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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. The first four perfect numbers are 6 (number), 6, 28 (number), 28, 496 (number), 496 and 8128 (number), 8128. The sum of proper divisors of a number is called its aliquot sum, so a perfect number is one that is equal to its aliquot sum. Equivalently, a perfect number is a number that is half the sum of all of its positive divisors; in symbols, \sigma_1(n)=2n where \sigma_1 is the sum-of-divisors function. This definition is ancient, appearing as early as Euclid's Elements, Euclid's ''Elements'' (VII.22) where it is called (''perfect'', ''ideal'', or ''complete number''). Euclid also proved a formation rule (IX.36) whereby \frac is an even perfect number whenever q ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perfect Number Cuisenaire Rods 6 Exact
Perfect commonly refers to: * Perfection; completeness, and excellence * Perfect (grammar), a grammatical category in some languages Perfect may also refer to: Film and television * Perfect (1985 film), ''Perfect'' (1985 film), a romantic drama * Perfect (2018 film), ''Perfect'' (2018 film), a science fiction thriller * Perfect (Doctors), "Perfect" (''Doctors''), a 2004 television episode Literature * Perfect (Friend novel), ''Perfect'' (Friend novel), a 2004 novel by Natasha Friend * Perfect (Hopkins novel), ''Perfect'' (Hopkins novel), a young adult novel by Ellen Hopkins * Perfect (Joyce novel), ''Perfect'' (Joyce novel), a 2013 novel by Rachel Joyce * Perfect (Shepard novel), ''Perfect'' (Shepard novel), a Pretty Little Liars novel by Sara Shepard * ''Perfect'', a young adult science fiction novel by Dyan Sheldon Music * Perfect interval, in music theory * Perfect Records, a record label Artists * Perfect (musician) (born 1980), reggae singer * Perfect (Polish band) * P ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nicomachus
Nicomachus of Gerasa (; ) was an Ancient Greek Neopythagorean philosopher from Gerasa, in the Roman province of Syria (now Jerash, Jordan). Like many Pythagoreans, Nicomachus wrote about the mystical properties of numbers, best known for his works ''Introduction to Arithmetic'' and ''Manual of Harmonics'', which are an important resource on Ancient Greek mathematics and Ancient Greek music in the Roman period. Nicomachus' work on arithmetic became a standard text for Neoplatonic education in Late antiquity, with philosophers such as Iamblichus and John Philoponus writing commentaries on it. A Latin paraphrase by Boethius of Nicomachus's works on arithmetic and music became standard textbooks in medieval education. Life Little is known about the life of Nicomachus except that he was a Pythagorean who came from Gerasa. His ''Manual of Harmonics'' was addressed to a lady of noble birth, at whose request Nicomachus wrote the book, which suggests that he was a respected schola ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Arithmetic
In mathematics, modular arithmetic is a system of arithmetic operations for integers, other than the usual ones from elementary arithmetic, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book '' Disquisitiones Arithmeticae'', published in 1801. A familiar example of modular arithmetic is the hour hand on a 12-hour clock. If the hour hand points to 7 now, then 8 hours later it will point to 3. Ordinary addition would result in , but 15 reads as 3 on the clock face. This is because the hour hand makes one rotation every 12 hours and the hour number starts over when the hour hand passes 12. We say that 15 is ''congruent'' to 3 modulo 12, written 15 ≡ 3 (mod 12), so that 7 + 8 ≡ 3 (mod 12). Similarly, if one starts at 12 and waits 8 hours, the hour hand will be at 8. If one instead waited twice as long, 16 hours, the hour hand would be on 4. This ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Marin Mersenne
Marin Mersenne, OM (also known as Marinus Mersennus or ''le Père'' Mersenne; ; 8 September 1588 – 1 September 1648) was a French polymath whose works touched a wide variety of fields. He is perhaps best known today among mathematicians for Mersenne prime numbers, those written in the form for some integer . He also developed Mersenne's laws, which describe the harmonics of a vibrating string (such as may be found on guitars and pianos), and his seminal work on music theory, '' Harmonie universelle'', for which he is referred to as the "father of acoustics". Mersenne, an ordained Catholic priest, had many contacts in the scientific world and has been called "the center of the world of science and mathematics during the first half of the 1600s" and, because of his ability to make connections between people and ideas, "the post-box of Europe". He was also a member of the ascetical Minim religious order and wrote and lectured on theology and philosophy. Life Mersenne was b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, or , involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorization, factorized as a product of primes that is unique up to their order. The property of being prime is called primality. A simple but slow primality test, method of checking the primality of a given number , called trial division, tests whether is a multiple of any integer between 2 and . Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pietro Cataldi
Pietro Antonio Cataldi (15 April 1548, Bologna – 11 February 1626, Bologna) was an Italian mathematician. A citizen of Bologna, he taught mathematics and astronomy and also worked on military problems. His work included the development of simple continued fractions and a method for their representation. He was one of many mathematicians who attempted to prove Euclid's fifth postulate. Cataldi discovered the sixth and seventh perfect numbers by 1588.Caldwell, Chris''The largest known prime by year'' His discovery of the 6th, that corresponding to p=17 in the formula Mp=2p-1, exploded a many-times repeated number-theoretical myth that the perfect numbers had units digits that invariably alternated between 6 and 8. (Until Cataldi, 19 authors going back to Nicomachus are reported to have made the claim, with a few more repeating this afterward, according to L.E.Dickson's '' History of the Theory of Numbers''). Cataldi's discovery of the 7th (for p=19) held the record for the lar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bayerische Staatsbibliothek
The Bavarian State Library (, abbreviated BSB, called ''Bibliotheca Regia Monacensis'' before 1919) in Munich is the central "State libraries of Germany, Landesbibliothek", i. e. the state library of the Free State of Bavaria, the biggest universal and research library in Germany and one of Europe's most important Universal library, universal libraries. With its collections currently comprising around 10.89 million books (as of 2019), it ranks among the leading research libraries worldwide. The furthermore is Europe's second-largest journals library (after the British Library). Furthermore, its historical holdings encompass one of the most important manuscript collections of the world, the largest collection of incunabula worldwide, as well as numerous further important special collections. Its collection of historical prints before 1850 totals almost one million units. The legal deposit law, still applicable today, has been in force since 1663 and requires that two copies ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ibn Fallus
Shams ad-Dīn Abû’t-Tāhir Ismāʽīl ibn-Ibrāhīm ibn-Ġāzī ibn-ʽAlī ibn Muhammad al-Ḥanafī al-Māridīnī (1194–1252), often called Ismāʽīl ibn-Fullūs or Ibn Fallus, was an Arab Egyptian mathematician of the Islamic Golden Age. Whilst on pilgrimage to Mecca, he tells of an epitome he wrote on number theory (extant in manuscript), which building on the work of Nicomachus, added three new perfect numbers (33,550,336; 8,589,869,056; and 137,438,691,328) to the four already discovered by the Greeks. His table also included seven other supposed perfect numbers which are now known to be incorrect. Roshdi Rashed believes the errors emerged from over-reliance on Nicomachus' method. This work did not reach Europe, and the three perfect numbers were only rediscovered there during the Renaissance, including by Pietro Cataldi.Scott, Paul"Just Perfect, part 2" ''The Australian Mathematics Teacher'', vol. 63, no. 2. p. 3. Further reading * Ismāʽīl ibn-FullūsIʽdād a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The City Of God
''On the City of God Against the Pagans'' (), often called ''The City of God'', is a book of Christian philosophy written in Latin by Augustine of Hippo in the early 5th century AD. Augustine wrote the book to refute allegations that Christianity initiated the decline of Rome and is considered one of his seminal works, standing alongside the '' Confessions'', the '' Enchiridion'', '' On Christian Doctrine'', and '' On the Trinity''. As a work of one of the most influential Church Fathers, ''The City of God'' is a cornerstone of Western thought, expounding on many questions of theology, such as the suffering of the righteous, the existence of evil, the conflict between free will and divine omniscience, and the doctrine of original sin. Background The sack of Rome by the Visigoths in 410 left Romans in a deep state of shock, and many Romans saw it as punishment for abandoning traditional Roman religion in favor of Christianity. In response to these accusations, and in order t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Augustine Of Hippo
Augustine of Hippo ( , ; ; 13 November 354 – 28 August 430) was a theologian and philosopher of Berber origin and the bishop of Hippo Regius in Numidia, Roman North Africa. His writings deeply influenced the development of Western philosophy and Western Christianity, and he is viewed as one of the most important Church Fathers of the Latin Church in the Patristic Period. His many important works include '' The City of God'', '' On Christian Doctrine'', and '' Confessions''. According to his contemporary, Jerome of Stridon, Augustine "established anew the ancient Faith". In his youth he was drawn to the Manichaean faith, and later to the Hellenistic philosophy of Neoplatonism. After his conversion to Christianity and baptism in 386, Augustine developed his own approach to philosophy and theology, accommodating a variety of methods and perspectives. Believing the grace of Christ was indispensable to human freedom, he helped formulate the doctrine of original sin and m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Didymus The Blind
Didymus the Blind ( Coptic: ; 398), alternatively spelled Dedimus or Didymous, was a Christian theologian in the Church of Alexandria, where he taught for about half a century. He was a student of Origen, and, after the Second Council of Constantinople condemned Origen, Didymus's works were not copied. Many of his writings are lost, but some of his commentaries and essays survive. He was seen as intelligent and a good teacher. Early life and education Didymus became blind at the age of four, before he had learned to read. He was a loyal follower of Origen, and opposed Arian and Macedonian teachings. Despite his blindness, Didymus excelled in scholarship because of his incredible memory. He found ways to help blind people to read, experimenting with carved wooden letters similar to Braille systems used by the blind today. He recalled and contemplated information while others slept. Teacher in Alexandria According to Rufinus, Didymus was "a teacher in the Church school", who was ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sources Chrétiennes
Sources Chrétiennes ( French "Christian sources") is a bilingual collection of patristic texts founded in Lyon in 1942 by the Jesuits Jean Daniélou, Claude Mondésert, and Henri de Lubac. Citations to the series are commonly made by the letters SC followed by the volume number. Overview The collection is edited by the ''Institut des Sources Chrétiennes'' (current director: Guillaume Bady) and published in Paris by ''Les Éditions du Cerf''. Each text is given on the left in Greek or Latin, with the French translation on the facing page. Over 600 works by Greek, Latin and occasionally Syriac authors have been published. Other oriental Christian (e.g. Armenian) writers have been published only in translation. An early decision was made not to exclude subsequently condemned authors (such as Origen of Alexandria). Clement of Alexandria, the Cappadocians (Basil of Caesarea and Gregory of Nazianzus) John Chrysostom, Theodoret especially are strongly represented. Most Latin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
