1.
Achilles
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In Greek mythology, Achilles was a Greek hero of the Trojan War and the central character and greatest warrior of Homers Iliad. His mother was the immortal nymph Thetis, and his father, Achilles’ most notable feat during the Trojan War was the slaying of the Trojan hero Hector outside the gates of Troy. Although the death of Achilles is not presented in the Iliad, other sources concur that he was killed near the end of the Trojan War by Paris, later legends state that Achilles was invulnerable in all of his body except for his heel. Alluding to these legends, the term Achilles heel has come to mean a point of weakness, Achilles name can be analyzed as a combination of ἄχος grief and λαός a people, tribe, nation. In other words, Achilles is an embodiment of the grief of the people, Achilles role as the hero of grief forms an ironic juxtaposition with the conventional view of Achilles as the hero of κλέος kleos. Laos has been construed by Gregory Nagy, following Leonard Palmer, to mean a corps of soldiers, a muster. With this derivation, the name would have a meaning in the poem, when the hero is functioning rightly, his men bring grief to the enemy. The poem is in part about the misdirection of anger on the part of leadership, R. S. P. Beekes has suggested a Pre-Greek origin of the name. The name Achilleus was a common and attested name among the Greeks soon after the 7th century BC. It was also turned into the female form Ἀχιλλεία attested in Attica in the 4th century BC and, in the form Achillia, Achilles was the son of the Nereid Thetis and Peleus, the king of the Myrmidons. Zeus and Poseidon had been rivals for the hand of Thetis until Prometheus, for this reason, the two gods withdrew their pursuit, and had her wed Peleus. Thetis, although a daughter of the sea-god Nereus, was brought up by Hera. According to the Achilleid, written by Statius in the 1st century AD, and to no surviving previous sources, however, he was left vulnerable at the part of the body by which she held him, his heel. It is not clear if this version of events was known earlier, in another version of this story, Thetis anointed the boy in ambrosia and put him on top of a fire, to burn away the mortal parts of his body. She was interrupted by Peleus and abandoned both father and son in a rage, however, none of the sources before Statius makes any reference to this general invulnerability. To the contrary, in the Iliad Homer mentions Achilles being wounded, in Book 21 the Paeonian hero Asteropaeus, son of Pelagon and he cast two spears at once, one grazed Achilles elbow, drawing a spurt of blood. Peleus entrusted Achilles to Chiron the Centaur, on Mt. Pelion, Achilles consuming rage is at times wavering, but at other times he cannot be cooled. Thetis foretold that her sons fate was either to gain glory and die young, or to live a long, Achilles chose the former, and decided to take part in the Trojan war

2.
Trojan War
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In Greek mythology, the Trojan War was waged against the city of Troy by the Achaeans after Paris of Troy took Helen from her husband Menelaus, king of Sparta. The war is one of the most important events in Greek mythology and has been narrated through many works of Greek literature, most notably through Homers Iliad. The Iliad relates four days in the year of the decade-long siege of Troy. Other parts of the war are described in a cycle of epic poems, episodes from the war provided material for Greek tragedy and other works of Greek literature, and for Roman poets including Virgil and Ovid. Zeus sent the goddesses to Paris, who judged that Aphrodite, as the fairest, in exchange, Aphrodite made Helen, the most beautiful of all women and wife of Menelaus, fall in love with Paris, who took her to Troy. Agamemnon, king of Mycenae and the brother of Helens husband Menelaus, led an expedition of Achaean troops to Troy and besieged the city for ten years because of Paris insult. After the deaths of heroes, including the Achaeans Achilles and Ajax, and the Trojans Hector and Paris. The Achaeans slaughtered the Trojans and desecrated the temples, thus earning the gods wrath, few of the Achaeans returned safely to their homes and many founded colonies in distant shores. The Romans later traced their origin to Aeneas, one of the Trojans, in 1868, however, the German archaeologist Heinrich Schliemann met Frank Calvert, who convinced Schliemann that Troy was a real city at what is now Hissarlik in Turkey. On the basis of excavations conducted by Schliemann and others, this claim is now accepted by most scholars, whether there is any historical reality behind the Trojan War remains an open question. The events of the Trojan War are found in works of Greek literature. There is no single, authoritative text which tells the events of the war. Instead, the story is assembled from a variety of sources, the most important literary sources are the two epic poems traditionally credited to Homer, the Iliad and the Odyssey, composed sometime between the 9th and 6th centuries BC. Each poem narrates only a part of the war, the Iliad covers a short period in the last year of the siege of Troy, while the Odyssey concerns Odysseuss return to his home island of Ithaca, following the sack of Troy. Other parts of the Trojan War were told in the poems of the Epic Cycle, also known as the Cyclic Epics, the Cypria, Aethiopis, Little Iliad, Iliou Persis, Nostoi, and Telegony. Though these poems survive only in fragments, their content is known from an included in Proclus Chrestomathy. The authorship of the Cyclic Epics is uncertain, both the Homeric epics and the Epic Cycle take origin from oral tradition. Even after the composition of the Iliad, Odyssey, and the Cyclic Epics, events and details of the story that are only found in later authors may have been passed on through oral tradition and could be as old as the Homeric poems

3.
Divisor
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In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some other integer to produce n. In this case one says also that n is a multiple of m, an integer n is divisible by another integer m if m is a divisor of n, this implies dividing n by m leaves no remainder. Under this definition, the statement m ∣0 holds for every m, as before, but with the additional constraint k ≠0. Under this definition, the statement m ∣0 does not hold for m ≠0, in the remainder of this article, which definition is applied is indicated where this is significant. Divisors can be negative as well as positive, although sometimes the term is restricted to positive divisors. For example, there are six divisors of 4, they are 1,2,4, −1, −2, and −4,1 and −1 divide every integer. Every integer is a divisor of itself, every integer is a divisor of 0. Integers divisible by 2 are called even, and numbers not divisible by 2 are called odd,1, −1, n and −n are known as the trivial divisors of n. A divisor of n that is not a divisor is known as a non-trivial divisor. A non-zero integer with at least one divisor is known as a composite number, while the units −1 and 1. There are divisibility rules which allow one to recognize certain divisors of a number from the numbers digits, the generalization can be said to be the concept of divisibility in any integral domain. 7 is a divisor of 42 because 7 ×6 =42 and it can also be said that 42 is divisible by 7,42 is a multiple of 7,7 divides 42, or 7 is a factor of 42. The non-trivial divisors of 6 are 2, −2,3, the positive divisors of 42 are 1,2,3,6,7,14,21,42. 5 ∣0, because 5 ×0 =0, if a ∣ b and b ∣ a, then a = b or a = − b. If a ∣ b and a ∣ c, then a ∣ holds, however, if a ∣ b and c ∣ b, then ∣ b does not always hold. If a ∣ b c, and gcd =1, then a ∣ c, if p is a prime number and p ∣ a b then p ∣ a or p ∣ b. A positive divisor of n which is different from n is called a proper divisor or a part of n. A number that does not evenly divide n but leaves a remainder is called an aliquant part of n, an integer n >1 whose only proper divisor is 1 is called a prime number

4.
Fundamental theorem of arithmetic
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For example,1200 =24 ×31 ×52 =3 ×2 ×2 ×2 ×2 ×5 ×5 =5 ×2 ×3 ×2 ×5 ×2 ×2 = etc. The requirement that the factors be prime is necessary, factorizations containing composite numbers may not be unique. This theorem is one of the reasons why 1 is not considered a prime number, if 1 were prime. Book VII, propositions 30,31 and 32, and Book IX, proposition 14 of Euclids Elements are essentially the statement, proposition 30 is referred to as Euclids lemma. And it is the key in the proof of the theorem of arithmetic. Proposition 31 is proved directly by infinite descent, proposition 32 is derived from proposition 31, and prove that the decomposition is possible. Book IX, proposition 14 is derived from Book VII, proposition 30, indeed, in this proposition the exponents are all equal to one, so nothing is said for the general case. Article 16 of Gauss Disquisitiones Arithmeticae is a modern statement. < pk are primes and the αi are positive integers and this representation is commonly extended to all positive integers, including one, by the convention that the empty product is equal to 1. This representation is called the representation of n, or the standard form of n. For example 999 = 33×37,1000 = 23×53,1001 = 7×11×13 Note that factors p0 =1 may be inserted without changing the value of n, allowing negative exponents provides a canonical form for positive rational numbers. However, as Integer factorization of large integers is much harder than computing their product, gcd or lcm, these formulas have, in practice, many arithmetical functions are defined using the canonical representation. In particular, the values of additive and multiplicative functions are determined by their values on the powers of prime numbers, the proof uses Euclids lemma, if a prime p divides the product of two natural numbers a and b, then p divides a or p divides b. We need to show that every integer greater than 1 is either prime or a product of primes, for the base case, note that 2 is prime. By induction, assume true for all numbers between 1 and n, if n is prime, there is nothing more to prove. Otherwise, there are integers a and b, where n = ab and 1 < a ≤ b < n, by the induction hypothesis, a = p1p2. pj and b = q1q2. qk are products of primes. But then n = ab = p1p2. pjq1q2. qk is a product of primes, assume that s >1 is the product of prime numbers in two different ways, s = p 1 p 2 ⋯ p m = q 1 q 2 ⋯ q n. We must show m = n and that the qj are a rearrangement of the pi, by Euclids lemma, p1 must divide one of the qj, relabeling the qj if necessary, say that p1 divides q1

5.
Prime number
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A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a number is called a composite number. For example,5 is prime because 1 and 5 are its only positive integer factors, the property of being prime is called primality. A simple but slow method of verifying the primality of a number n is known as trial division. It consists of testing whether n is a multiple of any integer between 2 and n, algorithms much more efficient than trial division have been devised to test the primality of large numbers. Particularly fast methods are available for numbers of forms, such as Mersenne numbers. As of January 2016, the largest known prime number has 22,338,618 decimal digits, there are infinitely many primes, as demonstrated by Euclid around 300 BC. There is no simple formula that separates prime numbers from composite numbers. However, the distribution of primes, that is to say, many questions regarding prime numbers remain open, such as Goldbachs conjecture, and the twin prime conjecture. Such questions spurred the development of branches of number theory. Prime numbers give rise to various generalizations in other domains, mainly algebra, such as prime elements. A natural number is called a number if it has exactly two positive divisors,1 and the number itself. Natural numbers greater than 1 that are not prime are called composite, among the numbers 1 to 6, the numbers 2,3, and 5 are the prime numbers, while 1,4, and 6 are not prime. 1 is excluded as a number, for reasons explained below. 2 is a number, since the only natural numbers dividing it are 1 and 2. Next,3 is prime, too,1 and 3 do divide 3 without remainder, however,4 is composite, since 2 is another number dividing 4 without remainder,4 =2 ·2. 5 is again prime, none of the numbers 2,3, next,6 is divisible by 2 or 3, since 6 =2 ·3. The image at the right illustrates that 12 is not prime,12 =3 ·4, no even number greater than 2 is prime because by definition, any such number n has at least three distinct divisors, namely 1,2, and n

6.
Pronic number
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A pronic number is a number which is the product of two consecutive integers, that is, a number of the form n. The study of these dates back to Aristotle. They are also called oblong numbers, heteromecic numbers, or rectangular numbers, however, the rectangular number name has also been applied to the composite numbers. The first few numbers are,0,2,6,12,20,30,42,56,72,90,110,132,156,182,210,240,272,306,342,380,420,462 …. The nth pronic number is also the difference between the odd square 2 and the st centered hexagonal number. The sum of the reciprocals of the numbers is a telescoping series that sums to 1,1 =12 +16 +112 ⋯ = ∑ i =1 ∞1 i. The partial sum of the first n terms in this series is ∑ i =1 n 1 i = n n +1, the nth pronic number is the sum of the first n even integers. It follows that all numbers are even, and that 2 is the only prime pronic number. It is also the only number in the Fibonacci sequence. The number of entries in a square matrix is always a pronic number. The fact that consecutive integers are coprime and that a number is the product of two consecutive integers leads to a number of properties. Each distinct prime factor of a number is present in only one of the factors n or n+1. Thus a pronic number is squarefree if and only if n and n +1 are also squarefree, the number of distinct prime factors of a pronic number is the sum of the number of distinct prime factors of n and n +1. If 25 is appended to the representation of any pronic number. This is because 2 =100 n 2 +100 n +25 =100 n +25

7.
On-Line Encyclopedia of Integer Sequences
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The On-Line Encyclopedia of Integer Sequences, also cited simply as Sloanes, is an online database of integer sequences. It was created and maintained by Neil Sloane while a researcher at AT&T Labs, Sloane continues to be involved in the OEIS in his role as President of the OEIS Foundation. OEIS records information on integer sequences of interest to professional mathematicians and amateurs, and is widely cited. As of 30 December 2016 it contains nearly 280,000 sequences, the database is searchable by keyword and by subsequence. Neil Sloane started collecting integer sequences as a student in 1965 to support his work in combinatorics. The database was at first stored on punched cards and he published selections from the database in book form twice, A Handbook of Integer Sequences, containing 2,372 sequences in lexicographic order and assigned numbers from 1 to 2372. The Encyclopedia of Integer Sequences with Simon Plouffe, containing 5,488 sequences and these books were well received and, especially after the second publication, mathematicians supplied Sloane with a steady flow of new sequences. The collection became unmanageable in book form, and when the database had reached 16,000 entries Sloane decided to go online—first as an e-mail service, as a spin-off from the database work, Sloane founded the Journal of Integer Sequences in 1998. The database continues to grow at a rate of some 10,000 entries a year, Sloane has personally managed his sequences for almost 40 years, but starting in 2002, a board of associate editors and volunteers has helped maintain the database. In 2004, Sloane celebrated the addition of the 100, 000th sequence to the database, A100000, in 2006, the user interface was overhauled and more advanced search capabilities were added. In 2010 an OEIS wiki at OEIS. org was created to simplify the collaboration of the OEIS editors and contributors, besides integer sequences, the OEIS also catalogs sequences of fractions, the digits of transcendental numbers, complex numbers and so on by transforming them into integer sequences. Sequences of rationals are represented by two sequences, the sequence of numerators and the sequence of denominators, important irrational numbers such as π =3.1415926535897. are catalogued under representative integer sequences such as decimal expansions, binary expansions, or continued fraction expansions. The OEIS was limited to plain ASCII text until 2011, yet it still uses a form of conventional mathematical notation. Greek letters are represented by their full names, e. g. mu for μ. Every sequence is identified by the letter A followed by six digits, sometimes referred to without the leading zeros, individual terms of sequences are separated by commas. Digit groups are not separated by commas, periods, or spaces, a represents the nth term of the sequence. Zero is often used to represent non-existent sequence elements, for example, A104157 enumerates the smallest prime of n² consecutive primes to form an n×n magic square of least magic constant, or 0 if no such magic square exists. The value of a is 2, a is 1480028129, but there is no such 2×2 magic square, so a is 0

8.
Highly composite number
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A highly composite number is a positive integer with more divisors than any smaller positive integer has. The term was coined by Ramanujan, the related concept of largely composite number refers to a positive integer which has at least as many divisors as any smaller positive integer. The initial or smallest 38 highly composite numbers are listed in the table below, the number of divisors is given in the column labeled d. The table below shows all the divisors of one of these numbers, the 15, 000th highly composite number can be found on Achim Flammenkamps website. Roughly speaking, for a number to be highly composite it has to have prime factors as small as possible, also, except in two special cases n =4 and n =36, the last exponent ck must equal 1. It means that 1,4, and 36 are the only square highly composite numbers, saying that the sequence of exponents is non-increasing is equivalent to saying that a highly composite number is a product of primorials. Note, that although the above described conditions are necessary, they are not sufficient for a number to be highly composite. For example,96 =25 ×3 satisfies the conditions and has 12 divisors but is not highly composite since there is a smaller number 60 which has the same number of divisors. If Q denotes the number of composite numbers less than or equal to x. The first part of the inequality was proved by Paul Erdős in 1944 and we have 1.13862 < lim inf log Q log log x ≤1.44 and lim sup log Q log log x ≤1.71. Highly composite numbers higher than 6 are also abundant numbers, one need only look at the three or four highest divisors of a particular highly composite number to ascertain this fact. It is false that all composite numbers are also Harshad numbers in base 10. The first HCN that is not a Harshad number is 245,044,800, which has a sum of 27. 10 of the first 38 highly composite numbers are highly composite numbers. The sequence of composite numbers is a subset of the sequence of smallest numbers k with exactly n divisors. A positive integer n is a composite number if d ≥ d for all m ≤ n. The counting function QL of largely composite numbers satisfies c ≤ log Q L ≤ d for positive c, d with 0.2 ≤ c ≤ d ≤0.5. Because the prime factorization of a composite number uses all of the first k primes

9.
Powerful number
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A powerful number is a positive integer m such that for every prime number p dividing m, p2 also divides m. Equivalently, a number is the product of a square and a cube, that is, a number m of the form m = a2b3. Powerful numbers are known as squareful, square-full, or 2-full. Paul Erdős and George Szekeres studied such numbers and Solomon W. Golomb named such numbers powerful, in the other direction, suppose that m is powerful, with prime factorization m = ∏ p i α i, where each αi ≥2. Define γi to be three if αi is odd, and zero otherwise, and define βi = αi - γi. Then, all values βi are nonnegative integers, and all values γi are either zero or three, so m = =23 supplies the desired representation of m as a product of a square. Informally, given the prime factorization of m, take b to be the product of the factors of m that have an odd exponent. Because m is powerful, each prime factor with an odd exponent has an exponent that is at least 3, in addition, each prime factor of m/b3 has an even exponent, so m/b3 is a perfect square, so call this a2, then m = a2b3. The representation m = a2b3 calculated in this way has the property that b is squarefree, the sum of the reciprocals of the powerful numbers converges. More generally, the sum of the reciprocals of the sth powers of the numbers is equal to ζ ζ ζ whenever it converges. Let k denote the number of numbers in the interval. Then k is proportional to the root of x. More precisely, c x 1 /2 −3 x 1 /3 ≤ k ≤ c x 1 /2, c = ζ / ζ =2.173 …, the two smallest consecutive powerful numbers are 8 and 9. However, one of the two numbers in a pair formed in this way must be a square. According to Guy, Erdős has asked whether there are many pairs of consecutive powerful numbers such as in which neither number in the pair is a square. Jaroslaw Wroblewski showed that there are indeed infinitely many such pairs by showing that 33c2 +1 = 73d2 has infinitely many solutions and it is a conjecture of Erdős, Mollin, and Walsh that there are no three consecutive powerful numbers. Any odd number is a difference of two squares,2 = k2 + 2k +1, so 2 − k2 = 2k +1. Similarly, any multiple of four is a difference of the squares of two numbers that differ by two,2 − k2 = 4k +4

10.
Perfect power
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In mathematics, a perfect power is a positive integer that can be expressed as an integer power of another positive integer. More formally, n is a perfect power if there exist natural numbers m >1, in this case, n may be called a perfect kth power. If k =2 or k =3, then n is called a square or perfect cube. Sometimes 1 is also considered a perfect power. The sum of the reciprocals of the perfect powers p without duplicates is, ∑ p 1 p = ∑ k =2 ∞ μ ≈0.874464368 … where μ is the Möbius function and this is sometimes known as the Goldbach-Euler theorem. Detecting whether or not a natural number n is a perfect power may be accomplished in many different ways. One of the simplest such methods is to all possible values for k across each of the divisors of n. This method can immediately be simplified by considering only prime values of k. This is because if n = m k for a composite k = a p p is prime. Because of this result, the value of k must necessarily be prime. As an example, consider n = 296·360·724, since gcd =12, n is a perfect 12th power. In 2002 Romanian mathematician Preda Mihăilescu proved that the pair of consecutive perfect powers is 23 =8 and 32 =9. Pillais conjecture states that for any positive integer k there are only a finite number of pairs of perfect powers whose difference is k. As an alternate way to perfect powers, the recursive approach has yet to be found useful. It is based on the observation that the difference between ab and b where a > b may not be constant, but if you take the difference of differences, b times. For example,94 =6561, and 104 is 10000, the difference between 84 and 94 is 2465, meaning the difference of differences is 974. A step further and you have 204, one step further, and you have 24, which is equal to 4. One step further and collating this key row from progressively larger exponents yields a similar to Pascals

11.
Arithmetic number
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In number theory, an arithmetic number is an integer for which the average of its positive divisors is also an integer. For instance,6 is a number because the average of its divisors is 1 +2 +3 +64 =3. However,2 is not a number because its only divisors are 1 and 2. It is known that the density of such numbers is 1, indeed. A number N is arithmetic if the number of divisors d divides the sum of divisors σ and it is known that the density of integers N obeying the stronger condition that d2 divides σ is 1/2