical career. During the London production of this musical, Chayefsky encountered Joshua Logan, a future collaborator, Garson Kanin, who invited Chayefsky to collaborate with him on a documentary of the Allied invasion, The True Glory. Returning to the United States, Chayefsky worked in his uncle's print shop, Regal Press, an experience which provided a background for his teleplay, Printer's Measure, as well as his story for the movie As Young as You Feel.

Kanin enabled Chayefsky to spend time working on his second play, Put Them All Together, but it was never produced. Producers Mike Gordon and Jerry Bressler gave him a junior writer's contract, he wrote The Great American Hoax, which sold to Good Housekeeping but was never published. Chayefsky went to Hollywood in 1947 with the aim of becoming a screenwriter, his friends Garson Kanin and Ruth Gordon found him a job in the accounting office of Universal Pictures. He studied acting at the Actor's Lab and Kanin got him a bit part in the film A Double Life, he returned to New York, submitted scripts, was hired as an apprentice scriptwriter by Universal. His script outlines were not accepted and he was fired after six weeks. After returning to New York, Chayefsky wrote the outline for a play that he submitted to the Wiilliam Morris Agency; the agency, treating it as a novella, submitted it to Good Housekeeping magazine. Movie rights were purchased by Twentieth Century Fox, Chayefsky was hired to write the script, he returned to Hollywood in 1948.

But Chayefsky was discouraged by the studio system, which involved rewrites and relegated writers to inferior roles, so he quit and moved back to New York, vowing not to return. During the late 1940s

Deer Woman

The Deer Woman, sometimes known as the Deer Lady, is a spirit in various forms of Native American mythology, associated with fertility and love. Though shown as a benign spirit, she is shown to lure promiscuous men to their death, she appears as either deer. Deer Woman stories are found in many Native American tribes, told to young children or by young adults and prepubescents in tribes like the Sioux, Ponca, the Omaha people, Muscogee, Choctaw, the Otoe tribe, the Pawnee people, the Iroquois - and those are only the few that have documented Deer Woman sightings. In Ojibwe tradition, she can be banished through the use of tobacco. Others claim. Other stories and traditions describe the sighting of Deer Woman as a sign of personal transformation or as a warning. Deer Woman is said to be fond of dancing and will sometimes join a communal dance unnoticed, leaving only when the drum beating ceases; the Deer Women show traits of both sirens and succubi. The siren, according to the Theoi Project, are monstrous sea nymphs that lure men to their deaths with their song.

Succubi, as defined by the Merriam-Webster dictionary, are "demons who take female form who have sexual intercourse with men in their sleep". Deer woman lure men with their beauty and magic, like the sirens, sleep with and kill men like succubi. Countless female spirits world wide have similar characteristics. Fiura, of the Chiloè region of Chilè, causes deformation in anyone who looks upon her and will cast spells to confuse young woodsmen into sleeping with her. La Patasola "one footed", is a shape-shifter from the Antioquia region of Colombia who takes the form of a beautiful woman to lure men with her cries of fear. La Tunda, another nature spirit from Colombia, lures people of all walks of life to them with their song and drains them of blood; the Brazilian Iara are beautiful warrior mermaids who - when found by a man - will charm him with her voice and beauty and either drown him or turn him into something like her and make him her lover. La Llorona, found in Mexico and the Southwest United States, is a female ghost who will kidnap the souls of children - killing them - and who's cries bring irrevocable sorrow and who the sighting of spells death for someone within the week.

While all these spirits will lure away and/or hurt others, they have physical deformity in common. The Deer Woman has hooves. Sirens are bird from the chest down. Succubi were portrayed as hideous and demonic; the Fiura is described as being hideous, La Patasola has no right leg from the pelvis down and her right breast is fused to her arm. La Tunda has a whisk for a leg, the Iara are half Brazilian guppy; the Deer Woman was featured as a character in an eponymous episode of the Showtime horror series Masters of Horror. It aired in North America on December 9, 2005 and was directed by John Landis. A short story entitled "Deer Woman" was published by Paula Gunn Allen A short story entitled "Memoir of a Deer Woman" was included in the book Holiday by M. Rickert In 2015, Anishinaabe writer Elizabeth LaPensée wrote Deer Woman: A Vignette Deer Woman And the Living Myth of the Dreamtime, article by Carolyn Dunn from the Endicott Journal of Mythic Arts 2003