Misplaced Pages

#477522

92-491: The first four perfect numbers are 6 , 28 , 496 and 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, σ 1 ( n ) = 2 n {\displaystyle \sigma _{1}(n)=2n} where σ 1 {\displaystyle \sigma _{1}}

184-879: A prime number , as follows: p = 2 : 2 1 ( 2 2 − 1 ) = 2 × 3 = 6 p = 3 : 2 2 ( 2 3 − 1 ) = 4 × 7 = 28 p = 5 : 2 4 ( 2 5 − 1 ) = 16 × 31 = 496 p = 7 : 2 6 ( 2 7 − 1 ) = 64 × 127 = 8128. {\displaystyle {\begin{aligned}p=2&:\quad 2^{1}(2^{2}-1)=2\times 3=6\\p=3&:\quad 2^{2}(2^{3}-1)=4\times 7=28\\p=5&:\quad 2^{4}(2^{5}-1)=16\times 31=496\\p=7&:\quad 2^{6}(2^{7}-1)=64\times 127=8128.\end{aligned}}} Prime numbers of

276-556: A 6 or an 8. Euclid proved that 2 p − 1 ( 2 p − 1 ) {\displaystyle 2^{p-1}(2^{p}-1)} is an even perfect number whenever 2 p − 1 {\displaystyle 2^{p}-1} is prime ( Elements , Prop. IX.36). For example, the first four perfect numbers are generated by the formula 2 p − 1 ( 2 p − 1 ) , {\displaystyle 2^{p-1}(2^{p}-1),} with p

368-418: A 6 that looks like a "b" is not practical. Just as in most modern typefaces , in typefaces with text figures the character for the digit 6 usually has an ascender , as, for example, in [REDACTED] . This digit resembles an inverted 9 . To disambiguate the two on objects and documents that can be inverted, the 6 has often been underlined, both in handwriting and on printed labels. Indeed, We created

460-404: A miracle. All even perfect numbers have a very precise form; odd perfect numbers either do not exist or are rare. There are a number of results on perfect numbers that are actually quite easy to prove but nevertheless superficially impressive; some of them also come under Richard Guy 's strong law of small numbers : The sum of proper divisors gives various other kinds of numbers. Numbers where

552-523: A morality of virtues without passions, such as lust/desire and anger, but with a "common human sympathy". Commentators can also infer from his mission to Caligula that Philo was involved in politics. However, the nature of his political beliefs, especially his viewpoint on the Roman Empire, is a matter of debate. Philo did suggest in his writings that a prudent man should withhold his genuine opinion about tyrants: he will of necessity take up caution as

644-585: A prime p are prime; for example, 2 − 1 = 2047 = 23 × 89 is not a prime number. In fact, Mersenne primes are very rare: of the primes p up to 68,874,199, 2 p − 1 {\displaystyle 2^{p}-1} is prime for only 48 of them. While Nicomachus had stated (without proof) that all perfect numbers were of the form 2 n − 1 ( 2 n − 1 ) {\displaystyle 2^{n-1}(2^{n}-1)} where 2 n − 1 {\displaystyle 2^{n}-1}

736-479: A proof that no odd perfect numbers exist. Any odd perfect number N must satisfy the following conditions: Furthermore, several minor results are known about the exponents e 1 , ...,  e k . In 1888, Sylvester stated: ... a prolonged meditation on the subject has satisfied me that the existence of any one such [odd perfect number]—its escape, so to say, from the complex web of conditions which hem it in on all sides—would be little short of

828-438: A shield, as a protection to prevent his suffering any sudden and unexpected evil; for as I imagine what a wall is to a city, that caution is to an individual. Do not these men then talk foolishly, are they not mad, who desire to display their inexperience and freedom of speech to kings and tyrants, at times daring to speak and to do things in opposition to their will? Do they not perceive that they have not only put their necks under

920-465: A total of six convex regular polytopes . In the classification of finite simple groups , twenty of twenty-six sporadic groups in the happy family are part of three families of groups which divide the order of the friendly giant , the largest sporadic group: five first generation Mathieu groups , seven second generation subquotients of the Leech lattice , and eight third generation subgroups of

1012-520: Is Free , § 8 [ii. 454]. Philo did not reject the subjective experience of ancient Judaism; yet, he repeatedly explained that the Septuagint cannot be understood as a concrete, objective history. Philo's allegorical interpretation of scripture allows him to grapple with morally disturbing events and impose a cohesive explanation of stories. Specifically, Philo interprets the characters of the Bible as aspects of

SECTION 10

#1733131865478

1104-491: Is a hexagon , one of the three regular polygons capable of tiling the plane . A hexagon also has 6 edges as well as 6 internal and external angles . 6 is the second smallest composite number . It is also the first number that is the sum of its proper divisors, making it the smallest perfect number . 6 is the first unitary perfect number , since it is the sum of its positive proper unitary divisors , without including itself. Only five such numbers are known to exist. 6

1196-413: Is a pernicious number . Every even perfect number is also a practical number (cf. Related concepts ). It is unknown whether any odd perfect numbers exist, though various results have been obtained. In 1496, Jacques Lefèvre stated that Euclid's rule gives all perfect numbers, thus implying that no odd perfect number exists, but Euler himself stated: "Whether ... there are any odd perfect numbers

1288-415: Is a practical number . By definition, a perfect number is a fixed point of the restricted divisor function s ( n ) = σ ( n ) − n , and the aliquot sequence associated with a perfect number is a constant sequence. All perfect numbers are also S {\displaystyle {\mathcal {S}}} -perfect numbers, or Granville numbers . A semiperfect number is a natural number that

1380-469: Is a "perfect ruler". The six exponentials theorem guarantees that under certain conditions one of a set of six exponentials is transcendental . The smallest non- abelian group is the symmetric group S 3 {\displaystyle \mathrm {S_{3}} } which has 3! = 6 elements. 6 the answer to the two-dimensional kissing number problem . A cube has 6 faces . A tetrahedron has 6 edges . In four dimensions , there are

1472-556: Is a Jewish work composed in Alexandria , Egypt , around the 1st century BCE, to bolster the faith of the Jewish community in a hostile Greek world. It is one of the seven Sapiential or Wisdom books included in the Septuagint . The Logos has a special relation to humankind. Philo seems to look at humans as a trichotomy of nous (mind), psyche (soul), and soma (body), which was common to

1564-485: Is a most difficult question". More recently, Carl Pomerance has presented a heuristic argument suggesting that indeed no odd perfect number should exist. All perfect numbers are also harmonic divisor numbers , and it has been conjectured as well that there are no odd harmonic divisor numbers other than 1. Many of the properties proved about odd perfect numbers also apply to Descartes numbers , and Pace Nielsen has suggested that sufficient study of those numbers may lead to

1656-578: Is a prime of the form 2 p − 1 {\displaystyle 2^{p}-1} for positive integer p {\displaystyle p} —what is now called a Mersenne prime . Two millennia later, Leonhard 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. In about 300 BC Euclid showed that if 2 − 1

1748-703: Is also described in Book 2, Chapter 5 of Eusebius 's Historia Ecclesiae Philo along with his brothers received a thorough education. They were educated in the Hellenistic culture of Alexandria and the culture of ancient Rome , to a degree in Ancient Egyptian religion and particularly in the traditions of Judaism , in the study of Jewish traditional literature and in Greek philosophy . In his works, Philo shows extensive influence not only from philosophers such as Plato and

1840-413: Is always the same (ἀΐδιος). God needs no other being (χρῄζει γὰρ οὐδενὸς τὸ παράπαν) for self-existence or the creation of material things, and God is self-sufficient (ἑαυτῷ ἱκανός). God can never perish (ἅφθαρτος), is self-existent (ὁ ὤν, τὸ ὄν), and has no relations with any other being (τὸ γὰρ ὄν, ᾗ ὄν ἐστιν, οὐχὶ τῶν πρός τι). Philo considered the anthropomorphism of the Bible to be an impiety that

1932-447: Is equal to the sum of all or some of its proper divisors. A semiperfect number that is equal to the sum of all its proper divisors is a perfect number. Most abundant numbers are also semiperfect; abundant numbers which are not semiperfect are called weird numbers . 6 (number) 6 ( six ) is the natural number following 5 and preceding 7 . It is a composite number and the smallest perfect number . A six-sided polygon

SECTION 20

#1733131865478

2024-547: Is followed by Origen , and by Didymus the Blind , who adds the observation that there are only four perfect numbers that are less than 10,000. (Commentary on Genesis 1. 14–19). St Augustine defines perfect numbers in City of God (Book XI, Chapter 30) 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 (1194–1252) mentioned

2116-516: Is interpreted by Philo as a manifestation of the Logos, which acts as Balaam's—or humankind's—conscience. As such, the Logos becomes the aspect of the divine that operates in the world through whom the world is created and sustained. Peter Schäfer argues that Philo's Logos was derived from his understanding of the "postbiblical Wisdom literature , in particular the Wisdom of Solomon ". The Wisdom of Solomon

2208-499: Is known that Philo came from a family which was noble, honourable and wealthy. It was either his father or paternal grandfather who was granted Roman citizenship from Roman dictator Gaius Julius Caesar . Jerome wrote that Philo came de genere sacerdotum (from a priestly family). His ancestors and family had social ties and connections to the priesthood in Judea , the Hasmonean dynasty ,

2300-448: Is likely that he used the word Κύριος when making a secondary reference to the divine name in his exposition". James Royse concludes: (1) the exegete [Philo] knows and reads biblical manuscripts in which the tetragram is written in palaeo-Hebrew or Aramaic script and not translated by kyrios and that (2) he quotes scriptures in the same way he would have pronounced it, that is, by translating it as kurios ." Philo represents

2392-454: Is prime (though he stated this somewhat differently), Ibn al-Haytham (Alhazen) circa AD 1000 was unwilling to go that far, declaring instead (also without proof) that the formula yielded only every even perfect number. It was not until the 18th century that Leonhard Euler proved that the formula 2 p − 1 ( 2 p − 1 ) {\displaystyle 2^{p-1}(2^{p}-1)} will yield all

2484-524: Is prime then 2(2 − 1) is perfect. The first four perfect numbers were the only ones known to early Greek mathematics , and the mathematician Nicomachus noted 8128 as early as around AD 100. In modern language, Nicomachus states without proof that every perfect number is of the form 2 n − 1 ( 2 n − 1 ) {\displaystyle 2^{n-1}(2^{n}-1)} where 2 n − 1 {\displaystyle 2^{n}-1}

2576-436: Is prime. He seems to be unaware that n itself has to be prime. He also says (wrongly) that the perfect numbers end in 6 or 8 alternately. (The first 5 perfect numbers end with digits 6, 8, 6, 8, 6; but the sixth also ends in 6.) 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

2668-1102: Is represented in binary form as p ones followed by p − 1 zeros; for example: 6 10 = 2 2 + 2 1 = 110 2 28 10 = 2 4 + 2 3 + 2 2 = 11100 2 496 10 = 2 8 + 2 7 + 2 6 + 2 5 + 2 4 = 111110000 2 8128 10 = 2 12 + 2 11 + 2 10 + 2 9 + 2 8 + 2 7 + 2 6 = 1111111000000 2 {\displaystyle {\begin{array}{rcl}6_{10}=&2^{2}+2^{1}&=110_{2}\\28_{10}=&2^{4}+2^{3}+2^{2}&=11100_{2}\\496_{10}=&2^{8}+2^{7}+2^{6}+2^{5}+2^{4}&=111110000_{2}\\8128_{10}=&\!\!2^{12}+2^{11}+2^{10}+2^{9}+2^{8}+2^{7}+2^{6}\!\!&=1111111000000_{2}\end{array}}} Thus every even perfect number

2760-581: Is specific; no appropriate predicates can be conceived. To Philo, God exists beyond time and space and does not make special interventions into the world because God already encompasses the entire cosmos. Philo also integrated select theology from the rabbinic tradition, including God's transcendence , and humankind's inability to behold an ineffable God. He argued that God has no attributes (ἁπλοῡς)—in consequence, no name (ἅρρητος)—and, therefore, that God cannot be perceived by man (ἀκατάληπτος). Furthermore, he posited that God cannot change (ἅτρεπτος): God

2852-461: Is still possible there may be others within this range, initial but exhaustive tests by GIMPS have revealed no other perfect numbers for p below 109332539. As of October 2024, 52 Mersenne primes are known, and therefore 52 even perfect numbers (the largest of which is 2 × (2 − 1) with 82,048,640 digits). It is not known whether there are infinitely many perfect numbers, nor whether there are infinitely many Mersenne primes. As well as having

Perfect number - Misplaced Pages Continue

2944-418: Is the sum-of-divisors function . 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 q ( q + 1 ) / 2 {\displaystyle q(q+1)/2} is an even perfect number whenever q {\displaystyle q}

3036-456: Is the largest of the four all-Harshad numbers . 6 is the 2nd superior highly composite number , the 2nd colossally abundant number , the 3rd triangular number , the 4th highly composite number , a pronic number , a congruent number , a harmonic divisor number , and a semiprime . 6 is also the first Granville number , or S {\displaystyle {\mathcal {S}}} -perfect number. A Golomb ruler of length 6

3128-549: Is untouched by unreasonable emotions, as appears in Exodus 32 :12, wherein Moses, torn by his feelings, perceives God alone to be calm. He is free from sorrow, pain, and other affections. But God is frequently represented as endowed with human emotions, and this serves to explain expressions referring to human repentance in the ancient Jewish context. Similarly, God cannot exist or change in space. He has no "where" (πού, obtained by changing

3220-479: The 2 p − 1 {\displaystyle 2^{p-1}} -th hexagonal number . Furthermore, each even perfect number except for 6 is the 2 p + 1 3 {\displaystyle {\tfrac {2^{p}+1}{3}}} -th centered nonagonal number and is equal to the sum of the first 2 p − 1 2 {\displaystyle 2^{\frac {p-1}{2}}} odd cubes (odd cubes up to

3312-581: The Herodian dynasty and the Julio-Claudian dynasty in Rome . Philo had one brother, Alexander Lysimachus, who was the general tax administrator of customs in Alexandria . He accumulated an immense amount of wealth, becoming not only the richest man in that city but also in the entire Hellenistic world. Alexander was so rich that he gave a loan to the wife of king Herod Agrippa , as well as gold and silver to overlay

3404-569: The Jewish Scriptures chiefly from the Septuagint , a Koine Greek translation of Hebraic texts later compiled as the Hebrew Bible and the deuterocanonical books . His numerous etymologies of Hebrew names, which are along the lines of the etymologic midrash to Genesis and of the earlier rabbinism , although not modern Hebrew philology , suggest some familiarity. Philo offers for some names three or four etymologies, sometimes including

3496-776: The Roman province of Egypt . The only event in Philo's life that can be decisively dated is his representation of the Alexandrian Jews in a delegation to the Roman emperor Caligula in 40 CE following civil strife between the Jewish and Greek communities of Alexandria. Philo was a leading writer of the Hellenistic Jewish community in Alexandria , Egypt. He wrote expansively in Koine Greek on

3588-639: The Roman province of Judaea . In Antiquities of the Jews , Josephus tells of Philo's selection by the Alexandrian Jewish community as their principal representative before the Roman emperor Gaius Caligula. He says that Philo agreed to represent the Alexandrian Jews about the civil disorder that had developed between the Jews and the Greeks. Josephus also tells us that Philo was skilled in philosophy and that he

3680-705: The Stoics , but also poets and orators, especially Homer , Euripides , and Demosthenes . Philo's largest philosophical influence was Plato, drawing heavily from the Timaeus and the Phaedrus , and also from the Phaedo , Theaetetus , Symposium , Republic , and Laws . The extent of Philo's knowledge of Hebrew, however, is debated. Philo was more fluent in Greek than in Hebrew and read

3772-650: The Torah (known in the Hellenic world as the Pentateuch ) but also include histories and comments on philosophy. Most of these were preserved in Greek by the Church Fathers ; some survive only through an Armenian translation, and a smaller number survive in a Latin translation. The exact dates of writing and original organization plans are unknown for many of the texts attributed to Philo. Most of Philo's surviving work deals with

Perfect number - Misplaced Pages Continue

3864-561: The Torah (the first five books of the Bible ). Within this corpus are three categories: Philo's commentary on the Pentateuch is usually classified into three genres. The Quaestiones explain the Pentateuch catechetically, in the form of questions and answers ("Zητήματα καὶ Λύσεις, Quæstiones et Solutiones"). Only the following fragments have been preserved: abundant passages in Armenian – possibly

3956-553: The Torah , with Greek philosophy was the first documented of its kind, and thereby often misunderstood. Many critics of Philo assumed his allegorical perspective would lend credibility to the notion of legend over historicity. Philo often advocated a literal understanding of the Torah and the historicity of such described events, while at other times favoring allegorical readings. Philo's dates of birth and death are unknown but can be judged by Philo's description of himself as "old" when he

4048-421: The nature of God ; he contrasted the nature of God with the nature of the physical world. Philo did not consider God similar to Heaven , the world , or man; he affirmed a transcendent God without physical features or emotional qualities resembling those of human beings. Following Plato, Philo equates matter to nothingness and sees its effect in fallacy, discord, damage, and decay of things. Only God's existence

4140-399: The "archetypal idea". Philo identified Plato's Ideas with the demiurge's thoughts. These thoughts make the contents of Logos; they were the seals for making sensual things during world creation. Logos resembles a book with creature paradigms. An Architect's design before the construction of a city serves to Philo as another simile of Logos. Since creation, Logos binds things together. As

4232-418: The "name of God," There are, in addition, Biblical elements: Philo connects his doctrine of the Logos with Scripture, first of all, based on Genesis 1:27, the relation of the Logos to God. He translates this passage as follows: "He made man after the image of God," concluding from that place that an image of God existed. The Logos is also designated as " high priest " in reference to the exalted position that

4324-562: The Alexander referenced in the Book of Acts , who presided over the Sanhedrin trial of John and Peter . Philo lived in an era of increasing ethnic tension in Alexandria, exacerbated by the new strictures of imperial rule . Some expatriate Hellenes (Greeks) in Alexandria condemned the Jews for a supposed alliance with Rome, even as Rome was seeking to suppress Jewish national and cultural identity in

4416-629: The Hebrew Bible, he interpreted the stories of the first five books as elaborate metaphors and symbols to demonstrate that Greek philosophers' ideas had preceded them in the Bible: Heraclitus 's concept of binary oppositions , according to Who is the Heir of Divine Things? § 43 [i. 503]; and the conception of the wise man expounded by Zeno of Citium , the founder of Stoicism , in Every Good Man

4508-558: The Hellenistic view of the mind-body relationship . In Philo's writings, however, mind and spirit are used interchangeably. The soul is the type; man is the copy. The similarity is found in the mind (νοῦς) of humans. For the shaping of the nous, the individual has the Logos for a pattern to follow. The latter officiates here also as "the divider" (τομεύς), separating and uniting. The Logos, as "interpreter," announces God's designs to humankind, acting in this respect as prophet and priest. As

4600-639: The Jewish embassage, a man eminent on all accounts, brother to Alexander the Alabarch, (30) and one not unskillful in philosophy, was ready to betake himself to make his defense against those accusations; but Gaius prohibited him, and bid him begone; he was also in such a rage, that it openly appeared he was about to do them some very great mischief. So Philo being thus affronted, went out, and said to those Jews who were about him, that they should be of good courage, since Gaius's words indeed showed anger at them, but in reality had already set God against himself. This event

4692-401: The Jewish inhabitants and the Greeks; and three ambassadors were chosen out of each party that were at variance, who came to Gaius. Now one of these ambassadors from the people of Alexandria was Apion , (29) who uttered many blasphemies against the Jews; and, among other things that he said, he charged them with neglecting the honors that belonged to Caesar; for that while all who were subject to

SECTION 50

#1733131865478

4784-568: The Law then follows in two sections. First come the biographies of the men who antedated the several written laws of the Torah, as Enos , Enoch , Noah , Abraham , Isaac , and Jacob . These were the Patriarchs, who were the living impersonations of the active law of virtue before there were any written laws. Then, the laws are discussed in detail: first, the chief ten commandments (the Decalogue), and then

4876-466: The Logos is directly related to the Middle Platonic view of God as unmoved and utterly transcendent; therefore, intermediary beings were necessary to bridge the enormous gap between God and the material world. The Logos was the highest of these intermediary beings and was called by Philo "the first-born of God." Philo also adapted Platonic elements in designating the Logos as the "idea of ideas" and

4968-523: The Logos is influenced by Heraclitus ' conception of the "dividing Logos" (λόγος τομεύς), which calls the various objects into existence by the combination of contrasts ("Quis Rerum Divinarum Heres Sit," § 43 [i. 503]), as well as the Stoic characterization of the Logos as the active and vivifying power. But Philo followed the Platonic distinction between imperfect matter and perfect Form, and Philo's conception of

5060-406: The Roman empire built altars and temples to Gaius, and in other regards universally received him as they received the gods, these Jews alone thought it a dishonorable thing for them to erect statues in honor of him, as well as to swear by his name. Many of these severe things were said by Apion, by which he hoped to provoke Gaius to anger at the Jews, as he was likely to be. But Philo, the principal of

5152-551: The accent in Genesis 3:9: "Adam, where [ποῡ] art thou?"), is not in any place. He is Himself the place; the dwelling-place of God means the same as God Himself, as in the Mishnah = "God is" (comp. Freudenthal, "Hellenistische Studien," p. 73), corresponding to the tenet of Greek philosophy that the existence of all things is summed up in God. God as such is motionless, as the Bible indicates by

5244-431: The apex of Jewish-Hellenistic syncretism . His work attempts to combine Plato and Moses into one philosophical system. Philo bases his doctrines on the Hebrew Bible , which he considers the source and standard not only of religious truth but of all truth. Its pronouncements are the ἱερὸς λόγος , θεῖος λόγος , and ὀρθὸς λόγος (holy word, godly word, righteous word), uttered sometimes directly and sometimes through

5336-634: The complete work – in explanation of Genesis and Exodus, an old Latin translation of a part of the "Genesis", and fragments from the Greek text in Eusebius , in the "Sacra Parallela", in the "Catena", and also in Ambrosius . The explanation is confined chiefly to determining the literal sense, although Philo frequently refers to the allegorical sense as the higher. Νόμων Ἱερῶν Ἀλληγορίαι, or "Legum Allegoriæ", deals, so far as it has been preserved, with selected passages from Genesis . According to Philo's original idea,

5428-558: The correct Hebrew root (e.g., Hebrew : י־ר־ד , romanized :  y-r-d , lit.   'descend' as the origin of the name Jordan ). However, his works do not display much understanding of Hebrew grammar , and they tend to follow the translation of the Septuagint more closely than the Hebrew version. . Philo identified the angel of the Lord (in the singular) with the Logos . In

5520-2193: The cube of 2 p + 1 2 − 1 {\displaystyle 2^{\frac {p+1}{2}}-1} ): 6 = 2 1 ( 2 2 − 1 ) = 1 + 2 + 3 , 28 = 2 2 ( 2 3 − 1 ) = 1 + 2 + 3 + 4 + 5 + 6 + 7 = 1 3 + 3 3 496 = 2 4 ( 2 5 − 1 ) = 1 + 2 + 3 + ⋯ + 29 + 30 + 31 = 1 3 + 3 3 + 5 3 + 7 3 8128 = 2 6 ( 2 7 − 1 ) = 1 + 2 + 3 + ⋯ + 125 + 126 + 127 = 1 3 + 3 3 + 5 3 + 7 3 + 9 3 + 11 3 + 13 3 + 15 3 33550336 = 2 12 ( 2 13 − 1 ) = 1 + 2 + 3 + ⋯ + 8189 + 8190 + 8191 = 1 3 + 3 3 + 5 3 + ⋯ + 123 3 + 125 3 + 127 3 {\displaystyle {\begin{alignedat}{3}6&=2^{1}(2^{2}-1)&&=1+2+3,\\[8pt]28&=2^{2}(2^{3}-1)&&=1+2+3+4+5+6+7\\&&&=1^{3}+3^{3}\\[8pt]496&=2^{4}(2^{5}-1)&&=1+2+3+\cdots +29+30+31\\&&&=1^{3}+3^{3}+5^{3}+7^{3}\\[8pt]8128&=2^{6}(2^{7}-1)&&=1+2+3+\cdots +125+126+127\\&&&=1^{3}+3^{3}+5^{3}+7^{3}+9^{3}+11^{3}+13^{3}+15^{3}\\[8pt]33550336&=2^{12}(2^{13}-1)&&=1+2+3+\cdots +8189+8190+8191\\&&&=1^{3}+3^{3}+5^{3}+\cdots +123^{3}+125^{3}+127^{3}\end{alignedat}}} Even perfect numbers (except 6) are of

5612-707: The digital root of 8128 is 1, because 8 + 1 + 2 + 8 = 19 , 1 + 9 = 10 , and 1 + 0 = 1 . This works with all perfect numbers 2 p − 1 ( 2 p − 1 ) {\displaystyle 2^{p-1}(2^{p}-1)} with odd prime p and, in fact, with all numbers of the form 2 m − 1 ( 2 m − 1 ) {\displaystyle 2^{m-1}(2^{m}-1)} for odd integer (not necessarily prime) m . Owing to their form, 2 p − 1 ( 2 p − 1 ) , {\displaystyle 2^{p-1}(2^{p}-1),} every even perfect number

SECTION 60

#1733131865478

5704-462: The efficient causes that not only represent the types of things, but also produce and maintain them. Philo endeavored to harmonize this conception with the Bible by designating these powers as angels. Philo conceives the powers both as independent hypostases and as immanent attributes of a Divine Being. In the same way, Philo contrasts the two divine attributes of goodness and power (ἄγαθότης and ἀρχή, δίναμις χαριστική and συγκολαστική) as expressed in

5796-707: The even perfect numbers. Thus, there is a one-to-one correspondence between even perfect numbers and Mersenne primes; each Mersenne prime generates one even perfect number, and vice versa. This result is often referred to as the Euclid–Euler theorem . An exhaustive search by the GIMPS distributed computing project has shown that the first 48 even perfect numbers are 2 p − 1 ( 2 p − 1 ) {\displaystyle 2^{p-1}(2^{p}-1)} for Four higher perfect numbers have also been discovered, namely those for which p = 74207281, 77232917, 82589933 and 136279841. Although it

5888-543: The form 2 p − 1 {\displaystyle 2^{p}-1} are known as Mersenne primes , after the seventeenth-century monk Marin Mersenne , who studied number theory and perfect numbers. For 2 p − 1 {\displaystyle 2^{p}-1} to be prime, it is necessary that p itself be prime. However, not all numbers of the form 2 p − 1 {\displaystyle 2^{p}-1} with

5980-439: The form 2 p − 1 ( 2 p − 1 ) {\displaystyle 2^{p-1}(2^{p}-1)} , each even perfect number is the ( 2 p − 1 ) {\displaystyle (2^{p}-1)} -th triangular number (and hence equal to the sum of the integers from 1 to 2 p − 1 {\displaystyle 2^{p}-1} ) and

6072-525: The form T 2 p − 1 = 1 + ( 2 p − 2 ) × ( 2 p + 1 ) 2 = 1 + 9 × T ( 2 p − 2 ) / 3 {\displaystyle T_{2^{p}-1}=1+{\frac {(2^{p}-2)\times (2^{p}+1)}{2}}=1+9\times T_{(2^{p}-2)/3}} with each resulting triangular number T 7 = 28 , T 31 = 496 , T 127 = 8128 (after subtracting 1 from

6164-473: The friendly giant. The remaining six sporadic groups do not divide the order of the friendly giant, which are termed the pariahs ( Ly , O'N , Ru , J 4 , J 3 , and J 1 ). Hexa is classical Greek for "six". Thus: Sex- is a Latin prefix meaning "six". Thus: The SI prefix for 1000 is exa- (E), and for its reciprocal atto- (a). The evolution of our modern digit 6 appears rather simple when compared with

6256-630: The heavens and the earth and everything in between in six Days, and We were not ˹even˺ touched with fatigue. Note 1: The word day is not always used in the Quran to mean a 24-hour period. According to Surah Al-Hajj (The Pilgrimage):47, a heavenly Day is 1000 years of our time. The Day of Judgment will be 50,000 years of our time - Surah Al-Maarij (The Ascending Stairways):4. Hence, the six Days of creation refer to six eons of time, known only by Allah. Note 2: Some Islamic scholars believe this verse comes in response to Exodus 31:17, which says, "The Lord made

6348-452: The heavens and the earth in six days, but on the seventh day He rested and was refreshed." Philo of Alexandria Philo of Alexandria ( / ˈ f aɪ l oʊ / ; Ancient Greek : Φίλων , romanized :  Phílōn ; Hebrew : יְדִידְיָה , romanized :  Yəḏīḏyāh ; c.  20 BCE  – c.   50 CE ), also called Philō Judæus , was a Hellenistic Jewish philosopher who lived in Alexandria , in

6440-520: The high priest occupied after the Exile as the physical center of the Jews' relationship with God. The Logos, like the high priest, is the expiator of the Jews' sins and the mediator and advocate for humankind before, and envoy to, God: ἱκέτης, and παράκλητος. He puts human minds in order. The right reason is an infallible law, the source of any other laws. The angel blocking Balaam 's way in Numbers 22:22–35

6532-507: The history of primal humanity is here considered a symbol of the religious and moral development of the human soul. This commentary included the following treatises: Philo wrote a systematic work on Moses and his laws, which is usually prefaced by the treatise " De Opificio Mundi ". The Creation is, according to Philo, the basis for the Mosaic legislation, which is in complete harmony with nature ("De Opificio Mundi", § 1 [i. 1]). The exposition of

6624-571: The human being and the stories of the Bible as episodes from universal human experience. For example, Adam represents the mind and Eve , the senses. Noah represents tranquility, a stage of "relative"—incomplete but progressing—righteousness. According to Josephus , Philo was inspired mainly in this by Aristobulus of Alexandria and the Alexandrian school . Philo frequently engaged in Pythagorean-inspired numerology , explaining at length

6716-494: The importance of the first 10 numerals: Philo also determines the values of the numbers 50, 70, 100, 12, and 120. There is also extensive symbolism of objects. Philo elaborates on the extensive symbolism of proper names, following the example of the Bible and the Midrash , to which he adds many new interpretations. Philo stated his theology both through the negation of opposing ideas and through detailed, positive explanations of

6808-584: The intersection of philosophy , politics , and religion in his time; specifically, he explored the connections between Greek Platonic philosophy and late Second Temple Judaism . For example, he maintained that the Greek-language Septuagint and the Jewish law still being developed by the rabbis of the period together serve as a blueprint for the pursuit of individual enlightenment. Philo's deployment of allegory to harmonize Jewish scripture, mainly

6900-401: The latter, the Logos softens punishments by making God's merciful power stronger than the punitive. The Logos has a special mystic influence upon the human soul, illuminating it and nourishing it with higher spiritual food, like the manna, of which the most diminutive piece has the same vitality as the whole. Philo's ethics were strongly influenced by Pythagoreanism and Stoicism , preferring

6992-483: The mouth of a prophet, and especially through Moses , whom Philo considers the true medium of revelation . However, he distinguishes between the words uttered by God himself, such as the Ten Commandments , and the edicts of Moses (as the special laws). Philo regards the Bible as the source not only of religious revelation but also of philosophical truth. By applying the Stoic mode of allegorical interpretation to

7084-450: The names of God; designating "Yhwh" as Goodness, Philo interpreted "Elohim" (LXX. Θεός) as designating the "cosmic power"; and as he considered the Creation the most important proof of divine goodness, he found the idea of goodness especially in Θεός. Philo also treats the divine powers of God as a single independent being, or demiurge , which he designates " Logos ". Philo's conception of

7176-544: The next three perfect numbers (33,550,336; 8,589,869,056; and 137,438,691,328) 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 (8,589,869,056) and the seventh (137,438,691,328) perfect numbers, and also proved that every perfect number obtained from Euclid's rule ends with

7268-508: The nine gates of the temple in Jerusalem . Due to his extreme wealth, Alexander was also influential in imperial Roman circles as a friend of emperor Claudius. Through Alexander, Philo had two nephews, Tiberius Julius Alexander and Marcus Julius Alexander . The latter was the first husband of the Herodian princess Berenice . Marcus died in 43 or 44. Some scholars identify Alexander Lysimachus as

7360-536: The other digits. The modern 6 can be traced back to the Brahmi numerals of India , which are first known from the Edicts of Ashoka c.  250 BCE . It was written in one stroke like a cursive lowercase e rotated 90 degrees clockwise. Gradually, the upper part of the stroke (above the central squiggle) became more curved, while the lower part of the stroke (below the central squiggle) became straighter. The Arabs dropped

7452-547: The part of the stroke below the squiggle. From there, the European evolution to our modern 6 was very straightforward, aside from a flirtation with a glyph that looked more like an uppercase G. On the seven-segment displays of calculators and watches, 6 is usually written with six segments. Some historical calculator models use just five segments for the 6, by omitting the top horizontal bar. This glyph variant has not caught on; for calculators that can display results in hexadecimal,

7544-488: The partly interpolated passages on the Essenes. In Legatio ad Gaium ( Embassy to Gaius ), Philo describes his diplomatic mission to Gaius Caligula , one of the few events in his life which is explicitly known. He relates that he was carrying a petition describing the sufferings of the Alexandrian Jews and asking the emperor to secure their rights. Philo describes their sufferings in more detail than Josephus's to characterize

7636-423: The perfect number and dividing the result by 9) ending in 3 or 5, the sequence starting with T 2 = 3 , T 10 = 55 , T 42 = 903 , T 2730 = 3727815, ... It follows that by adding the digits of any even perfect number (except 6), then adding the digits of the resulting number, and repeating this process until a single digit (called the digital root ) is obtained, always produces the number 1. For example,

7728-765: The phrase "God stands". Philo endeavored to find the Divine Being active and acting in the world, in agreement with Stoicism, yet his Platonic conception of Matter as evil required that he place God outside of the world in order to prevent God from having any contact with evil. Hence, he was obliged to separate from the Divine Being the activity displayed in the world and to transfer it to the divine powers, which accordingly were sometimes inherent in God and at other times exterior to God. In order to balance these Platonic and Stoic conceptions, Philo conceived of these divine attributes as types or patterns of actual things ("archetypal ideas") in keeping with Plato, but also regarded them as

7820-713: The precepts in amplification of each law. The work is divided into the following treatises: This exposition is more exoteric than allegorical and might have been intended for gentile audiences. Philo is also credited with writing: This is the second half of a work on the freedom of the just according to Stoic principles. The genuineness of this work has been disputed by Frankel (in "Monatsschrift", ii. 30 et seq., 61 et seq.), by Grätz ("Gesch." iii. 464 et seq.), and more recently by Ansfeld (1887), Hilgenfeld (in "Zeitschrift für Wissenschaftliche Theologie", 1888, pp. 49–71), and others. Now Wendland , Ohle , Schürer , Massebieau , and Krell consider it genuine, except

7912-417: The receptacle and holder of ideas, Logos is distinct from the material world. At the same time, Logos pervades the world, supporting it. This image of God is the type for all other things (the "Archetypal Idea" of Plato), a seal impressed upon things. The Logos is a kind of shadow cast by God, having the outlines but not the blinding light of the Divine Being. He calls the Logos "second god [deuteros theos]"

8004-429: The sum is less than the number itself are called deficient , and where it is greater than the number, abundant . These terms, together with perfect itself, come from Greek numerology . A pair of numbers which are the sum of each other's proper divisors are called amicable , and larger cycles of numbers are called sociable . A positive integer such that every smaller positive integer is a sum of distinct divisors of it

8096-496: The text attributed to Philo, he "consistently uses Κύριος as a designation for God". According to David B. Capes, "the problem for this case, however, is that Christian scholars are responsible for copying and transmitting Philo's words to later generations", and adds, George Howard surveys evidence and concludes: "Although it is improbable that Philo varied from the custom of writing the Tetragram when quoting from Scripture, it

8188-476: The yoke like brute beasts, but that they have also surrendered and betrayed their whole bodies and souls likewise, and their wives and their children, and their parents, and all the rest of the numerous kindred and community of their other relations? ... when an opportunity offers, it is a good thing to attack our enemies and put down their power; but when we have no such opportunity, it is better to be quiet The works of Philo are mostly allegorical interpretations of

8280-401: Was brother to the alabarch Alexander. According to Josephus, Philo and the larger Jewish community refused to treat the emperor as a god, to erect statues in honour of the emperor, and to build altars and temples to the emperor. Josephus says Philo believed that God actively supported this refusal. Josephus' complete comments about Philo: There was now a tumult arisen at Alexandria, between

8372-422: Was incompatible with the Platonic conception of "God in opposition to matter", instead interpreting the ascription to God of hands and feet, eyes and ears, tongue and windpipe, as allegories. In Philo's interpretation, Hebrew scripture adapts itself to human conceptions, and so God is occasionally represented as a man for pedagogic reasons. The same holds true for God's anthropopathic attributes. God, as such,

8464-493: Was part of the delegation to Gaius Caligula in 38 CE. Jewish history professor Daniel R. Schwartz estimates his birth year as sometime between 15 and 10 BCE. Philo's reference to an event under the reign of Emperor Claudius indicates that he died sometime between 45 and 50 CE. Philo also recounts that he visited the Second Temple in Jerusalem at least once in his lifetime. Although the names of his parents are unknown, it

#477522