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Encyclopedia > 4 (number)
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This article discusses the number Four. For the year 4 AD, see 4. For other uses of 4, see 4 (disambiguation)

4 (four) is a number, numeral, and glyph. It is the natural number following 3 and preceding 5. Four is the only number in the English language for which the number of letters in its name is equal to the number itself. This is also true in several other languages, such as German, Afrikaans, Dutch, and Flemish, as "vier".[citation needed] The binary numeral system, or base-2 number system, is a numeral system that represents numeric values using two symbols, usually 0 and 1. ... The octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7. ... The duodecimal (also known as base-12 or dozenal) system is a numeral system using twelve as its base. ... In mathematics and computer science, hexadecimal, base-16, or simply hex, is a numeral system with a radix, or base, of 16, usually written using the symbols 0â€“9 and Aâ€“F, or aâ€“f. ... The vigesimal or base-20 numeral system is based on twenty (in the same way in which the ordinary decimal numeral system is based on ten). ... Wiktionary (a portmanteau of wiki and dictionary) is a multilingual, Web-based project to create a free content dictionary, available in over 151 languages. ... For other uses, see 4 (disambiguation). ... Look up four in Wiktionary, the free dictionary. ... For other uses, see Number (disambiguation). ... A numeral is a symbol or group of symbols that represents a number. ... variant glyphs representing the character a (allographs of a) in the Zapfino typeface. ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... This article is about the number. ... Look up five in Wiktionary, the free dictionary. ... The English language is a West Germanic language that originates in England. ... For other uses, see Letter (disambiguation). ... In mathematics, two mathematical objects are considered equal if they are precisely the same in every way. ...

In mathematics

Four is the smallest composite number, its proper divisors being 1 and 2. Four is also a highly composite number. The next highly composite number is 6. A composite number is a positive integer which has a positive divisor other than one or itself. ... In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ... One redirects here. ... For other uses of 2, see 2 (disambiguation). ... A highly composite number is a positive integer which has more divisors than any positive integer below it. ... Look up six in Wiktionary, the free dictionary. ...

Four is the second square number, the second centered triangular number. In mathematics, a square number, sometimes also called a perfect square, is an integer that can be written as the square of some other integer; in other words, it is the product of some integer with itself. ... A centered triangular number is a centered figurate number that represents a triangle with a dot in the center and all other dots surrounding the center in successive triangular layers. ...

4 is the smallest squared prime (p2). It has an aliquot sum of 3 which is itself prime. The aliquot sequence of 4 has 4 members (4,3,1,0). Divisor function Ïƒ0(n) up to n=250 Sigma function Ïƒ1(n) up to n=250 Sum of the squares of divisors, Ïƒ2(n), up to n=250 Sum of cubes of divisors, Ïƒ3(n) up to n=250 In mathematics, and specifically in number theory, a divisor function is... For other uses, see 3 (disambiguation). ... In mathematics, an aliquot sequence is a recursive sequence which can be defined in the following way: if we write Ïƒ(n) = Ïƒ1(n) to be the divisor function normally, then, the aliquot sequence of k can be written: s0 = k sn = Ïƒ(snâˆ’1) âˆ’ snâˆ’1 For example, the aliquot sequence... This article is about the number. ...

The prime factorization of four is two times two.

Four is the smallest composite number that is equal to the sum of its prime factors. (As a consequence of this, it is the smallest Smith number). However, it is the largest (and only) composite number n for which $(n - 1)! equiv 0 ({rm mod} n)$ is false. A Smith number is a number which in a given base, the sum of its digits is equal to the sum of the digits in its factorization. ...

It is also a Motzkin number. In mathematics, a Motzkin number for a given number n is the number of different ways of drawing non-intersecting chords on a circle between n points. ...

In addition, $2 + 2 = 2 times 2 = 2^2 = 4$. Continuing the pattern in Knuth's up-arrow notation, $2 uparrowuparrow 2 = 2 uparrowuparrowuparrow 2 = 4$, and so on, for any number of up arrows. In mathematics, Knuths up-arrow notation is a notation for very large integers introduced by Donald Knuth in 1976. ...

A solid figure with four faces is a tetrahedron. The regular tetrahedron is the simplest Platonic solid. A tetrahedron, which can also be called a 3-simplex, has four triangular faces and four vertices. It is the only self-dual regular polyhedron. For the academic journal, see Tetrahedron (journal). ... In geometry, a Platonic solid is a convex regular polyhedron. ... For the academic journal, see Tetrahedron (journal). ... A 3-simplex or tetrahedron In geometry, a simplex (plural simplexes or simplices) or n-simplex is an n-dimensional analogue of a triangle. ... In mathematics, there are three related meanings of the term polyhedron: in the traditional meaning it is a 3-dimensional polytope, and in a newer meaning that exists alongside the older one it is a bounded or unbounded generalization of a polytope of any dimension. ...

Four-dimensional space has the largest number of dimensions used by more than three convex regular figures. There are infinitely many convex regular polygons (two-dimensional); five convex regular polyhedra (three-dimensional, the five Platonic Solids); six convex regular polychora (four-dimensional); and three regular convex polytopes occupying each higher-dimensional space. For other uses, see Fourth dimension (disambiguation). ... A regular pentagon A regular polygon is a simple polygon (a polygon which does not intersect itself anywhere) which is equiangular (all angles are equal) and equilateral (all sides have the same length). ... In mathematics, there are three related meanings of the term polyhedron: in the traditional meaning it is a 3-dimensional polytope, and in a newer meaning that exists alongside the older one it is a bounded or unbounded generalization of a polytope of any dimension. ... In geometry, a Platonic solid is a convex regular polyhedron. ... In mathematics, a convex regular 4-polytope (or polychoron) is 4-dimensional polytope which is both a regular and convex. ... In geometry polytope means, first, the generalization to any dimension of polygon in two dimensions, polyhedron in three dimensions, and polychoron in four dimensions. ...

The smallest non-cyclic group has four elements; it is the Klein four-group. Four is also the order of the smallest non-trivial groups that are not simple. In group theory, a cyclic group or monogenous group is a group that can be generated by a single element, in the sense that the group has an element g (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of... This picture illustrates how the hours on a clock form a group under modular addition. ... This article is about the mathematical group. ... In mathematics, a simple group is a group which is not the trivial group and whose only normal subgroups are the trivial group and the group itself. ...

Four is the maximum number of dimensions of a real division algebra (the quaternions), by a theorem of Ferdinand Georg Frobenius. In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division is possible. ... In mathematics, the quaternions are a non-commutative extension of the complex numbers. ... A picture of Frobenius Ferdinand Georg Frobenius (October 26, 1849 â€“ August 3, 1917) was a German mathematician, best-known for his contributions to the theory of differential equations and to group theory. ...

The four-color theorem states that a planar graph (or, equivalently, a flat map of two-dimensional regions such as countries) can be colored using four colors, so that adjacent vertices (or regions) are always different colors. Three colors are not, in general, sufficient to guarantee this. The largest planar complete graph has four vertices. Example of a four color map The four color theorem states that every possible geographical map can be colored with at most four colors in such a way that no two adjacent regions receive the same colour. ... In graph theory, a planar graph is a graph that can be drawn so that no edges intersect (or that can be embedded) in the plane. ... For other uses, see Map (disambiguation). ... In the mathematical field of graph theory, a complete graph is a simple graph where an edge connects every pair of distinct vertices. ...

Lagrange's four-square theorem states that every positive integer can be written as the sum of at most four square numbers. Three are not always sufficient; 7 for instance cannot be written as the sum of three squares. Lagranges four-square theorem, also known as Bachets conjecture, was proved in 1770 by Joseph Louis Lagrange. ... In mathematics, a square number, sometimes also called a perfect square, is an integer that can be written as the square of some other integer; in other words, it is the product of some integer with itself. ... 7 (seven) is the natural number following 6 and preceding 8. ...

Four is the first positive non-Fibonacci number. A tiling with squares whose sides are successive Fibonacci numbers in length In mathematics, the Fibonacci numbers are a sequence of numbers named after Leonardo of Pisa, known as Fibonacci. ...

Each natural number divisible by 4 is a difference of squares of two natural numbers, i.e. 4x = y2z2.

Four is an all-Harshad number and a semi-meandric number. A Harshad number, or Niven number, is an integer that is divisible by the sum of its digits in a given number base. ... In mathematics, a meander or closed meander is a self-avoiding closed curve which intersects a line a number of times. ...

List of basic calculations

Multiplication 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 50 100 1000
$4 times x$ 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100 200 400 4000
Division 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
$4 div x$ 4 2 $1.overline{3}$ 1 0.8 $0.overline{6}$ $0.overline{5}7142overline{8}$ 0.5 $0.overline{4}$ 0.4 $0.overline{3}overline{6}$ $0.overline{3}$ $0.overline{3}0769overline{2}$ $0.overline{2}8571overline{4}$ $0.2overline{6}$
$x div 4$ 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 3.25 3.5 3.75
Exponentiation 1 2 3 4 5 6 7 8 9 10 11 12 13
$4 ^ x,$ 4 16 64 256 1024 4096 16384 65536 262144 1048576 4194304 16777216 67108864
$x ^ 4,$ 1 16 81 256 625 1296 2401 4096 6561 10000 14641 20736 28561

In mathematics, multiplication is an elementary arithmetic operation. ... Look up eight in Wiktionary, the free dictionary. ... Look up twelve in Wiktionary, the free dictionary. ... 16 (sixteen) is the natural number following 15 and preceding 17. ... 20 (twenty) is the natural number following 19 and preceding 21. ... 24 (twenty-four) is the natural number following 23 and preceding 25. ... 28 (twenty-eight) is the natural number following 27 and preceding 29. ... 32 (thirty-two) is the natural number following 31 and preceding 33. ... 36 (thirty-six) is the natural number following 35 and preceding 37. ... 40 (forty) is the natural number following 39 and preceding 41. ... 44 (forty-four) is the natural number following 43 and preceding 45. ... 48 (forty-eight) is the natural number following 47 and preceding 49. ... 52 (fifty-two) is the natural number following 51 and preceding 53. ... 56 (fifty-six) is the natural number following 55 and preceding 57. ... 60 (sixty) is the natural number following 59 and preceding 61. ... 64 (sixty-four) is the natural number following 63 and preceding 65. ... 68 (sixty-eight) is the natural number following 67 and preceding 69 // Sixty-eight is a nontotient. ... 72 is the natural number following 71 and preceding 73. ... 76 is the natural number following 75 and preceding 77. ... 80 (eighty) is the natural number following 79 and preceding 81. ... 84 (eighty-four) is the natural number following 83 and preceding 85. ... 88 (eighty-eight) is the natural number following 87 and preceding 89. ... 92 (ninety-two) is the natural number following 91 and preceding 93. ... 96 is the natural number following 95 and preceding 97. ... 100 (one hundred) (the Roman numeral is C for centum) is the natural number following 99 and preceding 101. ... 200 is the natural number following 199 and preceding 201. ... Four hundred is the natural number following three hundred ninety-nine and preceding four hundred one. ... 4000 (four thousand) is the natural number following 3999 and preceding 4001. ... In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication. ... Zero redirects here. ... 25 (twenty-five) is the natural number following 24 and preceding 26. ... 75 (seventy-five) is the natural number following 74 and preceding 76. ... â€œExponentâ€ redirects here. ... 256 (two hundred [and] fifty-six, CCLVI) is the natural number following 255 and preceding 257. ... 81 is the natural number following 80 and preceding 82. ... Look up ten thousand in Wiktionary, the free dictionary. ...

Evolution of the glyph

Evolution of the numeral 4 from the Brahmin Indians to the Europeans I drew this characters by hand and scanned them. ...

Representing 1, 2 and 3 in as many lines as the number represents worked well enough, but writing four lines proved tiresome. The Brahmin Indians simplified 4 by joining its four lines into a cross that looks like our modern plus sign. The Sunga and other Indians would add a horizontal line on top of the numeral, and the Kshatrapa and Pallava evolved the numeral to a point where speed of writing was at best a secondary concern. The Arabs didn't have time for cursive fancy: their 4 still had the early concept of the cross, but for the sake of efficiency, was made in one stroke by connecting the "western" end to the "northern" end; the "eastern" end was finished off with a curve. The Europeans dropped off the finishing curve and gradually made the numeral less cursive, ending up with a glyph very close to the original Brahmin cross.[1] Note that the word Brahmin is also known as Brahman in English due to some translation issues between the Upanishads (Hindu Holy Texts) and modern English* Brahmin, in Hinduism, traditionally refers to the priestly caste or a member of this caste in the Hindu caste system. ... The Sunga Empire (or Shunga Empire) is a Magadha dynasty that controlled North-central and Eastern India from around 185 to 73 BCE. It was established after the fall of the Indian Mauryan empire. ... For other uses, see Arab (disambiguation). ...

While the shape of the 4 character has an ascender in most modern typefaces, in typefaces with text figures the character usually has a descender, as, for example, in . The ascenders are the parts of the characters that lie above the midline, highlighted in red. ... â€œFontâ€ redirects here. ... Hoefler Text, a contemporary font, uses hanging or old style text figures. ... The descenders are the parts of the characters that lie below the baseline. ... Image File history File links The numerals 1, 4 and 8 written in text figures. ...

On the seven-segment displays of pocket calculators and digital watches, 4 is seen with an open top. Image File history File links No higher resolution available. ... A seven-segment display (abbreviation: 7-seg(ment) display), less commonly known as a seven-segment indicator, is a form of display device that is an alternative to the more complex dot-matrix displays. ...

In science

In biology

Adenine Guanine Thymine Cytosine ... The structure of part of a DNA double helix Deoxyribonucleic acid, or DNA, is a nucleic acid molecule that contains the genetic instructions used in the development and functioning of all known living organisms. ... For other uses, see RNA (disambiguation). ... For the programming language Adenine, see Adenine (programming language). ... Guanine is one of the five main nucleobases found in the nucleic acids DNA and RNA; the others being adenine, cytosine, thymine, and uracil. ... Cytosine is one of the 5 main nucleobases used in storing and transporting genetic information within a cell in the nucleic acids DNA and RNA. It is a pyrimidine derivative, with a heterocyclic aromatic ring and two substituents attached (an amine group at position 4 and a keto group at... For the similarly-spelled vitamin compound, see Thiamine Thymine, also known as 5-methyluracil, is a pyrimidine nucleobase. ... Uracil is a pyrimidine which is common and naturally occurring. ... For other uses, see RNA (disambiguation). ... Classes See below Chordates (phylum Chordata) are a group of animals that includes the vertebrates, together with several closely related invertebrates. ... Classes Synapsida Sauropsida Amphibia A tetrapod (Greek tetrapoda, four-legged) is a vertebrate animal having four feet, legs or leglike appendages. ... Subclasses & Infraclasses Subclass â€ Allotheria* Subclass Prototheria Subclass Theria Infraclass â€ Trituberculata Infraclass Metatheria Infraclass Eutheria For the folk-rock band see The Mammals. ... The heart and lungs, from an older edition of Grays Anatomy. ... Subclasses & Infraclasses Subclass â€ Allotheria* Subclass Prototheria Subclass Theria Infraclass â€ Trituberculata Infraclass Metatheria Infraclass Eutheria For the folk-rock band see The Mammals. ... Families 17, See classification The diverse order Carnivora (IPA: or ; from Latin carÅ (stem carn-) flesh, + vorÄre to devour) includes over 260 species of placental mammals. ... Ungulates (meaning roughly hoofed or hoofed animal) make up several orders of mammals, of which six to eight survive: Artiodactyla: even-toed ungulates, cattle, sheep, goats, pigs, antelope, and many others Cetacea: whales and dolphins (which evolved from hoofed land animals) Perissodactyla: odd-toed ungulates such as horses and rhinos... Superfamilies and families Superfamily Hedyloidea: Hedylidae Superfamily Hesperioidea: Hesperiidae Superfamily Papilionoidea: Papilionidae Pieridae Nymphalidae Lycaenidae Riodinidae A butterfly is an insect of the order Lepidoptera. ... This article is about the insect. ... For other uses, see Wing (disambiguation). ... For other uses, see Blood (disambiguation). ... In mammalian oral anatomy, the canine teeth, also called cuspids, dogteeth, fangs, or (in the case of those of the upper jaw) eye teeth, are relatively long, pointed teeth. ... Incisors (from Latin incidere, to cut) are the first kind of tooth in heterodont mammals. ... Wisdom teeth are the third molars that usually appear between the ages of 16 and 24. ... Teeth redirects here. ...

In chemistry

• Valency of carbon (that is basis of life on the Earth) is four. Thanks to its tetrahedral crystal bond structure diamond (one of the natural allotropes of carbon) is the hardest known naturally occurring material. It is also the valence of silicon, whose compounds form the majority of the mass of the Earth's crust.
• The atomic number of beryllium
• There are four basic states of matter: solid, liquid, gas, and plasma.

For other uses, see Carbon (disambiguation). ... This article is about the mineral. ... Not to be confused with Silicone. ... See also: List of elements by atomic number In chemistry and physics, the atomic number (also known as the proton number) is the number of protons found in the nucleus of an atom. ... General Name, symbol, number beryllium, Be, 4 Chemical series alkaline earth metals Group, period, block 2, 2, s Appearance white-gray metallic Standard atomic weight 9. ... This article is about matter in physics and chemistry. ... This box:      For other uses, see Solid (disambiguation). ... For other uses, see Liquid (disambiguation). ... For other uses, see Gas (disambiguation). ... For other uses, see Plasma. ...

In physics

For a generally accessible and less technical introduction to the topic, see Introduction to special relativity. ... For a generally accessible and less technical introduction to the topic, see Introduction to general relativity. ... 2-dimensional renderings (ie. ... This article is about the concept of time. ... This article is about the idea of space. ... For other uses of this term, see Spacetime (disambiguation). ... An alpha particle is deflected by a magnetic field Alpha radiation consists of helium-4 nuclei and is readily stopped by a sheet of paper. ... A hadron, in particle physics, is a subatomic particle which experiences the nuclear force. ... A fundamental interaction is a mechanism by which particles interact with each other, and which cannot be explained by another more fundamental interaction. ... This box:      Electromagnetism is the physics of the electromagnetic field: a field which exerts a force on particles that possess the property of electric charge, and is in turn affected by the presence and motion of those particles. ... Gravity is a force of attraction that acts between bodies that have mass. ... The weak nuclear force or weak interaction is one of the four fundamental forces of nature. ... The strong nuclear force or strong interaction (also called color force or colour force) is a fundamental force of nature which affects only quarks and antiquarks, and is mediated by gluons in a similar fashion to how the electromagnetic force is mediated by photons. ...

In logic and philosophy

• The symbolic meanings of the number four are linked to those of the cross and the square. 'almost from prehistoric times, the number four was employed to signify what was solid, what could be touched and felt. its relationship to the cross (four points) made it an outstanding symbol of wholeness and universality, a symbol which drew all to itself'. where lines of latitude and longitude intersect, they divide the earth into four proportions. throughout the world kings and chieftains have been called 'lord of the four suns'...'lord of the four quarters of the earth'... by which is understood to the extent of their powers both territorially and in terms of total control of their subjects' doings.
• The Square of Opposition, in both its Aristotelian version and its Boolean version, consists of four forms: A ("All S is R"), I ("Some S is R"), E ("No S is R"), and O ("Some S is not R").
• Aristotle held that there are basically four causes in nature: the efficient cause, the matter, the end, and the form.
• Immanuel Kant expounded a table of judgments involving four three-way alternatives, in regard to 1. Quantity, 2. Quality, 3. Relation, 4. Modality, and, based thereupon, a table of four categories, named by the terms just listed, and each with three subcategories.
• Arthur Schopenhauer's doctoral thesis was On the Fourfold Root of the Principle of Sufficient Reason.
• C.S. Peirce, usually a trichotomist, discussed four basic methods of seeking to settle questions and arrive at firm beliefs: the method of tenacity (sticking to that which one is inclined to think), the method of authority, the a priori method, and the method of science. (See "The Fixation of Belief," 1877, Eprint.)
• Jonathan Lowe (E.J. Lowe) argues in The Four-Category Ontology, 2006, for four categories: objects, attributes, kinds, and modes (by "modes" he means property-particulars, also known as "tropes"). (See Lowe's "Recent Advances in Metaphysics," 2001, Eprint)

The Square of Opposition is a term from the study of Aristotelian logic or Term Logic in which the logical relationship between various types of sentences is spelled out. ... The Square of Opposition is a term from the study of Aristotelian logic or Term Logic in which the logical relationship between various types of sentences is spelled out. ... The Square of Opposition is a term from the study of Aristotelian logic or Term Logic in which the logical relationship between various types of sentences is spelled out. ... For other uses, see Aristotle (disambiguation). ... For other uses, see Aristotle (disambiguation). ... The efficient cause is a philosophical concept proposed by Aristotle. ... The Material Cause, that out of which the statue is made, is the marble or bronze. ... Final cause is one of Aristotles four forms of causation (the others being material, formal, and efficient). ... Formal cause is a concept used by Aristotle, and originates from the idea of the form by Plato and Socrates. ... Kant redirects here. ... In Kants philosophy, a category is a pure concept of the understanding. ... In Kants philosophy, a category is a pure concept of the understanding. ... Arthur Schopenhauer (February 22, 1788 â€“ September 21, 1860) was a German philosopher best known for his work The World as Will and Representation. ... Originally published as a doctoral dissertation in 1813, Arthur Schopenhauer revised this important work and re-published it in 1847. ... Charles Sanders Peirce Charles Sanders Peirce (September 10, 1839 – April 19, 1914) was an American logician, philosopher, scientist, and mathematician. ... Jonathan Lowe (E.J. Lowe) (born 1950) is currently Professor of Philosophy and Chair of the Examination Board of the Department of Philosophy at the University of Durham, England. ... The introduction to this article provides insufficient context for those unfamiliar with the subject matter. ...

In technology

• Most furniture has four legs - tables, chairs, etc.
• Four horses (quadriga) is the maximal number of horses in one row for carriage.
• Wide use of rectangles (with four angles and four sides) because they have effective form and capability for close adjacency to each other (houses, rooms, tables, bricks, sheets of paper, screens, film frames).
• On most phones, the 4 key is associated with the letters G, H, and I, but on the BlackBerry it is the key for D and F.
• In internet slang, "4" can replace the word "for" (as "four" and "for" are pronounced similarly). For example, typing "4u" instead of "for you".
• In Leetspeak, "4" may be used to replace the letter "A".
• In Expressing love, "4" may be used to replace the word "Love" (143).

In sports

In rugby union, the number of the lock forward, who usually jumps at number 2 in the line-out. For other uses, see Rugby (disambiguation). ... A normal rugby union team formation illustrating each of the positions and their respective numbers. ... A rugby lineout. ...

In rugby league, the number of one of the two centres. It is also the number of points awarded for a try. Rugby league football is a full-contact team sport played with a prolate spheroid-shaped ball by two teams of thirteen on a rectangular grass field. ... A typical rugby league team consists of thirteen players on the field plus four substitutes on the bench. ... This article refers to the use of the word Try in rugby football terminology. ...

In baseball, 4 represents the second baseman's position. It was also the jersey number of Lou Gehrig, Mel Ott, Joe Cronin, and Paul Molitor. It was the jersey number of Earl Weaver, long-time manager of the Baltimore Orioles. This article is about the sport. ... The position of the second baseman Second base redirects here. ... Henry Louis Lou Gehrig (June 19, 1903 â€“ June 2, 1941), born Ludwig Heinrich Gehrig[2], was an American baseball player in the 1920s and 1930s, who set several Major League records and was popularly called the The Iron Horse[2] for his durability. ... Melvin Thomas Mel Ott (March 2, 1909 â€“ November 21, 1958), nicknamed Master Melvin, was a Major League Baseball right fielder who played his entire career for the New York Giants (1926-1947). ... Joe Cronin Joseph Edward Cronin (October 12, 1906 â€“ September 7, 1984) was a Major League Baseball player from 1926 to 1945 and manager from 1933 to 1947. ... Reverse side of a Paul Molitor baseball card Paul Leo Molitor (born August 22, 1956 in St. ... Earl Sidney Weaver (born August 14, 1930 in St. ...

In basketball, 4 represents the power forward position. Also, The term Final Four refers to the last four teams remaining in the NCAA playoff tournament. If a player completes a three-pointer while being fouled, the player is awarded one free-throw for a possible 4-point-play. The number for the Toronto Raptors forward Chris Bosh. This article is about the sport. ... Power forward is a position in the sport of basketball. ... Final Four is a sports term that is commonly applied to the last four teams remaining in a playoff tournament. ... Sara Giauro shoots a three-point shot, FIBA Europe Cup for Women Finals 2007 In basketball, a three-point field goal, three-pointer, three-point shot, or simply three is a field goal made from beyond the three point line, a designated semi-ellipsoid arc radiating from the basket. ... For the professional wrestler, see Chris Bosh (wrestler). ...

In ice hockey, 4 was the jersey number worn by Hall of Famers Jean Beliveau, Bobby Orr, Scott Stevens and Rob Blake. Ice hockey, known simply as hockey in areas where it is more common than field hockey, is a team sport played on ice. ... Jean Arthur Béliveau, born August 31, 1931 in Trois-Rivières, Quebec, Canada was a professional ice hockey player. ... Robert Gordon Bobby Orr, OC (born March 20, 1948 in Parry Sound, Ontario) is a retired Canadian ice hockey defenseman, and is considered to be one of the greatest hockey players of all time. ... For other persons named Scott Stevens, see Scott Stevens (disambiguation). ... Robert Bowlby Rob Blake (born December 10, 1969, in Simcoe, Ontario) is a professional ice hockey defenseman in the NHL, playing for the Los Angeles Kings. ...

In American football, 4 is today most commonly identified with the jersey number of former Green Bay Packers quarterback Brett Favre. United States simply as football, is a competitive team sport that is both fast-paced and strategic. ... Packers redirects here. ... Brett Hillbilly Favre (pronounced Farv, born on October 10, 1969 in Gulfport, Mississippi [1]) is an American football player, currently starting quarterback for the Green Bay Packers of the National Football League (NFL). ...

In rowing, a four refers to a boat for four rowers, with or without coxswain. In rowing nomenclature 4- reprsents a coxless four and 4+ represents a coxed four. A coxless pair which is a sweep-oar boat. ... For other uses, see Boat (disambiguation). ... The coxswain (pronounced cox-É™n; often called the cox) is the person in charge of a boat, particularly its navigation and steering. ...

In soccer football, number 4 is often assigned to a centre back.and a number of luck. Soccer redirects here. ... In the sport of Association football, each of the eleven players in a team are assigned to a particular named position on the field of play. ...

In other fields

See also 4 (disambiguation). Image File history File links ICS_Four. ... Image File history File links ICS_Four. ... The system of international maritime signal flags is a way of representing individual letters of the alphabet on ships or in nautical situations. ... Look up four in Wiktionary, the free dictionary. ...

• Four is the universal number for women, as symbolically suggested in literature throughout history[citation needed]
• Four can be used as an obscenity in some technologically adept circles, replacing the common word for fornication. This usage derives from the fact that when counting in binary on ones fingers, 4, whose binary representation is 001002 is represented by raising the middle finger of one hand, yielding a vulgar gesture. A stronger variation uses the number 132, whose binary representation 0010000100 replicates this gesture on both hands.
• A four-letter word is used to describe most swear words in the English language, as most such terms do indeed possess four letters.
• Four (四, formal writing: 肆, pinyin sì) is considered an unlucky number in Chinese, Korean, and Japanese cultures because it sounds like the word "death" (死, pinyin sǐ). Due to that, many numbered product lines skip the "four": e.g. Nokia cell phones (there is no series beginning with a 4), Palm PDAs, the Leisure Suit Larry games, etc. Some buildings skip floor 4 or replace the number with the letter "F", particularly in heavily Asian areas. See tetraphobia and Numbers in Chinese culture.
• In the NATO phonetic alphabet, the digit 4 is called "fower".
• The number of characters in a canonical four-character idiom.
• The number of strings on a violin, a viola, a cello, double bass, a cuatro and a ukulele, and the number of string pairs on a mandolin.
• The designation of Interstate 4, a freeway in Florida.
• The number of arms possessed by video game villain Goro.
• The number of fingers that people have in many animated cartoons.
• Four, an album by Blues Traveler
• 4, an album by Foreigner
• The number of completed, numbered symphonies by Johannes Brahms
• In Astrology, Cancer is the 4th astrological sign of the Zodiac.
• 4 is one of the "Lost Numbers" on the television show, Lost, along with 8, 15, 16, 23, and 42.
• Arthur Conan Doyle wrote a book titled "The Sign of Four"
• In Tetris, every shape in the game is formed of 4 blocks each. Also the game was named after "tetra" the Greek word for 4.
• 4 is a symbol for for in computer chats.
• some people think the number 4 is an unlucky number.

In Music

In Western Music, common time is constructed of four beats. Western music is the genres of music originating in the Western world (Europe and its former colonies) including Western classical music, American Jazz, Country and Western, pop music and rock and roll. ... The time signature (also known as meter signature) is a notational device used in Western musical notation to specify how many beats are in each bar and which note value (minim, crotchet, eighth note and so on) constitutes one beat. ...

References

1. ^ Georges Ifrah, The Universal History of Numbers: From Prehistory to the Invention of the Computer transl. David Bellos et al. London: The Harvill Press (1998): 394, Fig. 24.64
• Wells, D. The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group. (1987): 55 - 58

Results from FactBites:

 Probability (783 words) The chances of landing on blue are 1 in 4, or one fourth. The chances of landing on red are 1 in 4, or one fourth. A number from 1 to 11 is chosen at random.
 4 (number) - Wikipedia, the free encyclopedia (1736 words) As a consequence of this, it is the smallest Smith number. The Arabs didn't have time for cursive fancy: their 4 still had the early concept of the cross, but for the sake of efficiency, was made in one stroke by connecting the "western" end to the "northern" end; the "eastern" end was finished off with a curve. The number of strings on a violin, a viola, a cello, double bass, and a ukulele, and the number of string pairs on a mandolin.
More results at FactBites »

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## Circles

circle is a planar figure with all points the same distance from a fixed point. That fixed point is called the center of the circle. Any segment that goes from the center to a point on the circle is called a radius of the circle. A diameter is any segment that passes through the center and has its endpoints on the circle. Obviously, a diameter is twice as long as a radius. In Figure 1O is the center, OB OC , and OA each a radius, and ACis a diameter.

Figure 1 A circle with center, radius, and diameter labeled.

## Finding the Circumference

In ancient times the Greeks discovered that for all circles, the circumference divided by the diameter always turns out to be the same constant value. The Greek letter π (pi) is now used to represent that value. In fractional or decimal form, the commonly used approximations of π are π ≈ 3.14 or π ≈ 22/7. The Greeks found the formula Ccircle/d = π, which is rewritten in the following form.

If you briefly regard a circle as a regular polygon with infinitely many infinitesimally small sides, you see that the apothem and radius become the same length (Figure 2).

Figure 2 Apothem and radius of a circle.

Looking at the area formula for a regular polygon and making the appropriate changes with regard to the circle,

That is, the formula for the area of a circle now becomes the following:

Example 1: Find the circumference and area for the circle in Figure 3. Use 3.14 as an approximation for π.

Figure 3 Finding the circumference and area of a circle.

Example 2: If the area of a circle is 81π ft2, find its circumference.

So the circumference is approximately 56.52 ft.

format: plain


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# Triangle: Let ABC is a triangle and M 1, M 2 and M 3 are medians of the given triangle. Then, # Inradius of triangle: Given, ABC is a triangle and a, b and c are the sides of given triangle. [2] 2018/03/12 11:01 Male / 60 years old level or over / An engineer / - / Purpose of use (2) If AB and CD are two chords intersecting each other at P. Inradius, perimeter, and area | Special properties and parts of triangles | Geometry | Khan Academy - Duration: 7:29. 7.5 cm. Speed with Accuracy for solving questions is the key to success in these tests. Note: the remaining two angles of an obtuse angled triangle are always acute. Inradius of a triangle given 3 exradii calculator uses Inradius of Triangle=1/(1/Exradius of excircle opposite ∠A+1/Exradius of excircle opposite ∠B+1/Exradius of excircle opposite ∠C) to calculate the Inradius of Triangle, The Inradius of a triangle given 3 exradii formula is given by relation 1/r = … Right Angled Triangle. Ø A, B and C are vertices and a, b, c are sides of triangle ABC.. Ø Sum of two sides of a triangle is always greater than third side of that triangle. Solution of Triangles: This problem involves understanding the formula of the inradius of a triangle. Then, the maximum possible area of the triangle will be R 2. The reason this is important is because a centroid divides each of the medians into two parts such that the distance from the centroid to the midpoint of the opposite … I need to find the inradius of a triangle with side lengths of $20$, $26$, and $24$. Ø Difference of two sides of a triangle is always smaller than the third side of that triangle. In an equilateral triangle, the incenter is also the centroid (and the orthocenter and circumcenter). Best Inradius Formula Of Equilateral Triangle Images. ‹ Derivation of Formula for Radius of Circumcircle up Derivation of Heron's / Hero's Formula for Area of Triangle › Log in or register to post comments 54292 reads ... For an equilateral triangle-Inradius = h/3 = a/2√3; Circumradius of a triangle given 3 exradii and inradius calculator uses Circumradius of Triangle=(Exradius of excircle opposite ∠A+Exradius of excircle opposite ∠B+Exradius of excircle opposite ∠C-Inradius of Triangle)/4 to calculate the Circumradius of Triangle, The Circumradius of a triangle given 3 exradii and inradius formula is given as R = (rA + rB + rC - r)/4. 8 cm. Proof of the formula relating the area of a triangle to its circumradius If you're seeing this message, it means we're having trouble loading external resources on our website. In the figure, for triangle ABC, a² =b² + c². A D 2 + B E 2 + C F 2 = B D 2 + C E 2 + A F 2. I know the semiperimeter is $35$, but how do I find the area without knowing the height? Let ‘r’ is the inradius of triangle. Triangle – Mensuration Formulas PDF: Triangle is a 2d geometric shape with 3 sides. In addition to the other answers, the calculation of a value for the inradius can be derived as follows: Let the inradius be $r$ and the triangle have area $A$ and sides $a$, $b$, $c$. Geometry Questions For SSC CHSL PDF SSC Stenographer Constable Geometry Question and Answers download PDF based on previous year question paper of SSC Stenographer exam. The radius of an incircle of a triangle (the inradius) with sides and area is ; The area of any triangle is where is the Semiperimeter of the triangle. Trigonometry formulas list for all the classes from 10 to 12 are accumulated here. The semi perimeter, s = \\frac{3a}{2}) In-radius, 'r' for any triangle = \\frac{A}{s}) R(circumradius) = hypotenuse/2. Formula 2: Area of a triangle if its inradius, r is known. There are various types of triangles. Area = rs (where r = in radius and s = (a+b+c)/2 where a,b and c are sides of the triangles) Definition : A closed figure formed by three sides. A = \\frac{\sqrt{3}}{4})a 2. With so many formulas to learn and remember, this section is going to take a lot of time to master. Pythagoras Theorem In the case of a right-angle triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Triangles: In radius of a right angle triangle. (1) Maximum area of triangle = R 2. Contributed by: Jay Warendorff (March 2011) Open content licensed under CC BY-NC-SA The formula above can be simplified with Heron's Formula, yielding ; The radius of an incircle of a right triangle (the inradius) with legs and hypotenuse is . 10 cm. Being a part of geometry, this topic demands that the aspirant be familiar not only with the different formulas of mensuration but also with the other theorems and concepts of geometry. If the given triangle is right angle triangle and its base is the diameter of the circle. Obtuse Triangle: If any one of the three angles of a triangle is obtuse (greater than 90°), then that particular triangle is said to be an obtuse angled triangle. Its radius, the inradius (usually denoted by r) is given by r = K/s, where K is the area of the triangle and s is the semiperimeter (a+b+c)/2 (a, b and c being the sides). Thank you. In this article we will discuss about the various centers of triangle. 2--Bisect each angle 3--The intersection of the bisectors is the center of the inscribed circle of the triangle with radius r.. 4--Let the center of this incircle be called O.and the three sides a, b and c. 5--Consider the three triangles AOB, BOC and COA If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. Calculating the radius []. Formulas. Download Topic Wise SSC CHSL Important Questions PDF SSC CHSL Study Material (FREE Tests) Question 1: Point R lies on line segment PQ […] Qualifying them is the main target of their aspirants. A D 2 + B E 2 + C F 2 = B D 2 + C E 2 + A F 2. Check SSC Tricks Triangle Formulas for all exams: SSC and IBPS exams are one of the most sought exams in India. A circumscribed circle or circumcircle of a triangle is a circle which passes through all the vertices of the triangle. In any triangle, the distance around the boundary of the triangle from a vertex to the point on the opposite edge touched by an excircle equals the semiperimeter.. Triangles - Inradius of right (angled) triangle: r - the inradius , c - hypotenuse , a,b - triangle sides If the length of the sides of a triangle are in the ratio 4 : 5 : 6 and the inradius of the triangle is 3 cm, then the altitude of the triangle corresponding to the largest side as base is. Any pedal triangle D E F DEF D E F satisfies. Formula 1: Area of an equilateral triangle if its side is known. 10 Very important Geometry questions for Stenographer Constable. Area A = r \\times) s, where r is the in radius and 's' is the semi perimeter. 1--Consider an acute triangle ABC, A lower left, B lower right and C at the top. 6 cm AD^2 + BE^2 + CF^2 = BD^2 + CE^2 + AF^2. # Circumradius of triangle: Given, ABC is a triangle and a, b and c are the sides of given triangle. The center of this circle is called the circumcenter and its radius is called the circumradius. In SSC CGL exams 1 or 2 question generally come in exam from this chapter. TRIANGLE. Orthocentre – Point of intersection of altitudes of a triangle ... you should immediately recall this formula] Over the years SSC has asked many questions based on this simple formula. The lines which are joining the midpoints of sides of a triangle to the opposite vertex are called medians. Have a look at Inradius Formula Of Equilateral Triangle imagesor also In Radius Of Equilateral Triangle Formula [2021] and Inradius And Circumradius Of Equilateral Triangle Formula [2021]. It only takes a minute to sign up. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Mensuration formulas based questions regularly feature in CAT, SSC CGL and other exams. Acute Triangle: If all the three angles of a triangle are acute i.e., less than 90°, then the triangle is an acute-angled triangle. To prove this, note that the lines joining the angles to the incentre divide the triangle into three smaller triangles, with bases a, b and c respectively and each with height r. The equilateral triangle will have the maximum area among all the triangles that can be formed with a given perimeter. List of all important formulas with reference to a right-triangle and a unit circle provided here at BYJU'S. 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# Fredholm Equation with Exponential Sum Kernel

I'm trying to solve the following integral equation to find the function $$f(x)$$ $$$$f(x) = K(x) - \int_0^\infty K(x-t)f(t)dt$$$$

where $$$$K(x) = \sum_{i=1}^N c_i e^{-\kappa_i |x|}, \quad \forall i\quad\kappa_i > 0$$$$ with the convenient property $$K(x-t)=K(t-x)=K(|x-t|)$$.

I know that this is an inhomogeneous Fredholm equation of the second kind, however I'm not sure where to begin solving this. Any references and suggestions where to start would be of great help.

• You need to show that K is a compact operator in the space you are interested in. – timur Feb 4 at 15:57
• Ok, I'm not sure what that means but I'm fairly confident that a solution exists. – Freelunch Feb 5 at 8:04


text:
# If f is a bounded function and Riemann integrable on a subset [a',b] of [a,b], prove that f is integrable on [a,b].

Let f : [a,b] $$\to$$ $$\mathbb{R}$$ be a bounded function such that f $$\epsilon$$ $$\mathcal{R}$$[a',b] (ie: f is Riemann integrable) for every a' $$\epsilon$$ (a,b).

Prove that f $$\epsilon$$ $$\mathcal{R}$$ [a,b].

I'm a little stuck as to how to start this one. It seems simple but the professor has it weighted kind of heavily in the assignment.


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senU {DOS} R Documentation

## Sensitivity Analysis for a New U Statistic

### Description

Sensitivity analysis for the new U statistic of Rosenbaum (2011). For m=m1=m2, this is the test of Stephenson (1981). The ranks proposed by Stephenson closely approximate the optimal ranks proposed by Conover and Salzburg (1988) for detecting a treatment that has a large effect on a small subpopulation and no effect on most of the population; see Rosenbaum (2007). The example reproduces some results from Chapter 16 of Design of Observational Studies (2010).

### Usage

senU(d, gamma = 1, m = 2, m1 = 2, m2 = 2, conf.int = FALSE,
alpha = 0.05, alternative = "greater", exact = NULL)

### Arguments

 d A vector of treated-minus-control matched pair differences in outcomes. gamma gamma >= 1 is the value of the sensitivity parameter. m See m2. m1 See m2. m2 If (m,m1,m2) are three integers such that 1 <= m1 <= m2 <= m, then the triple (m,m1,m2) defines a U statistic. If (m,m1,m2) = (1,1,1), then the U statistic is the sign test statistic. If (m,m1,m2) = (2,2,2), then it is the U statistic that closely approximates Wilcoxons signed rank test. If m=m1=m2, then the U statistic is the test of Stephenson (1981). The general U statistic is discussed in Rosenbaum (2011). conf.int If conf.int=TRUE, the a 1-alpha confidence interval and an interval of point estimates is returned in addition to the P-value testing no treatment effect. alpha Coverage rate of the confidence interval. With probability at least 1-alpha, the confidence interval will cover the treatment effect providing the bias in treatment assignment is at most gamma. alternative If alternative = "greater" or alternative = "less", then one-sided tests and intervals are returned. If alternative = "twosided", then both one sided tests are done, with the smaller P-value doubled to yield a two-sided P-value. If alternative = "twosided", the confidence interval is the intersection of two one-sided 1-alpha/2 confidence intervals. exact If exact is NULL, then exact is set to TRUE if length(d) <= 50, and is set to FALSE if length(d) > 50. The ranks used by the U statistic involve combinatorial coefficiencts that grow rapidly with increasing sample size. If exact=TRUE, these ranks are computed exactly using expression (8) in Rosenbaum (2011). If exact=FALSE, the ranks are computed by an asymptotic approximation that does not involve large combinatorial coefficients, specifically expression (9) in Rosenbaum (2012).

### Details

The senWilcox function uses a large sample Normal approximation to the distribution of Wilcoxon's signed rank statistic. When gamma=1, it should agree with the wilcox.test() function in the stats package with exact=FALSE and correct=FALSE. The example reproduces the example of the large-sample approximation in Section 3.5 of Design of Observational Studies. Note that the confidence intervals in Table 3.3 of that book are exact, not approximate, so they are slightly different.

### Value

 pval The upper bound on the P-value testing no effect in the presence of a bias in treatment assignment of at most gamma. If the bias in treatment assignment is at most gamma, and if there is no treatment effect, then there is at most an alpah chance that the P-value is less than alpha, this being true for all 0

### Note

The test of Stephenson (1981) uses ranks similar to those of Conover and Salzburg (1988) which were designed to have high power when most people are unaffected by treatment, but a small subpopulation is strongly affected; see Rosenbaum (2007). This is the situation discussed in Chapter 16 of Design of Observational Studies (2010). Even for pair differences d that are Normal with expectation tau and constant variance, the Wilcoxon test tends to exaggerate the degree of sensitivity to unmeasured bias. Compare the Wilcoxon test and the U statistic with (m,m1,m2) = (5,4,5) in the Normal situation. In a randomized experiment (gamma=1), the two tests have the same Pitman efficiency. However, as the number of pairs increases with tau=0.5, the Wilcoxon test has limiting sensitivity to bias of gamma=3.2 while (m,m1,m2) = (5,4,5) has limiting sensitivity 3.9, and (m,m1,m2) = (8,7,8) has limiting sensitivity 5.1. See Rosenbaum (2011) for specifics.

### Author(s)

Paul R. Rosenbaum

### References

Conover, W. J. and Salsburg, D. S. (1988). Locally most powerful tests for detecting treatment effects when only a subset of patients can be expected to" respond" to treatment. Biometrics, 189-196.

Hodges Jr, J. L. and Lehmann, E. L. (1963). Estimates of location based on rank tests. The Annals of Mathematical Statistics, 598-611.

Rosenbaum, P. R. (1993). Hodges-Lehmann point estimates of treatment effect in observational studies. Journal of the American Statistical Association, 88(424), 1250-1253.

Rosenbaum, P. R. (2007). Confidence intervals for uncommon but dramatic responses to treatment. Biometrics, 63(4), 1164-1171. <doi:10.1111/j.1541-0420.2007.00783.x> >

Rosenbaum, P. R. (2010). Design of Observational Studies. New York: Springer. The method and example are discussed in Chapter 16.

Rosenbaum, P. R. (2011). A new U statistic with superior design sensitivity in matched observational studies. Biometrics, 67(3), 1017-1027. <doi:10.1111/j.1541-0420.2010.01535.x>

Schoket, B., Phillips, D. H., Hewer, A. and Vincze, I. (1991). 32P-postlabelling detection of aromatic DNA adducts in peripheral blood lymphocytes from aluminium production plant workers. Mutation Research/Genetic Toxicology, 260(1), 89-98.

Stephenson, W. R. (1981). A general class of one-sample nonparametric test statistics based on subsamples. Journal of the American Statistical Association, 76(376), 960-966.

### Examples

data("schoket")
d<-schoket$adductsw-schoket$adductsc

# With the defaults, m=2, m1=2, m2=2, the U-statistic is very
# similar to Wilcoxon's signed rank statistic
senWilcox(d,gamma=1)
senU(d,gamma=1)

# With m=1, m1=1, m2=1, the U-statistic is the sign test
senU(d,gamma=1,m=1,m1=1,m2=1)
prop.test(sum(d>0),length(d),p=.5,
alternative="greater",correct=FALSE)$p.value # With m=m1=m2, this is the test of Stephenson (1981) whose ranks are similar to # those of Conover and Salzburg (1988); see Rosenbaum (2007). # The calculations that follow reproduce the sensitivity analysis for the # data of Schoket et al. () in Chapter 16 of Desgin of Observational Studies (2010). senU(d,gamma=1,m=2,m1=2,m2=2) senU(d,gamma=1,m=5,m1=5,m2=5) senU(d,gamma=1.5,m=2,m1=2,m2=2) senU(d,gamma=1.5,m=5,m1=5,m2=5) senU(d,gamma=1.8,m=2,m1=2,m2=2) senU(d,gamma=1.8,m=5,m1=5,m2=5) senU(d,gamma=2,m=2,m1=2,m2=2) senU(d,gamma=2,m=5,m1=5,m2=5) data(lead) # Reproduces parts of Table 2 in Rosenbaum (2011) senU(lead$dif,gamma=5.8,m=8,m1=5,m2=8)
senU(lead$dif,gamma=5,m=5,m1=4,m2=5) # m=2, m1=2, m2=2 is the U-statistic that closely # resembles Wilcoxon's signed rank test. Note # that the results are almost the same. senWilcox(lead$dif,gamma=5) # In Table 2


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# AES CBC encoding of decomposable OAuth2 tokens

We're building a stateless RESTful API that relies on the content of OAuth2 tokens to identify users and what they have access to. Users authenticate with other means and are given a token, which they then use to authenticate API requests for an hour or so till it expires.

Since we want it to be stateless, we can't use random tokens with a lookup table. The token basically consists of username + start timestamp + end timestamp. (The token is considered valid if it has a valid user and the current time is between the start and end timestamps.) We encrypt the token before sending it to the user and decrypt it each time we get it back in a request.

If we naively encrypt the token with AES in ECB mode, the ciphertexts for a given user's tokens end up very similar to one another, because the usernames encrypt to the same ciphertext every time, and I assume somebody who knew what they were doing could XOR a few tokens together and recover the key fairly easily.

If we encrypt them in CBC mode instead -- with a fixed IV, so we can decrypt them later -- I assume the attack is a little harder (maybe a lot harder? maybe not harder at all?), but the ciphertexts still end up very similar.

What we're currently doing to get around this is reverse the token (bytewise) before encrypting in CBC mode, so the highest-entropy bytes get encrypted first and feed into the rest of the chain, and there's no obvious similarity between ciphertexts, and at least we don't look stupid. But, as they say, anyone can design a cryptosystem they themselves can't break.

Questions, then:

• Is this actually doing any good?
• Is there a known best practice we should be following instead?
• If there isn't a known best practice, are there any obvious improvements?

(Note: All of this is happening over SSL so we're not particularly worried about man-in-the-middle attacks etc. But we don't want to encourage any otherwise authorized users to try to forge tokens, either.)

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A note: even with use of ECB mode, as long as you do not encipher (a derivative of) the key using itself, there is no way that knowledge of ciphertext and plaintext will allow an adversary to recover the key, or something that would allow to decipher or encipher some random thing. The best an adversary can do is recognize plaintext/ciphertext blocks that previously occurred, and recompose fake plaintexts by recomposing ciphertext blocks. A question: how is the key of your AES generated, stored (if it is across session), and used? That's a classical weak spot. – fgrieu Feb 9 '14 at 18:15

want to keep $\:$ username + start timestamp + end timestamp $\:$ confidential from someone


text:
Examples Using Euler's Theorem

# Examples Using Euler's Theorem

Recall that Euler's theorem states that if $(a, m) = 1$, then $a^{\phi (m)} \equiv 1 \pmod {m}$. We will now apply this property to some common questions.

## Example 1

Find the last digit of 555.

We first note that finding the last digit of 555 can be obtained by reducing 555 (mod 10), that is evaluating $55^{5} \pmod {10}$. We note that (10, 55) = 5, and hence this pair is not relatively prime, however, we know that 55 has a prime power decomposition of 55 = 5 x 11. (11, 10) = 1, hence it follows that $11^{\phi (10)} \equiv 1 \pmod {10}$. We note that $\phi (10) = 4$. Hence $11^4 \equiv 1 \pmod {10}$, and more appropriately:

(1)
\begin{align} 55^{5} = 5^5 \cdot 11^5 = 5^{5} \cdot 11^4 \cdot 11 \equiv 5^{12} \cdot (1)^4 \cdot 11 \equiv 34375 \equiv 5 \pmod {10} \end{align}

Hence the last digit of 555 is 5.

## Example 2

Find the last two digits of 33334444.

We first note that finding the last two digits of 33334444 can be obtained by reducing 33334444 (mod 100). Since (3333, 100) = 1, we can apply this theorem.

We first calculate that $\phi (100) = \phi (2^2) \phi(5^2) = (2)(5)(4) = 40$. Hence it follows from Euler's theorem that $3333^{40} \equiv 1 \pmod {100}$. Now let's apply the division algorithm on 4444 and 40 as follows:

(2)
$$4444 = 40(111) + 4$$

Hence it follows that:

(3)
\begin{align} 3333^{4444} \equiv (3333^{40})^{111} \cdot 3333^{4} \equiv (1)^{111} \cdot 3333^4 \pmod {100} \equiv 33^4 = 1185921 \equiv 21 \pmod {100} \\ \end{align}

Hence the last two digits of 33334444 are 2 and 1.

## Example 3

Find the remainder 29202 when divided by 13.

We first note that $(29, 13) = 1$. Hence we can apply Euler's Theorem to get that $29^{\phi (13)} \equiv 1 \pmod {13}$. Since 13 is prime, it follows that $\phi (13) = 12$, hence $29^{12} \equiv 1 \pmod {13}$. We can now apply the division algorithm between 202 and 12 as follows:

(4)
$$202 = 12(16) + 10$$

Hence it follows that $29^{202} = (29^{12})^{26} \cdot 29^{10} \equiv (1)^{26} \cdot 29^{10} \equiv 29^{10} \pmod{13}$.

Also we note that 29 can be reduced to 3 (mod 13), and hence:

(5)
\begin{align} 29^{10} \equiv 3^{10} = 59049 \equiv 3 \pmod {13}^2 \end{align}

Hence when 29202 is divided by 13, the remainder leftover is 3.

## Example 4

Find the remainder of 99999999 when divided by 23.

Once again we note that $(99, 23) = 1$, hence it follows that $99^{\phi (23)} \equiv 1 \pmod {23}$. Once again, since 23 is prime, it goes that $\phi (23) = 22$, and more appropriately $99^{22} \equiv 1 \pmod {23}$. We will now use the division algorithm between 999999 and 22 to get that:

(6)
$$999999 = 22(45454) + 11$$

Hence it follows that $99^{999999} = (99^{22})^{45454} \cdot 99^{11} \equiv 1^{45454} \cdot 99^{11} \equiv 7^{11} = 1977326743 \equiv 22 \pmod {23}$.

Hence the remainder of 99999999 when divided by 23 is 22.

Note that we can solve the final congruence a little differently as: $99^11 \equiv 7^{11} = (7^2)^5 \cdot 7 = (49)^5 \cdot 7 \equiv 3^5 \cdot 7 = 1701 \equiv 22 \pmod {23}$. There are many ways to evaluate these sort of congruences, some easier than others.