The sieve of Eratosthenes is an ancient method used to find what in mathematics?
"Rational sieve In mathematics, the rational sieve is a general algorithm for factoring integers into prime factors. It is a special case of the general number field sieve. While it is less efficient than the general algorithm, it is conceptually simpler. It serves as a helpful first step in understanding how the general number field sieve works. Suppose we are trying to factor the composite number ""n"". We choose a bound ""B"", and identify the ""factor base"" (which we will call ""P""), the set of all primes less than or equal to ""B"". Next, we search for positive integers ""z"""
"of a number of prime number sieves, is a simple, ancient algorithm for finding all prime numbers up to any given limit. It does so by iteratively marking as composite, ""i.e."", not prime, the multiples of each prime, starting with the multiples of 2. The multiples of a given prime are generated starting from that prime, as a sequence of numbers with the same difference, equal to that prime, between consecutive numbers. This is the sieve's key distinction from using trial division to sequentially test each candidate number for divisibility by each prime. Eratosthenes was one of the most pre-eminent"
"large error, which means the sieve only gives very weak bounds in most cases. For this reason it is almost never used in practice, having been superseded by other techniques such as the Brun sieve and Selberg sieve. However, since these more powerful sieves are extensions of the basic ideas of the Legendre sieve, it is useful to first understand how this sieve works. Legendre sieve In mathematics, the Legendre sieve, named after Adrien-Marie Legendre, is the simplest method in modern sieve theory. It applies the concept of the Sieve of Eratosthenes to find upper or lower bounds on the"
"sophisticated concepts from either algebraic number theory or analytic number theory. Nevertheless, the more advanced sieves can still get very intricate and delicate (especially when combined with other deep techniques in number theory), and entire textbooks have been devoted to this single subfield of number theory; a classic reference is and a more modern text is . The sieve methods discussed in this article are not closely related to the integer factorization sieve methods such as the quadratic sieve and the general number field sieve. Those factorization methods use the idea of the sieve of Eratosthenes to determine efficiently which"
"Eratosthenes Eratosthenes of Cyrene (; , ; – ) was a Greek mathematician, geographer, poet, astronomer, and music theorist. He was a man of learning, becoming the chief librarian at the Library of Alexandria. He invented the discipline of geography, including the terminology used today. He is best known for being the first person to calculate the circumference of the Earth, which he did by comparing altitudes of the mid-day sun at two places a known North-South distance apart. His calculation was remarkably accurate. He was also the first to calculate the tilt of the Earth's axis, again with remarkable"
"Generating primes In computational number theory, a variety of algorithms make it possible to generate prime numbers efficiently. These are used in various applications, for example hashing, public-key cryptography, and search of prime factors in large numbers. For relatively small numbers, it is possible to just apply trial division to each successive odd number. Prime sieves are almost always faster. A prime sieve or prime number sieve is a fast type of algorithm for finding primes. There are many prime sieves. The simple sieve of Eratosthenes (250s BCE), the sieve of Sundaram (1934), the still faster but more complicated sieve"
"Arithmetica Arithmetica () is an Ancient Greek text on mathematics written by the mathematician Diophantus in the 3rd century AD. It is a collection of 130 algebraic problems giving numerical solutions of determinate equations (those with a unique solution) and indeterminate equations. Equations in the book are presently called Diophantine equations. The method for solving these equations is known as Diophantine analysis. Most of the ""Arithmetica"" problems lead to quadratic equations. In Book 3, Diophantus solves problems of finding values which make two linear expressions simultaneously into squares or cubes. In book 4, he finds rational powers between given numbers."
"prominence of scientific enquiry in Europe at the time of their painting in 1668–69. The ancient Greeks discovered much of the knowledge that formed the basis for later scientific and geographical discoveries. Eratosthenes of Cyrene (born circa 276 BCE in Cyrene, Libya; died circa 194 BCE in Alexandria, Egypt) was a Greek writer, poet, astronomer, and geographer who created the first known measurements of the Earth's circumference. Eratosthenes used the angles at which the sun's rays fall on the ground to calculate the curvature of the Earth's surface, which he then used to determine the full horizontal length around the"
"scholarly figures of his time, and produced works covering a vast area of knowledge before and during his time at the Library. He wrote on many topics — geography, mathematics, philosophy, chronology, literary criticism, grammar, poetry, and even old comedies. Unfortunately, there are only fragments left of his works after the Destruction of the Library of Alexandria. Eratosthenes Eratosthenes of Cyrene (; , ; – ) was a Greek mathematician, geographer, poet, astronomer, and music theorist. He was a man of learning, becoming the chief librarian at the Library of Alexandria. He invented the discipline of geography, including the terminology"
"cull operations performed by the algorithm, following is a table showing the actually measured number of operations for a practical implementation of the sieve of Eratosthenes as compared to the number of operations predicted from the above expression with both expressed as a fraction of the range (rounded to four decimal places) for different sieve ranges and wheel factorizations (Note that the last column is a maximum practical wheel as to the size of the wheel gaps Look Up Table - almost 10 million values): The above table shows that the above expression is a very good approximation to the"
"Lattice sieving Lattice sieving is a technique for finding smooth values of a bivariate polynomial formula_1 over a large region. It is almost exclusively used in conjunction with the number field sieve. The original idea of the lattice sieve came from John Pollard. The algorithm implicitly involves the ideal structure of the number field of the polynomial; it takes advantage of the theorem that any prime ideal above some rational prime ""p"" can be written as formula_2. One then picks many prime numbers ""q"" of an appropriate size, usually just above the factor base limit, and proceeds by For the"
"this to be efficient you need to be able to store a number of updates at least comparable to the size of the sieve array, so this can be quite profligate in memory usage. Lattice sieving Lattice sieving is a technique for finding smooth values of a bivariate polynomial formula_1 over a large region. It is almost exclusively used in conjunction with the number field sieve. The original idea of the lattice sieve came from John Pollard. The algorithm implicitly involves the ideal structure of the number field of the polynomial; it takes advantage of the theorem that any prime"
"Viggo Brun Viggo Brun (13 October 1885 – 15 August 1978) was a Norwegian professor, mathematician and number theorist. Brun was born at Lier in Buskerud, Norway. He studied at the University of Oslo and began research at the University of Göttingen in 1910. In 1923, Brun became a professor at the Technical University in Trondheim and in 1946 a professor at the University of Oslo. In 1915, he introduced a new method, based on Legendre's version of the sieve of Eratosthenes, now known as the ""Brun sieve"", which addresses additive problems such as Goldbach's conjecture and the twin prime"
"Lehmer sieve Lehmer sieves are mechanical devices that implement sieves in number theory. Lehmer sieves are named for Derrick Norman Lehmer and his son Derrick Henry Lehmer. The father was a professor of mathematics at the University of California, Berkeley at the time, and his son followed in his footsteps as a number theorist and professor at Berkeley. A sieve in general is intended to find the numbers which are remainders when a set of numbers are divided by a second set. Generally, they are used in finding solutions of Diophantine equations or to factor numbers. A Lehmer sieve will"
"Archimedes Archimedes of Syracuse (; ; ) was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Generally considered the greatest mathematician of antiquity and one of the greatest of all time, Archimedes anticipated modern calculus and analysis by applying concepts of infinitesimals and the method of exhaustion to derive and rigorously prove a range of geometrical theorems, including the area of a circle, the surface area and volume of a sphere, and the area under a parabola. Other mathematical achievements"
"Scaphe The Scaphe () was a sundial said to have been invented by Aristarchus (3rd century BC). There are no original works still in existence by Aristarchus, but the adjacent image is an accurate image of what it might have looked like, only his would have been made of stone. It consisted of a hemispherical bowl which had a vertical gnomon placed inside it, with the top of the gnomon level with the edge of the bowl. Twelve gradations inscribed perpendicular to the hemisphere indicated the hour of the day. Using this measuring instrument, Eratosthenes of Cyrene (c. 220 BC)"
"important textbook in mathematics until the 19th century, built upon the work of the Hellenic era Pythagoreans. Euclid developed proofs for the Pythagorean Theorem, for the infinitude of primes, and worked on the five Platonic solids. Eratosthenes used his knowledge of geometry to measure the circumference of the Earth. His calculation was remarkably accurate. He was also the first to calculate the tilt of the Earth's axis (again with remarkable accuracy). Additionally, he may have accurately calculated the distance from the Earth to the Sun and invented the leap day. Known as the ""Father of Geography "", Eratosthenes also created"
"back to any school or period before that of the Ionian Greeks, but the subsequent history may be divided into periods, the distinctions between which are tolerably well marked. Greek mathematics, which originated with the study of geometry, tended from its commencement to be deductive and scientific. Since the fourth century AD, Pythagoras has commonly been given credit for discovering the Pythagorean theorem, a theorem in geometry that states that in a right-angled triangle the area of the square on the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares of"
"the axiomatic method still used in mathematics today, consisting of definition, axiom, theorem, and proof. His textbook ""Elements"" is widely considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is often held to be Archimedes (c. 287–212 BC) of Syracuse. He developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus. Other notable achievements of Greek mathematics are conic"
"involvement in the Prime Number Theorem. In chapter 7, the sieve of Eratosthenes is shown to be able to be simulated using the Zeta function. With this, the following statement which becomes the pillar stone of the book is asserted: Following the derivation of this finding, the book delves into how this is manipulated to expose the PNT's nature. The book was awarded the Mathematical Association of America's inaugural Euler Book Prize in 2007. Prime Obsession Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics (2003) is a historical book on mathematics by John Derbyshire, detailing the history"
"Selberg sieve In mathematics, in the field of number theory, the Selberg sieve is a technique for estimating the size of ""sifted sets"" of positive integers which satisfy a set of conditions which are expressed by congruences. It was developed by Atle Selberg in the 1940s. In terms of sieve theory the Selberg sieve is of ""combinatorial type"": that is, derives from a careful use of the inclusion–exclusion principle. Selberg replaced the values of the Möbius function which arise in this by a system of weights which are then optimised to fit the given problem. The result gives an ""upper"
"The Method of Mechanical Theorems The Method of Mechanical Theorems (), also referred to as The Method, is considered one of the major surviving works of the ancient Greek polymath Archimedes. ""The Method"" takes the form of a letter from Archimedes to Eratosthenes, the chief librarian at the Library of Alexandria, and contains the first attested explicit use of ""indivisibles"" (sometimes referred to as infinitesimals). The work was originally thought to be lost, but in 1906 was rediscovered in the celebrated Archimedes Palimpsest. The palimpsest includes Archimedes' account of the ""mechanical method"", so-called because it relies on the law of"
"crown on a scale with a pure gold reference sample of the same weight, then immersing the apparatus in water. The difference in density between the two samples would cause the scale to tip accordingly. Galileo considered it ""probable that this method is the same that Archimedes followed, since, besides being very accurate, it is based on demonstrations found by Archimedes himself."" In a 12th-century text titled ""Mappae clavicula"" there are instructions on how to perform the weighings in the water in order to calculate the percentage of silver used, and thus solve the problem. The Latin poem ""Carmen de"
"texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry. It is in Babylonian mathematics that elementary arithmetic (addition, subtraction, multiplication and division) first appear in the archaeological record. The Babylonians also possessed a place-value system, and used a sexagesimal numeral system, still in use today for measuring angles and time. Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Around 300 BC, Euclid introduced"
"Larger sieve In number theory, the larger sieve is a sifting device invented by P. X. Gallagher. The name denotes a heightening of the large sieve. In fact, combinatorial sieves like the Selberg sieve are strongest, when only a few residue classes are removed, while the term large sieve means that this sieve can take advantage of the removal of a large number of up to half of all residue classes. The larger sieve can exploit the deletion of an arbitrary number of classes. Suppose that formula_1 is a set of prime powers, ""N"" an integer, formula_2 a set of"
"The solution of the puzzle is a cycle containing twenty (in ancient Greek ""icosa"") edges (i.e. a Hamiltonian circuit on the dodecahedron). Icosian game The icosian game is a mathematical game invented in 1856 by William Rowan Hamilton. The game's object is finding a Hamiltonian cycle along the edges of a dodecahedron such that every vertex is visited a single time, and the ending point is the same as the starting point. The puzzle was distributed commercially as a pegboard with holes at the nodes of the dodecahedral graph and was subsequently marketed in Europe in many forms. The motivation"
"the Sieve of Eratosthenes or by an algorithm that tests each incremental ""m"" against all known primes < ). Then, before testing ""n"" for primality with a serious method, ""n"" can first be checked for divisibility by any prime from the list. If it is divisible by any of those numbers then it is composite, and any further tests can be skipped. A simple, but very inefficient primality test uses Wilson's theorem, which states that ""p"" is prime if and only if: Although this method requires about ""p"" modular multiplications, rendering it impractical, theorems about primes and modular residues form"
"numbers have been studied throughout recorded history. Euclid devoted one book of the ""Elements"" to the theory of primes; in it he proved the infinitude of the primes and the fundamental theorem of arithmetic, and presented the Euclidean algorithm for finding the greatest common divisor of two numbers. In 240 BC, Eratosthenes used the Sieve of Eratosthenes to quickly isolate prime numbers. But most further development of the theory of primes in Europe dates to the Renaissance and later eras. In 1796, Adrien-Marie Legendre conjectured the prime number theorem, describing the asymptotic distribution of primes. Other results concerning the distribution"
"195 BC. Losing the ability to read and to observe nature plagued and depressed him, leading him to voluntarily starve himself to death. He died in 194 BC at 82 in Alexandria. Eratosthenes calculated the Earth's circumference without leaving Egypt. He knew that at local noon on the summer solstice in Syene (modern Aswan, Egypt), the Sun was directly overhead. (Syene is at latitude 24°05′ North, near to the Tropic of Cancer, which was 23°42′ North in 100 BC) He knew this because the shadow of someone looking down a deep well at that time in Syene blocked the reflection"
"with equal base and height. He expressed the solution to the problem as an infinite geometric series, whose sum was . In ""The Sand Reckoner"", Archimedes set out to calculate the number of grains of sand that the universe could contain. In doing so, he challenged the notion that the number of grains of sand was too large to be counted, devising his own counting scheme based on the myriad, which denoted 10,000. Greek mathematics constitutes a major period in the history of mathematics, fundamental in respect of geometry and the idea of formal proof. Greek mathematics also contributed importantly"
"the same time, when in reality red, green, and blue are being cycled extremely quickly. Additionally, Archimedean spirals are used in food microbiology to quantify bacterial concentration through a spiral platter. They are also used to model the pattern that occurs in a roll of paper or tape of constant thickness wrapped around a cylinder. Archimedean spiral The Archimedean spiral (also known as the arithmetic spiral) is a spiral named after the 3rd century BC Greek mathematician Archimedes. It is the locus of points corresponding to the locations over time of a point moving away from a fixed point with"
"latter endeavor is really quite basic, and the calculations tedious rather than difficult. The availability of computer-generated graphics has, of course, completely altered the situation. A Fortran program was written for the VAX computer at the University of Leicester that enabled vertical or horizontal dials to be plotted for any latitude. Made famous by Eratosthenes of Cyrene, he calculated the circumference of the Earth without leaving Egypt. Eratosthenes knew that at local noon on the summer solstice in the Ancient Egyptian city of Swenet (known in ancient Greek as Syene, and now as Aswan) on the Tropic of Cancer, the"
"work as the sieve of Eratosthenes, it is generally more efficient to test for a divisor only those numbers for which it is not immediately clear whether they are prime or not. Typically, one may proceed by testing 2, 3, 5, and the numbers > 5, whose last digit is 1, 3, 7, 9 and the sum of digits is not a multiple of 3. This method works well for factoring small integers, but is inefficient for larger integers. For example, Pierre de Fermat was unable to discover that the 6th Fermat number is not a prime number. In fact,"