number theory

12-string guitar

12-string guitar.



One is the first positive integer and the first odd number; it is also known as unity. From the time of Euclid to the late 1500s, one wasn't generally considered to be a number but instead was thought of as the unit of which bona fide numbers were composed. The Old English (c.550–c.1100) ane served both for counting and as the indefinite article. Toward the end of the Old English period and the beginning of Middle English (c.1100-c.1500), ane developed two pronunciations, the first being used for 1 and the other for the indefinite article an, a. The existence of different words for the number one and the indefinite article seems to be unique to the English language. "One" can be traced back to the Latin unus and the Greek oine but probably came into English from the German eine.



Two is the first even number and the only even prime number. The word comes from the Greek dyo and the Latin duo through the Old English twa. Early languages often had both feminine and masculine forms for two and so there are a lot of diverse roots related to "two-ness". Many "two" words use the Greek root bi; biannual, binary, biscuit, and biceps, for examples. Others come from the Old English twa, such as between, twilight, twist, and twin. From duo we get dual, duet, dubious (of two minds), duplex (two layers), and double. The Latin di gives us diploma (two papers) and dihedral. The earlier Greek dyo produces dyad, composed of two parts. Two is the only positive real number that gives the same result when added to itself as when multiplied by itself. It is conjectured that 2 is the only even integer that cannot be written as the sum of two primes (see Goldbach's conjecture) and it has recently been proved that 2 is the largest value of n for which the equation x n + y n = z n has nonzero integer solutions (see Fermat's last theorem). Two is the base of the binary number system.



1 + 1 = 3, for large values of 1.


Three is the number of dimensions of the space in which we live; three is also the smallest odd prime number, the second triangular number, and a Fibonacci number. Three is often the number of repetitions in jokes and children's stories (for example, the tale of the Three Little Pigs and of the porridge-eating bears), because it is the minimum number needed to establish a pattern (such as a regular tempo) or to convey the impression of an ongoing sequence or succession. In the Christian tradition, three plays a crucial role: Christ represents one third of the Trinity (the Father, Son and Holy Ghost), was visited by the three wise men and, 33 years later, when Peter disowned him three times, rose on the third day after the crucifixion, having died at 3 p.m. Other things that come in threes: musketeers, primary colors, wishes, blind mice, bad luck, and London buses.



Four is the smallest composite number, the second smallest square number, the first non-Fibonacci number, the smallest Smith number, and the smallest number that can be written as the sum of two prime numbers. Four is the number of dimensions that make up spacetime (three of space and one of time). It is the most number of colors needed to color any map so that no two neighboring areas are the same color (see four-color problem). There are four cardinal points on the compass, four Riders of the Apocalypse, and four Gospels.



Five is the length of the hypotenuse of the smallest Pythagorean triangle (a right triangle having integral sides). Five is the only prime number that is a member of two pairs of twin primes. Every integer is the sum of five positive or negative cubes in an infinite number of ways. Five is the smallest degree of a polynomial equation for which there is no general formula for the solutions (see quintic).



Six is the smallest perfect number, the number of faces of a cube, and the number of sides of a hexagon. There are six players on a volleyball team, six kinds of chessmen, and six types of quark (not including antiquarks). A touchdown in American football earns six points and a hit across the boundary rope of a cricket field without bouncing scores six runs. Long ago people indicated a number by pointing to a part of their body; this is echoed in the New Guinea word for six, which is the same as that for "wrist."



Seven is a lucky number in the eyes of many people and one that has been given much spiritual significance. The early religious and cultural use of the seven-day week almost certainly stems from the fact that the Moon goes through its four phases in a bit over 28 days, which divides nicely into seven days per phase. There are seven moving objects in the sky visible to the naked eye (the Sun, the Moon, Mercury, Venus, Mars, Jupiter, and Saturn), seven seas, seven orders of architecture, seven deadly sins, and seven liberal arts and sciences, and seven dwarves. The seventh son of a seventh son is supposed to be born gifted (Donny Osmond was such a person). In the Bible, there were seven years of famine and seven years of plenty, and seven years were taken to construct King Solomon's Temple. The Pythagoreans were especially intrigued by the number as it is the sum of three and four, which are the number of sides of a triangle and a square-shapes of enormous importance to the sect. These links with Solomon's Temple and the Pythagoreans help explain the importance of seven in freemasonry.


Seven is the smallest positive integer whose reciprocal has a pattern of more than one repeating digit: 1/7 = 0.142857142857... and is the smallest number for which the digit sequence of 1/n is of length n-1 (the longest such a sequence can be). The next such numbers are 17, 19, 23, 29, 47, 59, 61, 97, 109, 113, ... Other curios: the citrus soda 7-UP, created in 1929, was so called because the original containers were 7 ounces and "up" was the direction of the bubbles, and seven is the maximum number of times you can fold any sheet of paper (try it!).



Eight is the second smallest cube number (after 13): 8 = 23 = 2 × 2 × 2. A queen or king in chess can move in eight different directions, in the same way that a compass has eight principal points: north, northeast, east, southeast, south, southwest, west, and northwest. In three dimensions, there are eight diagonal ways to move, corresponding to the eight octants into which three-dimensional space is divided by three mutually-perpendicular planes. Add a fourth dimension and movement becomes possible back and forth along four directions at right angles to each other: up and down, left and right, forward and back, and one other! The Spanish dollar was a gold coin with a value of eight reale, and was sometimes actually cut into eight wedge-shaped pieces – "pieces of eight" – to make change.



Nine is a number long considered to have strange, mystic properties. A phrase in a book written during the Dark Ages gave rise to the superstition that cats have nine lives. English author and satirist William Baldwin wrote in his Beware the Cat, "It is permitted for a witch to take her cat's body nine times." There were nine Muses, nine rivers of Hades, and nine heads on the Hydra. It took nine days for Vulcan to fall from the heavens. The phrase "nine days' wonder" comes from the proverb "a wonder lasts nine days and then the puppy's eyes are open." A cat-o'-nine-tails is a whip, usually made of nine knotted lines or cords fastened to a handle that produces scars like the scratches of a cat. Being on "cloud nine" may have its origin in Dante's ninth heaven of Paradise, whose inhabitants are blissful because they are closest to God. The term "the whole 9 yards" came from World War II fighter pilots in the Pacific. When arming their planes on the ground, the .50-caliber machine gun ammo belts measured exactly 27 feet, before being loaded into the fuselage. If the pilots fired all their ammo at a target, it got "the whole 9 yards." Less certain – though there is no shortage of theories – is the sources of the expression "dressed to the nines."


Nine is the largest single-digit number and the one that occurs least frequently in most situations; an exception is the tendency of businesses to set prices that end with one or more 9's. Because 9 is one less than the base of our number system, it is easy to see if a number is divisible by 9 by adding the digits (and repeating on the result if necessary). This process is sometimes called casting out nines. Similar processes can be developed for divisibility by 99, 999, etc. or any number that divides one of these numbers. Nine has many other interesting properties. For example, write down a number containing as many digits as you like, add these digits together, and deduct the sum from the first number. The sum of the digits of this new number will always be a multiple of nine.



Ten is the base of our familiar number system, which stems directly from the fact that we have ten fingers on which to count. Ten is the only triangular number that is a sum of consecutive odd squares (10 = 12 + 32) and the only composite integer such that all of its positive integer divisors other than 1 are of the form x2 + 1 (2 = 12 + 1, 5 = 22 + 1, 10 = 32 + 1). Strange but true: the lifespan of a taste bud is ten days.



Eleven is a palindromic number, the smallest integer that is not a Niven number, a prime number that is a member of a twin prime (11 and 13), and the largest integer that is not the sum of two or more distinct primes. There are 11 players on a soccer team and on a cricket team. Strange but true: the youngest pope was 11 years old.



Twelve is a number heavily used for grouping things (inches, hours, 12-packs), partly because it can be divided evenly in several different ways (by 2, 3, 4, and 6) and partly because there are roughly 12 cycles of the Moon for every one of the Sun. The Latin duodecim (two + ten) for twelve forms the root of dodecagon (originally duodecagon), meaning a 12-sided shape, and duodenum, the first part of the intestine that is about twelve inches long. Contracted and modified over the years, duodecim became "dozen." The number system based on 12 is called duodecimal.


Multiples of 12 have also been used by many cultures for various units and measures. A "shock" was 60 or five dozen (a dozen for each finger on one hand), and many cultures had a "great hundred" of 120 or ten dozen (a dozen for each finger on both hands). The Romans used a fraction system based on 12 and the smallest part, an uncil, became our word for an ounce. The French emperor Charlemagne established a monetary system that had a base of twelve and twenty, the remnants of which persist. Until 1970, the English pound sterling consisted of 20 shillings, and each shilling contained 12 pence. In 1944, The Duodecimal Society (which later changed its named to the Dozenal Society) was formed in New York with the purpose of proposing a switch to base 12 for all scientific work. There are 12 signs of the zodiac and there were 12 apostles of Christ.


Twelve is the smallest abundant number, a Niven number, and a semiperfect number (because 12 = 1 + 2 + 3 + 6; see perfect number).



Thirteen is the unluckiest of numbers – if you happen to be superstitious. This belief has a couple of historical roots. According to Biblical tradition, there were 13 people at Christ's Last Supper, and Christ was crucified on Friday 13th. Further back in time, Alexander the Great decided he wanted to be the thirteenth god alongside the 12 that already stood for each month of the year, so he had a thirteenth statue built on the place of his capital. His death shortly after gave the number a bad name. Many buildings don't have a floor labeled 13 and many hotels will have room numbered 12A instead of 13. There is even a name for a morbid fear of 13: triskaidekaphobia. Fresh disasters involving the number hardly help triskaidekaphobics overcome their affliction. The most notorious of these involved the Apollo 13 Moon mission, which was launched on April 11, 1970 (the sum of 4, 11 and 70 equals 85, the digital sum of which is 13), from Pad 39 (three times 13) at 13:13 local time, and suffered an explosion on April 13. (The astronauts did, however, make it home safely, which could be considered good luck.) There is always at least one Friday 13th in each year; in some years, there are two and rarely three (for example, 1998 and 2009). There were 13 original U.S. colonies (hence the 13 stripes on the American flag) and 13 signers of the Declaration of Independence. In Japan, the numbers 4 and 9 are considered unlucky, not because 13 can be represented as sum of these two perfect squares but because of their pronunciation. In Japanese, four is shi, which is pronounced the same as the word for death; nine is ku, which sounds the same as the word for torture. And speaking of torture, it was not unusual in times past for bakers to come in for stiff punishment if they shortchanged their customers. In ancient Egypt, someone found selling light loaves might end up with his ear nailed to a doorpost, while in medieval Britain the punishment was likely to be a spell in the pillory. This led to the custom of adding a thirteenth loaf to every batch of 12 to be on the safe side, and hence the expression "a baker's dozen."


Mathematically, the reverse of the square of 13 is the same as the square of the reverse of 13: 132 = 169; the reverse of 169 is 961 and the reverse of 13 is 31; 312 = 961.


Thirteen is the smallest prime number that can be expressed as the sum of the squares of two prime numbers: 13 = 22 + 32. Also the sum of all prime numbers up to 13 (2 + 3 + 5 + 7 + 11 + 13) is equal to the thirteenth prime number (41), and this is the largest such number. On the anagrammatical front is this nice equation: ELEVEN + TWO = TWELVE + ONE.



Seventeen is the number most often picked in response to the request "Pick a random number from 1 to 20." Seventeen is a Fermat prime (a prime number of the form 22n + 1, where n is a positive integer), the exponent of a Mersenne prime (a prime p for which 2p - 1 is prime), and the only prime that is the sum of four consecutive primes (2 + 3 + 5 + 7). Seventeen is also the smallest number for which the sum of the digits of its cube is equal to the number: 173 = 4913, 4 + 9 + 1 + 3 = 17, and the smallest number that can be written as A2 + B3 in two different ways: 17 = 32 + 23 = 42 + 13. The pair (8, 9), whose sum is 17, is the only pair of consecutive numbers where one is a square and the other is a cube (a result proved by Leonhard Euler.) There are 17 planar crystallographic groups. The minimum number of faces on a convex polyhedron that has only one stable face is 17. (A stable face is one that the figure can rest on without falling over; most polygons have more than one such face.) Seventeen is also the answer to the follow problem: At a party where any two people have previously met each other in one of three other places, what is the least number of people who must be at the party to guarantee that there is at least one group of three people who have met each other before in the same place?



Many early cultures, including some in Europe, the Mayans of Central America, and the Ainu, the indigenous people of the Japanese islands, used a base of twenty for counting. The base 20 system was retained until about 1970 in the British monetary system, in which there were 20 shillings to the pound. Two of the five Platonic solids involve 20: the icosahedron has 20 triangular faces and the dodecahedron has 20 vertices. Twenty is a Niven number, a semiperfect number, and a practical number. It is also a tetrahedral number – the sum of consecutive triangular numbers (1 + 3 + 6 + 10).


A score is a group of 20 items. The word comes from the Old Norse skor for a heavy mark used to indicate a string of twenty smaller marks; skor, in turn, is descended from the Indo-European sker, for cutting or slicing. From about 1400, score was also the word for a record or an amount due – the total of the score marks on a tally. It became a common word for the total of a tradesman's or innkeeper's account. So, to settle the score originally meant just to pay one's bill. But it acquired the figurative sense of taking revenge on somebody, and that's usually what is meant by the expression now. The more general meaning of score, as a tally, is used daily when the results of sports competitions are reported.