100 or one hundred is the natural number following 99 and preceding 101. In medieval contexts, it may be described as the short hundred or five score in order to differentiate the English and Germanic use of "hundred" to describe the long hundred of six score or 120. 100 is the square of 10. The standard SI prefix for a hundred is "hecto-". 100 is the basis of percentages. 100 is the sum of the first nine prime numbers, as well as the sum of some pairs of prime numbers e.g. 3 + 97, 11 + 89, 17 + 83, 29 + 71, 41 + 59, 47 + 53. 100 is the sum of the cubes of the first four integers. This is related by Nicomachus's theorem to the fact that 100 equals the square of the sum of the first four integers: 100 = 102 = 2.26 + 62 = 100, thus 100 is a Leyland number.100 is an 18-gonal number. It is divisible by 25, the number of primes below it, it can not be expressed as the difference between any integer and the total of coprimes below it, making it a noncototient. It can be expressed as a sum of some of its divisors.
100 is a Harshad number in base 10, in base 4, in that base it is a self-descriptive number. There are 100 prime numbers whose digits are in ascending order. 100 is the smallest number. One hundred is the atomic number of fermium, an actinide and the first of the heavy metals that cannot be created through neutron bombardment. On the Celsius scale, 100 degrees is the boiling temperature of pure water at sea level; the Kármán line lies at an altitude of 100 kilometres above the Earth's sea level and is used to define the boundary between Earth's atmosphere and outer space. There are 100 blasts of the Shofar heard in the service of the Jewish New Year. A religious Jew is expected to utter at least 100 blessings daily. In the Hindu book of the Mahabharata, the king Dhritarashtra had 100 sons known as the Kauravas; the United States Senate has 100 Senators. Most of the world's currencies are divided into 100 subunits; the 100 Euro banknotes feature a picture of a Rococo gateway on the obverse and a Baroque bridge on the reverse.
The U. S. hundred-dollar bill has Benjamin Franklin's portrait. S. bill in print. American savings bonds of $100 have Thomas Jefferson's portrait, while American $100 treasury bonds have Andrew Jackson's portrait. One hundred is also: The number of years in a century; the number of pounds in an American short hundredweight. In Greece, India and Nepal, 100 is the police telephone number. In Belgium, 100 is the firefighter telephone number. In United Kingdom, 100 is the operator telephone number; the HTTP status code indicating that the client should continue with its request. The 100 The age at which a person becomes a centenarian; the number of yards in an American football field. The number of runs required for a cricket batsman to score a significant milestone; the number of points required for a snooker player to score a century break, a significant milestone. The record number of points scored in one NBA game by a single player, set by Wilt Chamberlain of the Philadelphia Warriors on March 2, 1962.
1 vs. 100 AFI's 100 Years... Hundred Hundred Hundred Days Hundred Years' War List of highways numbered 100 Top 100 Greatest 100 Wells, D; the Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group.: 133 Chisholm, Hugh, ed.. "Hundred". Encyclopædia Britannica. Cambridge University Press. On the Number 100
In mathematics, factorization or factoring consists of writing a number or another mathematical object as a product of several factors smaller or simpler objects of the same kind. For example, 3 × 5 is a factorization of the integer 15, is a factorization of the polynomial x2 – 4. Factorization is not considered meaningful within number systems possessing division, such as the real or complex numbers, since any x can be trivially written as × whenever y is not zero. However, a meaningful factorization for a rational number or a rational function can be obtained by writing it in lowest terms and separately factoring its numerator and denominator. Factorization was first considered by ancient Greek mathematicians in the case of integers, they proved the fundamental theorem of arithmetic, which asserts that every positive integer may be factored into a product of prime numbers, which cannot be further factored into integers greater than 1. Moreover, this factorization is unique up to the order of the factors.
Although integer factorization is a sort of inverse to multiplication, it is much more difficult algorithmically, a fact, exploited in the RSA cryptosystem to implement public-key cryptography. Polynomial factorization has been studied for centuries. In elementary algebra, factoring a polynomial reduces the problem of finding its roots to finding the roots of the factors. Polynomials with coefficients in the integers or in a field possess the unique factorization property, a version of the fundamental theorem of arithmetic with prime numbers replaced by irreducible polynomials. In particular, a univariate polynomial with complex coefficients admits a unique factorization into linear polynomials: this is a version of the fundamental theorem of algebra. In this case, the factorization can be done with root-finding algorithms; the case of polynomials with integer coefficients is fundamental for computer algebra. There are efficient computer algorithms for computing factorizations within the ring of polynomials with rational number coefficients.
A commutative ring possessing the unique factorization property is called a unique factorization domain. There are number systems, such as certain rings of algebraic integers, which are not unique factorization domains. However, rings of algebraic integers satisfy the weaker property of Dedekind domains: ideals factor uniquely into prime ideals. Factorization may refer to more general decompositions of a mathematical object into the product of smaller or simpler objects. For example, every function may be factored into the composition of a surjective function with an injective function. Matrices possess many kinds of matrix factorizations. For example, every matrix has a unique LUP factorization as a product of a lower triangular matrix L with all diagonal entries equal to one, an upper triangular matrix U, a permutation matrix P. By the fundamental theorem of arithmetic, every integer greater than 1 has a unique factorization into prime numbers, which are those integers which cannot be further factorized into the product of integers greater than one.
For computing the factorization of an integer n, one needs an algorithm for finding a divisor q of n or deciding that n is prime. When such a divisor is found, the repeated application of this algorithm to the factors q and n / q gives the complete factorization of n. For finding a divisor q of n, if any, it suffices to test all values of q such that 1 < q and q2 ≤ n. In fact, if r is a divisor of n such that r2 > n q = n / r is a divisor of n such that q2 ≤ n. If one tests the values of q in increasing order, the first divisor, found is a prime number, the cofactor r = n / q cannot have any divisor smaller than q. For getting the complete factorization, it suffices thus to continue the algorithm by searching a divisor of r, not smaller than q and not greater than √r. There is no need to test all values of q for applying the method. In principle, it suffices to test only prime divisors; this needs to have a table of prime numbers that may be generated for example with the sieve of Eratosthenes.
As the method of factorization does the same work as the sieve of Eratosthenes, it is more efficient to test for a divisor only those numbers for which it is not clear whether they are prime or not. One may proceed by testing 2, 3, 5, the numbers > 5, whose last digit is 1, 3, 7, 9 and the sum of digits is not a multiple of 3. This method is inefficient for larger integers. For example, Pierre de Fermat was unable to discover that the 6th Fermat number 1 + 2 2 5 = 1 + 2 32 = 4 294 967 297 is not a prime number. In fact, applying the above method would require more than 10000 divisions, for a number that has 10 decimal digits. There are more efficient factoring algorithms; however they remain inefficient, as, with the present state of the art, one cannot factorize with the more powerful computers, a number of 500 decimal digits, the product of two randomly chosen prime numbers. This insures the security of the RSA cryptosystem, used for secure internet communication. For fa
A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. A natural number greater than 1, not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 6 is composite because it is the product of two numbers that are both smaller than 6. 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 factorized as a product of primes, unique up to their order; the property of being prime is called primality. A simple but slow method of checking the primality of a given number n, called trial division, tests whether n is a multiple of any integer between 2 and n. Faster algorithms include the Miller–Rabin primality test, fast but has a small chance of error, the AKS primality test, which always produces the correct answer in polynomial time but is too slow to be practical.
Fast methods are available for numbers of special forms, such as Mersenne numbers. As of December 2018 the largest known prime number has 24,862,048 decimal digits. There are infinitely many primes, as demonstrated by Euclid around 300 BC. No known simple formula separates prime numbers from composite numbers. However, the distribution of primes within the natural numbers in the large can be statistically modelled; the first result in that direction is the prime number theorem, proven at the end of the 19th century, which says that the probability of a randomly chosen number being prime is inversely proportional to its number of digits, that is, to its logarithm. Several historical questions regarding prime numbers are still unsolved; these include Goldbach's conjecture, that every integer greater than 2 can be expressed as the sum of two primes, the twin prime conjecture, that there are infinitely many pairs of primes having just one number between them. Such questions spurred the development of various branches of number theory, focusing on analytic or algebraic aspects of numbers.
Primes are used in several routines in information technology, such as public-key cryptography, which relies on the difficulty of factoring large numbers into their prime factors. In abstract algebra, objects that behave in a generalized way like prime numbers include prime elements and prime ideals. A natural number is called a prime number if it is greater than 1 and cannot be written as a product of two natural numbers that are both smaller than it; the numbers greater than 1 that are not prime are called composite numbers. In other words, n is prime if n items cannot be divided up into smaller equal-size groups of more than one item, or if it is not possible to arrange n dots into a rectangular grid, more than one dot wide and more than one dot high. For example, among the numbers 1 through 6, the numbers 2, 3, 5 are the prime numbers, as there are no other numbers that divide them evenly. 1 is not prime, as it is excluded in the definition. 4 = 2 × 2 and 6 = 2 × 3 are both composite. The divisors of a natural number n are the numbers.
Every natural number has both itself as a divisor. If it has any other divisor, it cannot be prime; this idea leads to a different but equivalent definition of the primes: they are the numbers with two positive divisors, 1 and the number itself. Yet another way to express the same thing is that a number n is prime if it is greater than one and if none of the numbers 2, 3, …, n − 1 divides n evenly; the first 25 prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97. No number n greater than 2 is prime because any such number can be expressed as the product 2 × n / 2. Therefore, every prime number other than 2 is an odd number, is called an odd prime; when written in the usual decimal system, all prime numbers larger than 5 end in 1, 3, 7, or 9. The numbers that end with other digits are all composite: decimal numbers that end in 0, 2, 4, 6, or 8 are and decimal numbers that end in 0 or 5 are divisible by 5; the set of all primes is sometimes denoted by P or by P.
The Rhind Mathematical Papyrus, from around 1550 BC, has Egyptian fraction expansions of different forms for prime and composite numbers. However, the earliest surviving records of the explicit study of prime numbers come from Ancient Greek mathematics. Euclid's Elements proves the infinitude of primes and the fundamental theorem of arithmetic, shows how to construct a perfect number from a Mersenne prime. Another Greek invention, the Sieve of Eratosthenes, is still used to construct lists of primes. Around 1000 AD, the Islamic mathematician Alhazen found Wilson's theorem, characterizing the prime numbers as the numbers n that evenly divide
In mathematics, the natural numbers are those used for counting and ordering. In common mathematical terminology, words colloquially used for counting are "cardinal numbers" and words connected to ordering represent "ordinal numbers"; the natural numbers can, at times, appear as a convenient set of codes. Some definitions, including the standard ISO 80000-2, begin the natural numbers with 0, corresponding to the non-negative integers 0, 1, 2, 3, …, whereas others start with 1, corresponding to the positive integers 1, 2, 3, …. Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, but in other writings, that term is used instead for the integers; the natural numbers are a basis from which many other number sets may be built by extension: the integers, by including the neutral element 0 and an additive inverse for each nonzero natural number n. These chains of extensions make the natural numbers canonically embedded in the other number systems.
Properties of the natural numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics. In common language, for example in primary school, natural numbers may be called counting numbers both to intuitively exclude the negative integers and zero, to contrast the discreteness of counting to the continuity of measurement, established by the real numbers; the most primitive method of representing a natural number is to put down a mark for each object. A set of objects could be tested for equality, excess or shortage, by striking out a mark and removing an object from the set; the first major advance in abstraction was the use of numerals to represent numbers. This allowed systems to be developed for recording large numbers; the ancient Egyptians developed a powerful system of numerals with distinct hieroglyphs for 1, 10, all the powers of 10 up to over 1 million.
A stone carving from Karnak, dating from around 1500 BC and now at the Louvre in Paris, depicts 276 as 2 hundreds, 7 tens, 6 ones. The Babylonians had a place-value system based on the numerals for 1 and 10, using base sixty, so that the symbol for sixty was the same as the symbol for one, its value being determined from context. A much advance was the development of the idea that 0 can be considered as a number, with its own numeral; the use of a 0 digit in place-value notation dates back as early as 700 BC by the Babylonians, but they omitted such a digit when it would have been the last symbol in the number. The Olmec and Maya civilizations used 0 as a separate number as early as the 1st century BC, but this usage did not spread beyond Mesoamerica; the use of a numeral 0 in modern times originated with the Indian mathematician Brahmagupta in 628. However, 0 had been used as a number in the medieval computus, beginning with Dionysius Exiguus in 525, without being denoted by a numeral; the first systematic study of numbers as abstractions is credited to the Greek philosophers Pythagoras and Archimedes.
Some Greek mathematicians treated the number 1 differently than larger numbers, sometimes not as a number at all. Independent studies occurred at around the same time in India and Mesoamerica. In 19th century Europe, there was mathematical and philosophical discussion about the exact nature of the natural numbers. A school of Naturalism stated that the natural numbers were a direct consequence of the human psyche. Henri Poincaré was one of its advocates, as was Leopold Kronecker who summarized "God made the integers, all else is the work of man". In opposition to the Naturalists, the constructivists saw a need to improve the logical rigor in the foundations of mathematics. In the 1860s, Hermann Grassmann suggested a recursive definition for natural numbers thus stating they were not natural but a consequence of definitions. Two classes of such formal definitions were constructed. Set-theoretical definitions of natural numbers were initiated by Frege and he defined a natural number as the class of all sets that are in one-to-one correspondence with a particular set, but this definition turned out to lead to paradoxes including Russell's paradox.
Therefore, this formalism was modified so that a natural number is defined as a particular set, any set that can be put into one-to-one correspondence with that set is said to have that number of elements. The second class of definitions was introduced by Charles Sanders Peirce, refined by Richard Dedekind, further explored by Giuseppe Peano, it is based on an axiomatization of the properties of ordinal numbers: each natural number has a
In mathematics and digital electronics, a binary number is a number expressed in the base-2 numeral system or binary numeral system, which uses only two symbols: "0" and "1". The base-2 numeral system is a positional notation with a radix of 2; each digit is referred to as a bit. Because of its straightforward implementation in digital electronic circuitry using logic gates, the binary system is used by all modern computers and computer-based devices; the modern binary number system was studied in Europe in the 16th and 17th centuries by Thomas Harriot, Juan Caramuel y Lobkowitz, Gottfried Leibniz. However, systems related to binary numbers have appeared earlier in multiple cultures including ancient Egypt and India. Leibniz was inspired by the Chinese I Ching; the scribes of ancient Egypt used two different systems for their fractions, Egyptian fractions and Horus-Eye fractions. Horus-Eye fractions are a binary numbering system for fractional quantities of grain, liquids, or other measures, in which a fraction of a hekat is expressed as a sum of the binary fractions 1/2, 1/4, 1/8, 1/16, 1/32, 1/64.
Early forms of this system can be found in documents from the Fifth Dynasty of Egypt 2400 BC, its developed hieroglyphic form dates to the Nineteenth Dynasty of Egypt 1200 BC. The method used for ancient Egyptian multiplication is closely related to binary numbers. In this method, multiplying one number by a second is performed by a sequence of steps in which a value is either doubled or has the first number added back into it; this method can be seen in use, for instance, in the Rhind Mathematical Papyrus, which dates to around 1650 BC. The I Ching dates from the 9th century BC in China; the binary notation in the I Ching is used to interpret its quaternary divination technique. It is based on taoistic duality of yin and yang.eight trigrams and a set of 64 hexagrams, analogous to the three-bit and six-bit binary numerals, were in use at least as early as the Zhou Dynasty of ancient China. The Song Dynasty scholar Shao Yong rearranged the hexagrams in a format that resembles modern binary numbers, although he did not intend his arrangement to be used mathematically.
Viewing the least significant bit on top of single hexagrams in Shao Yong's square and reading along rows either from bottom right to top left with solid lines as 0 and broken lines as 1 or from top left to bottom right with solid lines as 1 and broken lines as 0 hexagrams can be interpreted as sequence from 0 to 63. The Indian scholar Pingala developed a binary system for describing prosody, he used binary numbers in the form of long syllables, making it similar to Morse code. Pingala's Hindu classic titled Chandaḥśāstra describes the formation of a matrix in order to give a unique value to each meter; the binary representations in Pingala's system increases towards the right, not to the left like in the binary numbers of the modern, Western positional notation. The residents of the island of Mangareva in French Polynesia were using a hybrid binary-decimal system before 1450. Slit drums with binary tones are used to encode messages across Asia. Sets of binary combinations similar to the I Ching have been used in traditional African divination systems such as Ifá as well as in medieval Western geomancy.
In the late 13th century Ramon Llull had the ambition to account for all wisdom in every branch of human knowledge of the time. For that purpose he developed a general method or ‘Ars generalis’ based on binary combinations of a number of simple basic principles or categories, for which he has been considered a predecessor of computing science and artificial intelligence. In 1605 Francis Bacon discussed a system whereby letters of the alphabet could be reduced to sequences of binary digits, which could be encoded as scarcely visible variations in the font in any random text. For the general theory of binary encoding, he added that this method could be used with any objects at all: "provided those objects be capable of a twofold difference only. John Napier in 1617 described a system he called location arithmetic for doing binary calculations using a non-positional representation by letters. Thomas Harriot investigated several positional numbering systems, including binary, but did not publish his results.
The first publication of the system in Europe was by Juan Caramuel y Lobkowitz, in 1700. Leibniz studied binary numbering in 1679. Leibniz's system uses 1, like the modern binary numeral system. An example of Leibniz's binary numeral system is as follows: 0 0 0 1 numerical value 20 0 0 1 0 numerical value 21 0 1 0 0 numerical value 22 1 0 0 0 numerical value 23Leibniz interpreted the hexagrams of the I Ching as evidence of binary calculus; as a Sinophile, Leibniz was aware of
In number theory, a perfect number is a positive integer, equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself. Equivalently, a perfect number is a number, half the sum of all of its positive divisors i.e. σ1 = 2n. 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 perfect number is 6. Its proper divisors are 1, 2, 3, 1 + 2 + 3 = 6. Equivalently, the number 6 is equal to half the sum of all its positive divisors: ÷ 2 = 6; the next perfect number is 28: 28 = 1 + 2 + 4 + 7 + 14. This is followed by the perfect numbers 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 had noted 8128 as early as 100 AD. 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. In a manuscript written between 1456 and 1461, an unknown mathematician recorded the earliest European reference to a fifth perfect number, with 33,550,336 being identified for the first time.
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. Nicomachus conjectured. Ibn al-Haytham circa 1000 AD conjectured 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 infinitely many perfect numbers, nor whether there are infinitely many Mersenne primes; as well as having the form 2p−1, each perfect number is the th triangular number and the 2p−1th hexagonal number.
Furthermore, each perfect number except for 6 is the th centered nonagonal number and is equal to the sum of the first 2/2 odd cubes: 6 = 2 1 = 1 + 2 +
In mathematics, parity is the property of an integer's inclusion in one of two categories: or odd. An integer is if it is divisible by two and odd if it is not even. For example, 6 is because there is no remainder when dividing it by 2. By contrast, 3, 5, 7, 21 leave a remainder of 1 when divided by 2. Examples of numbers include −4, 0, 82 and 178. In particular, zero is an number; some examples of odd numbers are −5, 3, 29, 73. A formal definition of an number is that it is an integer of the form n = 2k, where k is an integer, it is important to realize that the above definition of parity applies only to integer numbers, hence it cannot be applied to numbers like 1/2 or 4.201. See the section "Higher mathematics" below for some extensions of the notion of parity to a larger class of "numbers" or in other more general settings; the sets of and odd numbers can be defined as following: Even = Odd = A number expressed in the decimal numeral system is or odd according to whether its last digit is or odd.
That is, if the last digit is 1, 3, 5, 7, or 9 it is odd. The same idea will work using any base. In particular, a number expressed in the binary numeral system is odd if its last digit is 1 and if its last digit is 0. In an odd base, the number is according to the sum of its digits – it is if and only if the sum of its digits is even; the following laws can be verified using the properties of divisibility. They are a special case of rules in modular arithmetic, are used to check if an equality is to be correct by testing the parity of each side; as with ordinary arithmetic and addition are commutative and associative in modulo 2 arithmetic, multiplication is distributive over addition. However, subtraction in modulo 2 is identical to addition, so subtraction possesses these properties, not true for normal integer arithmetic. Even ± = even; the division of two whole numbers does not result in a whole number. For example, 1 divided by 4 equals 1/4, neither nor odd, since the concepts and odd apply only to integers.
But when the quotient is an integer, it will be if and only if the dividend has more factors of two than the divisor. The ancient Greeks considered 1, the monad, to be neither odd nor even; some of this sentiment survived into the 19th century: Friedrich Wilhelm August Fröbel's 1826 The Education of Man instructs the teacher to drill students with the claim that 1 is neither nor odd, to which Fröbel attaches the philosophical afterthought, It is well to direct the pupil's attention here at once to a great far-reaching law of nature and of thought. It is this, that between two different things or ideas there stands always a third, in a sort of balance, seeming to unite the two. Thus, there is here between odd and numbers one number, neither of the two. In form, the right angle stands between the acute and obtuse angles. A thoughtful teacher and a pupil taught to think for himself can scarcely help noticing this and other important laws. Integer coordinates of points in Euclidean spaces of two or more dimensions have a parity defined as the parity of the sum of the coordinates.
For instance, the face-centered cubic lattice and its higher-dimensional generalizations, the Dn lattices, consist of all of the integer points whose sum of coordinates is even. This feature manifests itself in chess, where the parity of a square is indicated by its color: bishops are constrained to squares of the same parity; this form of parity was famously used to solve the mutilated chessboard problem: if two opposite corner squares are removed from a chessboard the remaining board cannot be covered by dominoes, because each domino covers one square of each parity and there are two more squares of one parity than of the other. The parity of an ordinal number may be defined to be if the number is a limit ordinal, or a limit ordinal plus a finite number, odd otherwise. Let R be a commutative ring and let I be an ideal of R whose index is 2. Elements of the coset 0 + I may be called while elements of the coset 1 + I may be called odd; as an example, let R = Z be the localization of Z at the prime ideal.
An element of R is or odd if and only if its numerator is so in Z. The numbers form an ideal in the ring of integers, but the odd numbers do not — this is clear from the fact that the identity element for addition, zero, is an element of the numbers only. An integer is if it is congruent to 0 modulo this ideal, in other words if it is congruent to 0 modulo 2, odd if it is congruent to 1 modulo 2. All prime numbers are odd, with one exception: the prime number 2. All known perfect numbers are even. Goldbach's conjecture states that every integer greater than 2 can be represented as a sum of two prime numbers. Modern computer calculations have shown this conjecture to