1.
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
2.
Integer
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An integer is a number that can be written without a fractional component. For example,21,4,0, and −2048 are integers, while 9.75, 5 1⁄2, the set of integers consists of zero, the positive natural numbers, also called whole numbers or counting numbers, and their additive inverses. This is often denoted by a boldface Z or blackboard bold Z standing for the German word Zahlen, ℤ is a subset of the sets of rational and real numbers and, like the natural numbers, is countably infinite. The integers form the smallest group and the smallest ring containing the natural numbers, in algebraic number theory, the integers are sometimes called rational integers to distinguish them from the more general algebraic integers. In fact, the integers are the integers that are also rational numbers. Like the natural numbers, Z is closed under the operations of addition and multiplication, that is, however, with the inclusion of the negative natural numbers, and, importantly,0, Z is also closed under subtraction. The integers form a ring which is the most basic one, in the following sense, for any unital ring. This universal property, namely to be an object in the category of rings. Z is not closed under division, since the quotient of two integers, need not be an integer, although the natural numbers are closed under exponentiation, the integers are not. The following lists some of the properties of addition and multiplication for any integers a, b and c. In the language of algebra, the first five properties listed above for addition say that Z under addition is an abelian group. As a group under addition, Z is a cyclic group, in fact, Z under addition is the only infinite cyclic group, in the sense that any infinite cyclic group is isomorphic to Z. The first four properties listed above for multiplication say that Z under multiplication is a commutative monoid. However, not every integer has an inverse, e. g. there is no integer x such that 2x =1, because the left hand side is even. This means that Z under multiplication is not a group, all the rules from the above property table, except for the last, taken together say that Z together with addition and multiplication is a commutative ring with unity. It is the prototype of all objects of algebraic structure. Only those equalities of expressions are true in Z for all values of variables, note that certain non-zero integers map to zero in certain rings. The lack of zero-divisors in the means that the commutative ring Z is an integral domain
3.
Smooth number
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In number theory, a smooth number is an integer which factors completely into small prime numbers. The term seems to have coined by Leonard Adleman. Smooth numbers are important in cryptography relying on factorization. The 2-smooth numbers are just the powers of 2, a positive integer is called B-smooth if none of its prime factors is greater than B. For example,1,620 has prime factorization 22 ×34 ×5 and this definition includes numbers that lack some of the smaller prime factors, for example, both 10 and 12 are 5-smooth, despite the fact that they miss out prime factors 3 and 5 respectively. Note that B does not have to be a prime factor, if the largest prime factor of a number is p then the number is B-smooth for any B ≥ p. Usually B is given as a prime, but composite numbers work as well, a number is B-smooth if and only if it is p-smooth, where p is the largest prime less than or equal to B. By using B-smooth numbers, one ensures that the cases of this recursion are small primes. 5-smooth or regular numbers play a role in Babylonian mathematics. They are also important in theory, and the problem of generating these numbers efficiently has been used as a test problem for functional programming. Smooth numbers have a number of applications to cryptography, although most applications involve cryptanalysis, the VSH hash function is one example of a constructive use of smoothness to obtain a provably secure design. Let Ψ denote the number of y-smooth integers less than or equal to x, if the smoothness bound B is fixed and small, there is a good estimate for Ψ, Ψ ∼1 π. ∏ p ≤ B log x log p. where π denotes the number of less than or equal to B. Otherwise, define the parameter u as u = log x / log y, then, Ψ = x ⋅ ρ + O where ρ is the Dickman function. The average size of the part of a number of given size is known as ζ. Further, m is called B-powersmooth if all prime powers p ν dividing m satisfy, for example,720 is 5-smooth but is not 5-powersmooth. It is 16-powersmooth since its greatest prime factor power is 24 =16, the number is also 17-powersmooth, 18-powersmooth, etc. B-smooth and B-powersmooth numbers have applications in number theory, such as in Pollards p −1 algorithm, for example, the Pohlig–Hellman algorithm for computing discrete logarithms has a running time of O for groups of B-smooth order
4.
Regular polygon
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In Euclidean geometry, a regular polygon is a polygon that is equiangular and equilateral. Regular polygons may be convex or star, in the limit, a sequence of regular polygons with an increasing number of sides becomes a circle, if the perimeter is fixed, or a regular apeirogon, if the edge length is fixed. These properties apply to all regular polygons, whether convex or star, a regular n-sided polygon has rotational symmetry of order n. All vertices of a regular polygon lie on a common circle and that is, a regular polygon is a cyclic polygon. Together with the property of equal-length sides, this implies that every regular polygon also has a circle or incircle that is tangent to every side at the midpoint. Thus a regular polygon is a tangential polygon, a regular n-sided polygon can be constructed with compass and straightedge if and only if the odd prime factors of n are distinct Fermat primes. The symmetry group of a regular polygon is dihedral group Dn, D2, D3. It consists of the rotations in Cn, together with reflection symmetry in n axes that pass through the center, if n is even then half of these axes pass through two opposite vertices, and the other half through the midpoint of opposite sides. If n is odd then all pass through a vertex. All regular simple polygons are convex and those having the same number of sides are also similar. An n-sided convex regular polygon is denoted by its Schläfli symbol, for n <3 we have two degenerate cases, Monogon, degenerate in ordinary space. Digon, a line segment, degenerate in ordinary space. In certain contexts all the polygons considered will be regular, in such circumstances it is customary to drop the prefix regular. For instance, all the faces of uniform polyhedra must be regular, for n >2 the number of diagonals is n 2, i. e.0,2,5,9. for a triangle, square, pentagon, hexagon. The diagonals divide the polygon into 1,4,11,24, for a regular n-gon inscribed in a unit-radius circle, the product of the distances from a given vertex to all other vertices equals n. For a regular simple n-gon with circumradius R and distances di from a point in the plane to the vertices. For a regular n-gon, the sum of the distances from any interior point to the n sides is n times the apothem. This is a generalization of Vivianis theorem for the n=3 case, the sum of the perpendiculars from a regular n-gons vertices to any line tangent to the circumcircle equals n times the circumradius
5.
Conic section
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In mathematics, a conic section is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse, the circle is a special case of the ellipse, and is of sufficient interest in its own right that it was sometimes called a fourth type of conic section. The conic sections have been studied by the ancient Greek mathematicians with this work culminating around 200 BC, the conic sections of the Euclidean plane have various distinguishing properties. Many of these have used as the basis for a definition of the conic sections. The type of conic is determined by the value of the eccentricity, in analytic geometry, a conic may be defined as a plane algebraic curve of degree 2, that is, as the set of points whose coordinates satisfy a quadratic equation in two variables. This equation may be written in form, and some geometric properties can be studied as algebraic conditions. In the Euclidean plane, the conic sections appear to be different from one another. By extending the geometry to a projective plane this apparent difference vanishes, further extension, by expanding the real coordinates to admit complex coordinates, provides the means to see this unification algebraically. The conic sections have been studied for thousands of years and have provided a source of interesting. A conic is the curve obtained as the intersection of a plane, called the cutting plane and we shall assume that the cone is a right circular cone for the purpose of easy description, but this is not required, any double cone with some circular cross-section will suffice. Planes that pass through the vertex of the cone will intersect the cone in a point and these are called degenerate conics and some authors do not consider them to be conics at all. Unless otherwise stated, we assume that conic refers to a non-degenerate conic. There are three types of conics, the ellipse, parabola, and hyperbola, the circle is a special kind of ellipse, although historically it had been considered as a fourth type. The circle and the ellipse arise when the intersection of the cone and plane is a closed curve, if the cutting plane is parallel to exactly one generating line of the cone, then the conic is unbounded and is called a parabola. In the remaining case, the figure is a hyperbola, in this case, the plane will intersect both halves of the cone, producing two separate unbounded curves. A property that the conic sections share is often presented as the following definition, a conic section is the locus of all points P whose distance to a fixed point F is a constant multiple of the distance from P to a fixed line L. For 0 < e <1 we obtain an ellipse, for e =1 a parabola, a circle is a limiting case and is not defined by a focus and directrix, in the plane. The eccentricity of a circle is defined to be zero and its focus is the center of the circle, an ellipse and a hyperbola each have two foci and distinct directrices for each of them
6.
Sign (mathematics)
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In mathematics, the concept of sign originates from the property of every non-zero real number of being positive or negative. Zero itself is signless, although in some contexts it makes sense to consider a signed zero, along with its application to real numbers, change of sign is used throughout mathematics and physics to denote the additive inverse, even for quantities which are not real numbers. Also, the sign can indicate aspects of mathematical objects that resemble positivity and negativity. A real number is said to be if its value is greater than zero. The attribute of being positive or negative is called the sign of the number, zero itself is not considered to have a sign. Also, signs are not defined for complex numbers, although the argument generalizes it in some sense, in common numeral notation, the sign of a number is often denoted by placing a plus sign or a minus sign before the number. For example, +3 denotes positive three, and −3 denotes negative three, when no plus or minus sign is given, the default interpretation is that a number is positive. Because of this notation, as well as the definition of numbers through subtraction. In this context, it makes sense to write − = +3, any non-zero number can be changed to a positive one using the absolute value function. For example, the value of −3 and the absolute value of 3 are both equal to 3. In symbols, this would be written |−3| =3 and |3| =3, the number zero is neither positive nor negative, and therefore has no sign. In arithmetic, +0 and −0 both denote the same number 0, which is the inverse of itself. Note that this definition is culturally determined, in France and Belgium,0 is said to be both positive and negative. The positive resp. negative numbers without zero are said to be strictly positive resp, in some contexts, such as signed number representations in computing, it makes sense to consider signed versions of zero, with positive zero and negative zero being different numbers. One also sees +0 and −0 in calculus and mathematical analysis when evaluating one-sided limits and this notation refers to the behaviour of a function as the input variable approaches 0 from positive or negative values respectively, these behaviours are not necessarily the same. Because zero is positive nor negative, the following phrases are sometimes used to refer to the sign of an unknown number. A number is negative if it is less than zero, a number is non-negative if it is greater than or equal to zero. A number is non-positive if it is less than or equal to zero, thus a non-negative number is either positive or zero, while a non-positive number is either negative or zero
7.
2 (number)
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2 is a number, numeral, and glyph. It is the number following 1 and preceding 3. The number two has many properties in mathematics, an integer is called even if it is divisible by 2. For integers written in a system based on an even number, such as decimal and hexadecimal. If it is even, then the number is even. In particular, when written in the system, all multiples of 2 will end in 0,2,4,6. In numeral systems based on an odd number, divisibility by 2 can be tested by having a root that is even. Two is the smallest and first prime number, and the only prime number. Two and three are the two consecutive prime numbers. 2 is the first Sophie Germain prime, the first factorial prime, the first Lucas prime, the first Ramanujan prime, and it is an Eisenstein prime with no imaginary part and real part of the form 3n −1. It is also a Stern prime, a Pell number, the first Fibonacci prime, and it is the third Fibonacci number, and the second and fourth Perrin numbers. Despite being prime, two is also a highly composite number, because it is a natural number which has more divisors than any other number scaled relative to the number itself. The next superior highly composite number is six, vulgar fractions with only 2 or 5 in the denominator do not yield infinite decimal expansions, as is the case with all other primes, because 2 and 5 are factors of ten, the decimal base. Two is the number x such that the sum of the reciprocals of the powers of x equals itself. In symbols ∑ k =0 ∞12 k =1 +12 +14 +18 +116 + ⋯ =2. This comes from the fact that, ∑ k =0 ∞1 n k =1 +1 n −1 for all n ∈ R >1, powers of two are central to the concept of Mersenne primes, and important to computer science. Two is the first Mersenne prime exponent, the square root of 2 was the first known irrational number. The smallest field has two elements, in the set-theoretical construction of the natural numbers,2 is identified with the set
8.
3 (number)
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3 is a number, numeral, and glyph. It is the number following 2 and preceding 4. Three is the largest number still written with as many lines as the number represents, to this day 3 is written as three lines in Roman and Chinese numerals. This was the way the Brahmin Indians wrote it, and the Gupta made the three lines more curved, the Nagari started rotating the lines clockwise and ending each line with a slight downward stroke on the right. Eventually they made these strokes connect with the lines below, and it was the Western Ghubar Arabs who finally eliminated the extra stroke and created our modern 3. ٣ While the shape of the 3 character has an ascender in most modern typefaces, in typefaces with text figures the character usually has a descender, as, for example, in some French text-figure typefaces, though, it has an ascender instead of a descender. A common variant of the digit 3 has a flat top and this form is sometimes used to prevent people from fraudulently changing a 3 into an 8. It is usually found on UPC-A barcodes and standard 52-card decks,3 is, a rough approximation of π and a very rough approximation of e when doing quick estimates. The first odd prime number, and the second smallest prime, the only number that is both a Fermat prime and a Mersenne prime. The first unique prime due to the properties of its reciprocal, the second triangular number and it is the only prime triangular number. Both the zeroth and third Perrin numbers in the Perrin sequence, the smallest number of sides that a simple polygon can have. The only prime which is one less than a perfect square, any other number which is n2 −1 for some integer n is not prime, since it is. This is true for 3 as well, but in case the smaller factor is 1. If n is greater than 2, both n −1 and n +1 are greater than 1 so their product is not prime, the number of non-collinear points needed to determine a plane and a circle. Also, Vulgar fractions with 3 in the denominator have a single digit repeating sequences in their decimal expansions,0.000, a natural number is divisible by three if the sum of its digits in base 10 is divisible by 3. For example, the number 21 is divisible by three and the sum of its digits is 2 +1 =3, because of this, the reverse of any number that is divisible by three is also divisible by three. For instance,1368 and its reverse 8631 are both divisible by three and this works in base 10 and in any positional numeral system whose base divided by three leaves a remainder of one. Three of the five regular polyhedra have triangular faces – the tetrahedron, the octahedron, also, three of the five regular polyhedra have vertices where three faces meet – the tetrahedron, the hexahedron, and the dodecahedron
9.
5 (number)
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5 is a number, numeral, and glyph. It is the number following 4 and preceding 6. Five is the prime number. Because it can be written as 221 +1, five is classified as a Fermat prime, therefore a regular polygon with 5 sides is constructible with compass and unmarked straightedge. 5 is the third Sophie Germain prime, the first safe prime, the third Catalan number, Five is the first Wilson prime and the third factorial prime, also an alternating factorial. Five is the first good prime and it is an Eisenstein prime with no imaginary part and real part of the form 3n −1. It is also the number that is part of more than one pair of twin primes. Five is conjectured to be the only odd number and if this is the case then five will be the only odd prime number that is not the base of an aliquot tree. Five is also the only prime that is the sum of two primes, namely 2 and 3. The number 5 is the fifth Fibonacci number, being 2 plus 3,5 is also a Pell number and a Markov number, appearing in solutions to the Markov Diophantine equation. Whereas 5 is unique in the Fibonacci sequence, in the Perrin sequence 5 is both the fifth and sixth Perrin numbers,5 is the length of the hypotenuse of the smallest integer-sided right triangle. In bases 10 and 20,5 is a 1-automorphic number,5 and 6 form a Ruth–Aaron pair under either definition. There are five solutions to Známs problem of length 6 and this is related to the fact that the symmetric group Sn is a solvable group for n ≤4 and not solvable for n ≥5. While all graphs with 4 or fewer vertices are planar, there exists a graph with 5 vertices which is not planar, K5, Five is also the number of Platonic solids. A polygon with five sides is a pentagon, figurate numbers representing pentagons are called pentagonal numbers. Five is also a square pyramidal number, Five is the only prime number to end in the digit 5, because all other numbers written with a 5 in the ones-place under the decimal system are multiples of five. As a consequence of this,5 is in base 10 a 1-automorphic number, vulgar fractions with 5 or 2 in the denominator do not yield infinite decimal expansions, unlike expansions with all other prime denominators, because they are prime factors of ten, the base. When written in the system, all multiples of 5 will end in either 5 or 0
10.
7 (number)
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7 is the natural number following 6 and preceding 8. Seven, the prime number, is not only a Mersenne prime. It is also a Newman–Shanks–Williams prime, a Woodall prime, a prime, a lucky prime, a happy number, a safe prime. Seven is the lowest natural number that cannot be represented as the sum of the squares of three integers, Seven is the aliquot sum of one number, the cubic number 8 and is the base of the 7-aliquot tree. N =7 is the first natural number for which the statement does not hold, Two nilpotent endomorphisms from Cn with the same minimal polynomial. 7 is the only number D for which the equation 2n − D = x2 has more than two solutions for n and x natural, in particular, the equation 2n −7 = x2 is known as the Ramanujan–Nagell equation. 7 is the dimension, besides the familiar 3, in which a vector cross product can be defined. 7 is the lowest dimension of an exotic sphere, although there may exist as yet unknown exotic smooth structures on the 4-dimensional sphere. 999,999 divided by 7 is exactly 142,857, for example, 1/7 =0.142857142857. and 2/7 =0.285714285714. In fact, if one sorts the digits in the number 142857 in ascending order,124578, the remainder of dividing any number by 7 will give the position in the sequence 124578 that the decimal part of the resulting number will start. For example,628 ÷7 =89 5/7, here 5 is the remainder, so in this case,628 ÷7 =89.714285. Another example,5238 ÷7 =748 2/7, hence the remainder is 2, in this case,5238 ÷7 =748.285714. A seven-sided shape is a heptagon, the regular n-gons for n ≤6 can be constructed by compass and straightedge alone, but the regular heptagon cannot. Figurate numbers representing heptagons are called heptagonal numbers, Seven is also a centered hexagonal number. Seven is the first integer reciprocal with infinitely repeating sexagesimal representation, There are seven frieze groups, the groups consisting of symmetries of the plane whose group of translations is isomorphic to the group of integers. There are seven types of catastrophes. When rolling two standard six-sided dice, seven has a 6 in 36 probability of being rolled, the greatest of any number, the Millennium Prize Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute in 2000. Currently, six of the problems remain unsolved, in quaternary,7 is the smallest prime with a composite sum of digits
11.
17 (number)
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17 is the natural number following 16 and preceding 18. In spoken English, the numbers 17 and 70 are sometimes confused because they sound similar, when carefully enunciated, they differ in which syllable is stressed,17 /sɛvənˈtiːn/ vs 70 /ˈsɛvənti/. However, in such as 1789 or when contrasting numbers in the teens, such as 16,17,18. The number 17 has wide significance in pure mathematics, as well as in applied sciences, law, music, religion, sports,17 is the sum of the first 4 prime numbers. In a 24-hour clock, the hour is in conventional language called five or five oclock. Seventeen is the 7th prime number, the next prime is nineteen, with which it forms a twin prime. 17 is the sixth Mersenne prime exponent, yielding 131071,17 is an Eisenstein prime with no imaginary part and real part of the form 3n −1. 17 is the third Fermat prime, as it is of the form 22n +1, specifically with n =2, since 17 is a Fermat prime, regular heptadecagons can be constructed with compass and unmarked ruler. This was proven by Carl Friedrich Gauss,17 is the only positive Genocchi number that is prime, the only negative one being −3. It is also the third Stern prime,17 is the average of the first two Perfect numbers. 17 is the term of the Euclid–Mullin sequence. Seventeen is the sum of the semiprime 39, and is the aliquot sum of the semiprime 55. There are exactly 17 two-dimensional space groups and these are sometimes called wallpaper groups, as they represent the seventeen possible symmetry types that can be used for wallpaper. Like 41, the number 17 is a prime that yields primes in the polynomial n2 + n + p, the maximum possible length of such a sequence is 17. Either 16 or 18 unit squares can be formed into rectangles with equal to the area. 17 is the tenth Perrin number, preceded in the sequence by 7,10,12, in base 9, the smallest prime with a composite sum of digits is 17. 17 is the least random number, according to the Hackers Jargon File and it is a repunit prime in hexadecimal. 17 is the possible number of givens for a sudoku puzzle with a unique solution
12.
19 (number)
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19 is the natural number following 18 and preceding 20. In a 24-hour clock, the hour is in conventional language called seven or seven oclock. 19 is the 8th prime number, the sequence continues 23,29,31,37. 19 is the seventh Mersenne prime exponent,19 is the fifth happy number and the third happy prime. 19 is the sum of two odd discrete semiprimes,65 and 77 and is the base of the 19-aliquot tree. 19 is the number of fourth powers needed to sum up to any natural number. It is the value of g.19 is the lowest prime centered triangular number, a centered hexagonal number. The only non-trivial normal magic hexagon contains 19 hexagons,19 is the first number with more than one digit that can be written from base 2 to base 19 using only the digits 0 to 9, the other number is 20. 19 is The TCP/IP port used for chargen, astronomy, Every 19 years, the solar year and the lunar year align in whats known as the metonic cycle. Quran code, There have been claims that patterns of the number 19 are present a number of times in the Quran. The Number of Verse and Sura together in the Quran which announces Jesus son of Maryams birth, in the Bábí and Baháí faiths, a group of 19 is called a Váhid, a Unity. The numerical value of this word in the Abjad numeral system is 19, the Baháí calendar is structured such that a year contains 19 months of 19 days each, as well as a 19-year cycle and a 361-year supercycle. The Báb and his disciples formed a group of 19, There were 19 Apostles of Baháulláh. With a similar name and anti-Vietnam War theme, I Was Only Nineteen by the Australian group Redgum reached number one on the Australian charts in 1983, in 2005 a hip hop version of the song was produced by The Herd. 19 is the name of Adeles 2008 debut album, so named since she was 19 years old at the time, hey Nineteen is a song by American jazz rock band Steely Dan, written by members Walter Becker and Donald Fagen, and released on their 1980 album Gaucho. Nineteen has been used as an alternative to twelve for a division of the octave into equal parts and this idea goes back to Salinas in the sixteenth century, and is interesting in part because it gives a system of meantone tuning, being close to 1/3 comma meantone. Some organs use the 19th harmonic to approximate a minor third and they refer to the ka-tet of 19, Directive Nineteen, many names add up to 19,19 seems to permeate every aspect of Roland and his travelers lives. In addition, the ends up being a powerful key
13.
37 (number)
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37 is the natural number following 36 and preceding 38. Thirty-seven is the 12th prime number, a prime with 73. It is a hexagonal number and a star number. Every positive integer is the sum of at most 37 fifth powers,37 appears in the Padovan sequence, preceded by the terms 16,21, and 28. Since the greatest prime factor of 372 +1 =1370 is 137, the atomic number of rubidium The normal human body temperature in degrees Celsius Messier object M37, a magnitude 6. The duration of Saros series 37 was 1298.1 years, the Saros number of the lunar eclipse series which began on -1492 April 3 and ended on -194 May 22. The duration of Saros series 37 was 1298.1 years, kepler-37b is the smallest known planet. The New York Yankees, also for Stengel and this honor made him the first manager to have had his number retired by two different teams. In the NFL, The Detroit Lions, for Doak Walker, the San Francisco 49ers, for Jimmy Johnson. Thirty-seven is, The number of plays William Shakespeare is thought to have written, today the +37 prefix is shared by Lithuania, Latvia, Estonia, Moldova, Armenia, Belarus, Andorra, Monaco, San Marino and Vatican City. A television channel reserved for radio astronomy in the United States The number people are most likely to state when asked to give a number between 0 and 100. The inspiration for the album 37 Everywhere by Punchline List of highways numbered 37 Number Thirty-Seven, Pennsylvania, unincorporated community in Cambria County, Pennsylvania I37
14.
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
15.
Mersenne prime
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In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a number that can be written in the form Mn = 2n −1 for some integer n. They are named after Marin Mersenne, a French Minim friar, the first four Mersenne primes are 3,7,31, and 127. If n is a number then so is 2n −1. The definition is therefore unchanged when written Mp = 2p −1 where p is assumed prime, more generally, numbers of the form Mn = 2n −1 without the primality requirement are called Mersenne numbers. The smallest composite pernicious Mersenne number is 211 −1 =2047 =23 ×89, Mersenne primes Mp are also noteworthy due to their connection to perfect numbers. As of January 2016,49 Mersenne primes are known, the largest known prime number 274,207,281 −1 is a Mersenne prime. Since 1997, all newly found Mersenne primes have been discovered by the “Great Internet Mersenne Prime Search”, many fundamental questions about Mersenne primes remain unresolved. It is not even whether the set of Mersenne primes is finite or infinite. The Lenstra–Pomerance–Wagstaff conjecture asserts that there are infinitely many Mersenne primes,23 | M11,47 | M23,167 | M83,263 | M131,359 | M179,383 | M191,479 | M239, and 503 | M251. Since for these primes p, 2p +1 is congruent to 7 mod 8, so 2 is a quadratic residue mod 2p +1, since p is a prime, it must be p or 1. The first four Mersenne primes are M2 =3, M3 =7, M5 =31, a basic theorem about Mersenne numbers states that if Mp is prime, then the exponent p must also be prime. This follows from the identity 2 a b −1 = ⋅ = ⋅ and this rules out primality for Mersenne numbers with composite exponent, such as M4 =24 −1 =15 =3 ×5 = ×. Though the above examples might suggest that Mp is prime for all p, this is not the case. The evidence at hand does suggest that a randomly selected Mersenne number is more likely to be prime than an arbitrary randomly selected odd integer of similar size. Nonetheless, prime Mp appear to grow increasingly sparse as p increases, in fact, of the 2,270,720 prime numbers p up to 37,156,667, Mp is prime for only 45 of them. The lack of any simple test to determine whether a given Mersenne number is prime makes the search for Mersenne primes a difficult task, the Lucas–Lehmer primality test is an efficient primality test that greatly aids this task. The search for the largest known prime has somewhat of a cult following, consequently, a lot of computer power has been expended searching for new Mersenne primes, much of which is now done using distributed computing
16.
Andrew M. Gleason
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Gleasons theorem in quantum logic and the Greenwood–Gleason graph, an important example in Ramsey theory, are named for him. Gleasons entire academic career was at Harvard University, from which he retired in 1992 and his numerous academic and scholarly leadership posts included chairmanship of the Harvard Mathematics Department and Harvard Society of Fellows, and presidency of the American Mathematical Society. He continued to advise the United States government on security. Gleason won the Newcomb Cleveland Prize in 1952 and the Gung–Hu Distinguished Service Award of the American Mathematical Society in 1996 and he was a member of the National Academy of Sciences and of the American Philosophical Society, and held the Hollis Chair of Mathematics and Natural Philosophy at Harvard. He was fond of saying that mathematical proofs really arent there to convince you that something is there to show you why it is true. His older brother Henry, Jr. became a linguist and he grew up in Bronxville, New York, where his father was the curator of the New York Botanical Garden. After briefly attending Berkeley High School he graduated from Roosevelt High School in Yonkers, so I learned first year calculus and second year calculus and became the consultant to one end of the whole Old Campus. I used to do all the homework for all the sections of, I got plenty of practice in doing elementary calculus problems. I dont think there exists a problem—the classical kind of pseudo reality problem which first, one month later he enrolled in a differential equations course as well. When Einar Hille temporarily replaced the regular instructor, Gleason found Hilles style unbelievably different and he had a view of mathematics that was just vastly different. That was an important experience for me. So after that I took a lot of courses from Hille including, in his sophomore year, starting with that course with Hille, I began to have some sense of what mathematics is about. While at Yale he competed three times in the recently founded William Lowell Putnam Mathematical Competition, always placing among the top five entrants in the country. After the Japanese attacked Pearl Harbor during his year, Gleason applied for a commission in the US Navy. In 1946, at the recommendation of Navy colleague Donald Howard Menzel and he returned to Harvard in the fall of 1952, and soon after published the most important of his results on Hilberts fifth problem. Harvard awarded him tenure the following year, in January 1959 he married Jean Berko whom he had met at a party featuring the music of Tom Lehrer. Berko, a psycholinguist, worked for years at Boston University. In 1969 Gleason took the Hollis Chair of Mathematics and Natural Philosophy and he died in 2008 from complications following surgery
17.
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
18.
Raphael M. Robinson
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Raphael Mitchel Robinson was an American mathematician. Born in National City, California, Robinson was the youngest of four children of a lawyer and he was awarded from the University of California, Berkeley in mathematics, the BA, MA, and Ph. D. His Ph. D. thesis, on analysis, was titled Some results in the theory of Schlicht functions. In 1941, Robinson married his former student Julia Bowman and she became his Berkeley colleague and the first woman president of the American Mathematical Society. Robinson worked on logic, set theory, geometry, number theory. In 1937 he set out a simpler and more version of the John von Neumann 1923 axiomatic set theory. In 1950 Robinson proved that an essentially undecidable theory need not have a number of axioms by coming up with a counterexample. Q is finitely axiomatizable because it lacks Peano arithmetics axiom schema of induction, nevertheless Q, like Peano arithmetic, is incomplete, Robinson worked in number theory, even employing very early computers to obtain results. For example, he coded the Lucas-Lehmer primality test to determine whether 2n −1 was prime for all prime n <2304 on a SWAC. In 1952, he showed that these Mersenne numbers were all composite except for 17 values of n =2,3,5,7,13,17,19,31,61,89,107,127,521,607,1279,2203,2281. He discovered the last five of these Mersenne primes, the largest ones known at the time, alfred Tarski, A. Mostowski, and R. M. Robinson,1953. Leon Henkin,1995, In memoriam, Raphael Mitchell Robinson, in memoriam, Raphael Mitchell Robinson, Modern Logic 5,329. OConnor, John J. Robertson, Edmund F. Raphael M. Robinson, MacTutor History of Mathematics archive, the source for much of this entry. Raphael M. Robinson at the Mathematics Genealogy Project
19.
John B. Cosgrave
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Cosgrave is an Irish mathematician specialising in number theory. Educated at Royal Holloway College, London, he lectured in Carysfort College, in January 1999, while preparing some work for his students, he identified a highly structured prime number with exactly two thousand digits. He donated his royalties to the Irish Cancer Society. The third of those continued to stand until it was broken in June 2011. Cosgrave, John B. and Dilcher, Karl, the Multiplicative Orders of Certain Gauss Factorials, International Journal of Number Theory, Volume 7, Number 1, February 2011. Cosgrave, John B. and Dilcher, Karl, mod p^3 analogues of theorems of Gauss and Jacobi on binomial coefficients, Acta Arithmetica, Vol.142, No. Cosgrave, John B. and Dilcher, Karl, extensions of the Gauss-Wilson theorem, Integers, Electronic Journal of Combinatorial Number Theory, Vol.8, #A39,2008. Number Theory and Cryptography, in David Joyner USNA, Coding Theory and Cryptography, From Enigma to Geheimschreiber to Quantum Theory, Springer-Verlag,2000, a Prime for the Millennium, published by Folding Landscapes. An Introduction to Number Theory with Talented Youth, USA School Science and Mathematics, Vol 99, No 6, October 1999. From divisibility by 6 to the Euclidean Algorithm and the RSA cryptographic method, The American Mathematical Association of Two-Year Colleges Review, Vol 19, No 1, Fall 1997, 38-45. Teaching Mathematics by Questioning – The Socratic Method, Newsletter of Irish Mathematics Teachers Association, Nos 81-82,1993, a Halmos Problem and a Related Problem, American Mathematical Monthly, Vol.101, No. A Remark on Euclids Proof of the Infinitude of Primes, American Mathematical Monthly, Vol.96, transcendental numbers in the p-adic domain. An application of Wilsons theorem to prove that 2^p =2 when p is prime, A theorem about certain sequences, a new proof of Wilsons Theorem and A new proof of Wilsons theorem for primes of the form, Mathematical Gazette, Feb.1967. Prime number Fermat numbers Number theory John B, cosgraves web site The website for the Folding Landscapes booklet A Prime For The Millennium, written by Dr. John B. Cosgrave, with an introduction by Tim Robinson Science. ie article on Dr. John B
20.
PrimeGrid
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PrimeGrid is a distributed computing project for searching for prime numbers of world-record size. It makes use of the Berkeley Open Infrastructure for Network Computing platform, PrimeGrid started in June 2005 under the name Message@home and tried to decipher text fragments hashed with MD5. Message@home was a test to port the BOINC scheduler to Perl to obtain greater portability, after a while the project attempted the RSA factoring challenge trying to factor RSA-640. After RSA-640 was factored by a team in November 2005. With the chance to succeed too small, it discarded the RSA challenges, was renamed to PrimeGrid, at 210,000,000,000 the primegen subproject was stopped. In June 2006, dialog started with Riesel Sieve to bring their project to the BOINC community, PrimeGrid provided PerlBOINC support and Riesel Sieve was successful in implementing their sieve as well as a prime finding application. With collaboration from Riesel Sieve, PrimeGrid was able to implement the LLR application in partnership with another prime finding project, in November 2006, the TPS LLR application was officially released at PrimeGrid. Less than two months later, January 2007, the twin was found by the original manual project. PrimeGrid and TPS then advanced their search for even larger twin primes, the summer of 2007 was very active as the Cullen and Woodall prime searches were launched. In the Fall, more prime searches were added through partnerships with the Prime Sierpinski Problem, additionally, two sieves were added, the Prime Sierpinski Problem combined sieve which includes supporting the Seventeen or Bust sieve, and the combined Cullen/Woodall sieve. In the Fall of 2007, PrimeGrid migrated its systems from PerlBOINC to standard BOINC software, since September 2008, PrimeGrid is also running a Proth prime sieving subproject. In January 2010 the subproject Seventeen or Bust was added, the calculations for the Riesel problem followed in March 2010. In addition, PrimeGrid is helping test for a record Sophie Germain prime. As of March 2016, PrimeGrid is working on or has worked on the projects,321 Prime Search is a continuation of Paul Underwoods 321 Search which looked for primes of the form 3 · 2n −1. PrimeGrid added the +1 form and continues the search up to n = 25M, the search was successful in April 2010 with the finding of the first known AP26,43142746595714191 +23681770 · 23# · n is prime for n =0. 23# = 2·3·5·7·11·13·17·19·23 =223092870, or 23 primorial, is the product of all primes up to 23, PrimeGrid is also running a search for Cullen prime numbers, yielding the two largest known Cullen primes. The first one being the 14th largest known prime at the time of discovery, as of 9 March 2014 PrimeGrid has eliminated 14 values of k from the Riesel problem and is continuing the search to eliminate the 50 remaining numbers. Primegrid then worked with the Twin Prime Search to search for a twin prime at approximately 58700 digits
21.
Mathematics of paper folding
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The art of origami or paper folding has received a considerable amount of mathematical study. Fields of interest include a given paper models flat-foldability and the use of paper folds to solve mathematical equations, in 1893, Indian mathematician T. Sundara Rao published Geometric Exercises in Paper Folding which used paper folding to demonstrate proofs of geometrical constructions. This work was inspired by the use of origami in the kindergarten system and this book had an approximate trisection of angles and implied construction of a cube root was impossible. In 1936 Margharita P. Beloch showed that use of the Beloch fold, later used in the sixth of the Huzita–Hatori axioms, in 1949, R C Yeates book Geometric Methods described three allowed constructions corresponding to the first, second, and fifth of the Huzita–Hatori axioms. The axioms were discovered by Jacques Justin in 1989, but were overlooked until the first six were rediscovered by Humiaki Huzita in 1991. The first International Meeting of Origami Science and Technology was held in 1989 in Ferrara, the construction of origami models is sometimes shown as crease patterns. The major question about such crease patterns is whether a given crease pattern can be folded to a model, and if so, how to fold them. Related problems when the creases are orthogonal are called map folding problems, there are three mathematical rules for producing flat-foldable origami crease patterns, Maekawas theorem, at any vertex the number of valley and mountain folds always differ by two. It follows from this that every vertex has an number of creases. Kawasakis theorem, at any vertex, the sum of all the odd angles adds up to 180 degrees, a sheet can never penetrate a fold. Paper exhibits zero Gaussian curvature at all points on its surface, curved surfaces that cant be flattened can be produced using a non-folded crease in the paper, as is easily done with wet paper or a fingernail. Assigning a crease pattern mountain and valley folds in order to produce a model has been proven by Marshall Bern. Further references and technical results are discussed in Part II of Geometric Folding Algorithms, Paper fold strips can be constructed to solve equations up to degree 4. The Huzita–Hatori axioms are an important contribution to field of study. These describe what can be constructed using a sequence of creases with at most two point or line alignments at once, complete methods for solving all equations up to degree 4 by applying methods satisfying these axioms are discussed in detail in Geometric Origami. Methods for folding most regular polygons up to and including the regular 19-gon have been developed, the side of a square can be divided at an arbitrary rational fraction in a variety of ways. Hagas theorems say that a set of constructions can be used for such divisions. Surprisingly few folds are necessary to generate large odd fractions, for instance 1⁄5 can be generated with three folds, first halve a side, then use Hagas theorem twice to produce first 2⁄3 and then 1⁄5
22.
Cubic function
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In algebra, a cubic function is a function of the form f = a x 3 + b x 2 + c x + d, where a is nonzero. Setting f =0 produces an equation of the form. The solutions of this equation are called roots of the polynomial f, If all of the coefficients a, b, c, and d of the cubic equation are real numbers then there will be at least one real root. All of the roots of the equation can be found algebraically. The roots can also be found trigonometrically, alternatively, numerical approximations of the roots can be found using root-finding algorithms like Newtons method. The coefficients do not need to be complex numbers, much of what is covered below is valid for coefficients of any field with characteristic 0 or greater than 3. The solutions of the cubic equation do not necessarily belong to the field as the coefficients. For example, some cubic equations with rational coefficients have roots that are complex numbers. Cubic equations were known to the ancient Babylonians, Greeks, Chinese, Indians, Babylonian cuneiform tablets have been found with tables for calculating cubes and cube roots. The Babylonians could have used the tables to solve cubic equations, the problem of doubling the cube involves the simplest and oldest studied cubic equation, and one for which the ancient Egyptians did not believe a solution existed. Methods for solving cubic equations appear in The Nine Chapters on the Mathematical Art, in the 3rd century, the Greek mathematician Diophantus found integer or rational solutions for some bivariate cubic equations. In the 11th century, the Persian poet-mathematician, Omar Khayyám, in an early paper, he discovered that a cubic equation can have more than one solution and stated that it cannot be solved using compass and straightedge constructions. He also found a geometric solution, in the 12th century, the Indian mathematician Bhaskara II attempted the solution of cubic equations without general success. However, he gave one example of an equation, x3 + 12x = 6x2 +35. He used what would later be known as the Ruffini-Horner method to approximate the root of a cubic equation. He also developed the concepts of a function and the maxima and minima of curves in order to solve cubic equations which may not have positive solutions. He understood the importance of the discriminant of the equation to find algebraic solutions to certain types of cubic equations. Leonardo de Pisa, also known as Fibonacci, was able to approximate the positive solution to the cubic equation x3 + 2x2 + 10x =20
23.
Compass
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A compass is an instrument used for navigation and orientation that shows direction relative to the geographic cardinal directions, or points. Usually, a called a compass rose shows the directions north, south, east. When the compass is used, the rose can be aligned with the geographic directions, so, for example. Frequently, in addition to the rose or sometimes instead of it, North corresponds to zero degrees, and the angles increase clockwise, so east is 90 degrees, south is 180, and west is 270. These numbers allow the compass to show azimuths or bearings, which are stated in this notation. The magnetic compass was first invented as a device for divination as early as the Chinese Han Dynasty, the first usage of a compass recorded in Western Europe and the Islamic world occurred around the early 13th century. The magnetic compass is the most familiar compass type and it functions as a pointer to magnetic north, the local magnetic meridian, because the magnetized needle at its heart aligns itself with the horizontal component of the Earths magnetic field. The needle is mounted on a pivot point, in better compasses a jewel bearing. When the compass is level, the needle turns until, after a few seconds to allow oscillations to die out. In navigation, directions on maps are usually expressed with reference to geographical or true north, the direction toward the Geographical North Pole, the rotation axis of the Earth. Depending on where the compass is located on the surface of the Earth the angle between north and magnetic north, called magnetic declination can vary widely with geographic location. The local magnetic declination is given on most maps, to allow the map to be oriented with a parallel to true north. The location of the Earths magnetic poles slowly change with time, the effect of this means a map with the latest declination information should be used. Some magnetic compasses include means to compensate for the magnetic declination. The first compasses in ancient Han dynasty China were made of lodestone, the compass was later used for navigation during the Song Dynasty of the 11th century. Later compasses were made of iron needles, magnetized by striking them with a lodestone, dry compasses began to appear around 1300 in Medieval Europe and the Islamic world. This was supplanted in the early 20th century by the magnetic compass. Modern compasses usually use a needle or dial inside a capsule completely filled with a liquid
24.
Straightedge
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A straightedge is a tool with an edge free from curves, or straight, used for transcribing straight lines, or checking the straightness of lines. If it has equally spaced markings along its length, it is called a ruler. Straightedges are used in the service and machining industry to check the flatness of machined mating surfaces. A pair of straightedges called winding sticks are used in woodworking to amplify twist in pieces of wood, an idealized straightedge is used in compass-and-straightedge constructions in plane geometry. It may be used, Given two points, to draw the line connecting them, Given a point and a circle, to draw either tangent. Given two circles, to any of their common tangents. It may not be marked or used together with the compass so as to transfer the length of one segment to another and it is possible to do all compass and straightedge constructions without the straightedge. That is, it is possible, using only a compass and it is not, however, possible to do all constructions using only a straightedge. It is possible to do them with straightedge alone given one circle, chalk line Geometrography Wayne R. Moore, Foundations of Mechanical Accuracy, Moore Special Tool Company, Bridgeport, CT Making Accurate Straight-Edges from Scratch
25.
Angle trisection
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Angle trisection is a classical problem of compass and straightedge constructions of ancient Greek mathematics. It concerns construction of an equal to one third of a given arbitrary angle. The problem as stated is generally impossible to solve, as proved by Pierre Wantzel in 1837, however, although there is no way to trisect an angle in general with just a compass and a straightedge, some special angles can be trisected. For example, it is straightforward to trisect a right angle. It is possible to trisect an angle by using tools other than straightedge. For example, neusis construction, also known to ancient Greeks, involves simultaneous sliding and rotation of a marked straightedge, other techniques were developed by mathematicians over the centuries. Because it is defined in terms, but complex to prove unsolvable. These solutions often involve mistaken interpretations of the rules, or are simply incorrect, three problems proved elusive, specifically, trisecting the angle, doubling the cube, and squaring the circle. The problem of angle trisection reads, Construct an angle equal to one-third of an arbitrary angle. Pierre Wantzel published a proof of the impossibility of trisecting an arbitrary angle in 1837. Wantzels proof, restated in modern terminology, uses the algebra of field extensions. However Wantzel published these results earlier than Galois and did not use the connection between field extensions and groups that is the subject of Galois theory itself. The problem of constructing an angle of a given measure θ is equivalent to constructing two segments such that the ratio of their length is cos θ. From a solution to one of two problems, one may pass to a solution of the other by a compass and straightedge construction. The triple-angle formula gives an expression relating the cosines of the angle and its trisection. It follows that, given a segment that is defined to have unit length and this equivalence reduces the original geometric problem to a purely algebraic problem. Every irrational number which is constructible in a step from some given numbers is a root of a polynomial of degree 2 with coefficients in the field generated by these numbers. Therefore, any number which is constructible by a sequence of steps is a root of a polynomial whose degree is a power of two
26.
Euler's totient function
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In number theory, Eulers totient function counts the positive integers up to a given integer n that are relatively prime to n. It is written using the Greek letter phi as φ or ϕ and it can be defined more formally as the number of integers k in the range 1 ≤ k ≤ n for which the greatest common divisor gcd is equal to 1. The integers k of this form are referred to as totatives of n. For example, the totatives of n =9 are the six numbers 1,2,4,5,7 and 8. They are all relatively prime to 9, but the three numbers in this range,3,6, and 9 are not, because gcd = gcd =3. As another example, φ =1 since for n =1 the only integer in the range from 1 to n is 1 itself, Eulers totient function is a multiplicative function, meaning that if two numbers m and n are relatively prime, then φ = φφ. This function gives the order of the group of integers modulo n. It also plays a key role in the definition of the RSA encryption system, leonhard Euler introduced the function in 1763. However, he did not at that time choose any specific symbol to denote it. In a 1784 publication, Euler studied the function further, choosing the Greek letter π to denote it, he wrote πD for the multitude of less than D. This definition varies from the current definition for the totient function at D =1 but is otherwise the same, the now-standard notation φ comes from Gausss 1801 treatise Disquisitiones Arithmeticae. Although Gauss didnt use parentheses around the argument and wrote φA, thus, it is often called Eulers phi function or simply the phi function. In 1879, J. J. Sylvester coined the term totient for this function, so it is referred to as Eulers totient function. Jordans totient is a generalization of Eulers, the cototient of n is defined as n − φ. It counts the number of positive integers less than or equal to n that have at least one factor in common with n. There are several formulas for computing φ and it states φ = n ∏ p ∣ n, where the product is over the distinct prime numbers dividing n. The proof of Eulers product formula depends on two important facts and this means that if gcd =1, then φ = φ φ. If p is prime and k ≥1, then φ = p k − p k −1 = p k −1 = p k, proof, since p is a prime number the only possible values of gcd are 1, p, p2
27.
Hendecagon
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In geometry, a hendecagon or 11-gon is an eleven-sided polygon. A regular hendecagon is represented by Schläfli symbol, a regular hendecagon has internal angles of 147.27 degrees. The area of a regular hendecagon with side length a is given by A =114 a 2 cot π11 ≃9.36564 a 2, as 11 is not a Fermat prime, the regular hendecagon is not constructible with compass and straightedge. Because 11 is not a Pierpont prime, construction of a regular hendecagon is still impossible even with the usage of an angle trisector and it can, however, be constructed via neusis construction. Close approximations to the regular hendecagon can be constructed, however, for instance, the ancient Greek mathematicians approximated the side length of a hendecagon inscribed in a unit circle as being 14/25 units long. The following construction description is given by T, the regular hendecagon has Dih11 symmetry, order 22. Since 11 is a number there is one subgroup with dihedral symmetry, Dih1, and 2 cyclic group symmetries, Z11. These 4 symmetries can be seen in 4 distinct symmetries on the hendecagon, john Conway labels these by a letter and group order. Full symmetry of the form is r22 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices or edges, cyclic symmetries in the middle column are labeled as g for their central gyration orders. Each subgroup symmetry allows one or more degrees of freedom for irregular forms, only the g11 subgroup has no degrees of freedom but can seen as directed edges. The Canadian dollar coin, the loonie, is similar to, but not exactly, the cross-section of a loonie is actually a Reuleaux hendecagon. Anthony dollar has a hendecagonal outline along the inside of its edges
28.
American Mathematical Monthly
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The American Mathematical Monthly is a mathematical journal founded by Benjamin Finkel in 1894. It is published ten times each year by the Mathematical Association of America, the American Mathematical Monthly is an expository journal intended for a wide audience of mathematicians, from undergraduate students to research professionals. Articles are chosen on the basis of their broad interest and reviewed and edited for quality of exposition as well as content, in this the American Mathematical Monthly fulfills a different role from that of typical mathematical research journals. The American Mathematical Monthly is the most widely read journal in the world according to records on JSTOR. Since 1997, the journal has been available online at the Mathematical Association of Americas website, the MAA gives the Lester R. Ford Awards annually to authors of articles of expository excellence published in the American Mathematical Monthly