The AKS primality test (also known as Agrawal–Kayal–Saxena primality test and cyclotomic AKS test) is a deterministic primality-proving algorithm created and published by Manindra Agrawal, Neeraj Kayal, and Nitin Saxena, computer scientists at the Indian Institute of Technology Kanpur, on August 6, 2002, in a paper titled "PRIMES is in P".[1] The authors received many accolades, including the 2006 Gödel Prize and the 2006 Fulkerson Prize, for this work.

The algorithm determines whether a number is prime or composite within polynomial time. Contents

1 Importance
2 Concepts
3 History and running time
4 Algorithm
5 References
6 External links

Importance

AKS is the first primality-proving algorithm to be simultaneously general, polynomial, deterministic, and unconditional. Previous algorithms had been developed for centuries but achieved three of these properties at most, but not all four.

The AKS algorithm can be used to verify the primality of any general number given. Many fast primality tests are known that work only for numbers with certain properties. For example, the Lucas–Lehmer test for Mersenne numbers works only for Mersenne numbers, while Pépin's test can be applied to Fermat numbers only. The maximum running time of the algorithm can be expressed as a polynomial over the number of digits in the target number. ECPP and APR conclusively prove or disprove that a given number is prime, but are not known to have polynomial time bounds for all inputs. The algorithm is guaranteed to distinguish deterministically whether the target number is prime or composite. Randomized tests, such as Miller–Rabin and Baillie–PSW, can test any given number for primality in polynomial time, but are known to produce only a probabilistic result. The correctness of AKS is not conditional on any subsidiary unproven hypothesis. In contrast, the Miller test is fully deterministic and runs in polynomial time over all inputs,...

...arithmetic. With his scores, he stood first in the district.[18] That year, Ramanujan entered Town Higher Secondary School where he encountered formal mathematics for the first time.[18]
By age 11, he had exhausted the mathematical knowledge of two college students who were lodgers at his home. He was later lent a book on advanced trigonometry written by S. L. Loney.[19][20] He completely mastered this book by the age of 13 and discovered sophisticated theorems on his own. By 14, he was receiving merit certificates and academic awards which continued throughout his school career and also assisted the school in the logistics of assigning its 1200 students (each with their own needs) to its 35-odd teachers.[21] He completed mathematical exams in half the allotted time, and showed a familiarity with infinite series. In 1903 when he was 16, Ramanujan obtained from a friend a library-loaned copy of a book by G. S. Carr.[22][23] The book was titled A Synopsis of Elementary Results in Pure and Applied Mathematics and was a collection of 5000 theorems. Ramanujan reportedly studied the contents of the book in detail.[24] The book is generally acknowledged as a key element in awakening the genius of Ramanujan.[24] The next year, he had independently developed and investigated the Bernoulli numbers and had calculated Euler's constant up to 15 decimal places.[25] His peers at the time commented that they "rarely understood him" and "stood in respectful awe"...

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Pythagorean Theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle (right-angled triangle). In terms of areas, it states:
In any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle).
The theorem can be written as an equation relating the lengths of the sides a, b and c, often called the Pythagorean equation:[1]
where c represents the length of the hypotenuse, and a and b represent the lengths of the other two sides.
These two formulations show two fundamental aspects of this theorem: it is both a statement about areas and about lengths. Tobias Dantzig refers to these as areal and metric interpretations.[2][3] Some proofs of the theorem are based on one interpretation, some upon the other. Thus, Pythagoras' theorem stands with one foot in geometry and the other in algebra, a connection made clear originally byDescartes in his work La Géométrie, and extending today into other branches of mathematics.[4]
The Pythagorean theorem has been modified to apply outside its original domain. A number of these generalizations are described below, including...

...bernoulli's theorem
ABSTRACT / SUMMARY
The main purpose of this experiment is to investigate the validity of the Bernoulli equation when applied to the steady flow of water in a tape red duct and to measure the flow rate and both static and total pressure heads in a rigid convergent/divergent tube of known geometry for a range of steady flow rates. The apparatus used is Bernoulli’s Theorem Demonstration Apparatus, F1-15. In this experiment, the pressure difference taken is from h1- h5. The time to collect 3 L water in the tank was determined. Lastly the flow rate, velocity, dynamic head, and total head were calculated using the readings we got from the experiment and from the data given for both convergent and divergent flow. Based on the results taken, it has been analysed that the velocity of convergent flow is increasing, whereas the velocity of divergent flow is the opposite, whereby the velocity decreased, since the water flow from a narrow areato a wider area. Therefore, Bernoulli’s principle is valid for a steady flow in rigid convergent and divergent tube of known geometry for a range of steady flow rates, and the flow rates, static heads and total heads pressure are as well calculated. The experiment was completed and successfully conducted.
INTRODUCTION
In fluid dynamics, Bernoulli’s principle is best explained in the application that involves in viscid flow, whereby the speed of the moving fluid is increased...

...Fermat's Last Theorem
Fermat's Last Theorem states that no three positive integers, for example (x,y,z), can satisfy the equation x^n+y^n=z^n if the integer value of n is greater than 2. Fermat's Last Theorem is an example a Diophantine equation(Weisstein). A Diophantine equation is a polynomial equation in which the solution must be an integers. These equations came from the works of Diophantus who was a mathematician who worked methods on solving these equations. Fermat's Last Theorem was based on Diophantus's work. A more common Diophantine equation would be Pythagorean Theorem, where the solution would be the the Pythagorean triples(Weisstein). However, unlike Pythagorean Theorem, Fermat's Last Theorem has no practical real world applications.
Fermat had scribbled on the margin of Arithmetica, the book that inspired his theorem, that he had a proof that would not fit on the margin of a book. From the 1600's-mid 1900's this proof remained unsolved. It was eventually solved by Andrew Wiles. Andrew Wiles as a child always loved math, he would always make up problems and challenge himself....

...Negative Externalities and the Coase Theorem
As Adam Smith explained, selfishness leads markets to produce whatever people want. To get rich, you have to sell what the public wants to buy. Voluntary exchange will only take place if both parties perceive that they are better off. Positive externalities result in beneficial outcomes for others, whereas negative externalities impose costs on others. The Coase Theorem is most easily explained via an example
This paper addresses a classic example of a negative externality (pollution), and describes three possible solutions for the problem: government regulation, taxation and property right – a better solution to overcome the externality as described by economist Ronald Coase.
Imagine being a corn farmer and growing corn. What are the private costs that you face that help you determine production? Things like fuel, seed, fertilizer; these are your private costs. But it turns out that every spring and summer when you lay down the fertilizer some of this flows into the stream nearby and flows into a lake downstream, oftentimes resulting in large fish kills. All those downstream, the fisherman, the recreationist, and the landowners all incur a negative externality.
There are three ways in which we can address these externalities:
1- Government Regulation:
a) First, direct regulation is applied through technology-specific methods. This is where the government requires producers to use a...

...BINOMIAL THEOREM :
AKSHAY MISHRA
XI A , K V 2 , GWALIOR
In elementary algebra, the binomial theorem describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the power (x + y)n into a sum involving terms of the form axbyc, where the coefficient of each term is a positive integer, and the sum of the exponents of x and y in each term is n. For example: The coefficients appearing in the binomial expansion are known as binomial coefficients. They are the same as the entries of Pascal's triangle, and can be determined by a simple formula involving factorials. These numbers also arise in combinatorics, where the coefficient of xn−kyk is equal to the number of different combinations of k elements that can be chosen from an n-element set.
HISTORY :
HISTORY This formula and the triangular arrangement of the binomial coefficients are often attributed to Blaise Pascal, who described them in the 17th century, but they were known to many mathematicians who preceded him. The 4th century B.C. Greek mathematician Euclid mentioned the special case of the binomial theorem for exponent 2 as did the 3rd century B.C. Indian mathematician Pingala to higher orders. A more general binomial theorem and the so-called "Pascal's triangle" were known in the 10th-century A.D. to Indian mathematician Halayudha and Persian mathematician Al-Karaji, and in the 13th century...

...The Coase Theorem
In “The Problem of Social Cost,” Ronald Coase introduced a different way of thinking about externalities, private property rights and government intervention. The student will briefly discuss how the Coase Theorem, as it would later become known, provides an alternative to government regulation and provision of services and the importance of private property in his theorem.
In his book The Economics of Welfare, Arthur C. Pigou, a British economist, asserted that the existence of externalities, which are benefits conferred or costs imposed on others that are not taken into account by the person taking the action (innocent bystander?), is sufficient justification for government intervention. He advocated subsidies for activities that created positive externalities and, when negative externalities existed, he advocated a tax on such activities to discourage them. (The Concise, n.d.). He asserted that when negative externalities are present, which indicated a divergence between private cost and social cost, the government had a role to tax and/or regulate activities that caused the externality to align the private cost with the social cost (Djerdingen, 2003, p. 2). He advocated that government regulation can enhance efficiency because it can correct imperfections, called “market failures” (McTeer, n.d.).
In contrast, Ronald Coase challenged the idea that the government had a role in taking action targeted...

...successfully employed in mechanism like the carburetor and the atomizer.
The study focuses on Bernoulli’s Theorem in Fluid Application. A fluid is any substance which when acted upon by a shear force, however small, cause a continuous or unlimited deformation, but at a rate proportional to the applied force. As a matter of fact, if a fluid is moving horizontally along a streamline, the increase in speed can be explained due the fluid that moves from a region of high pressure to a lower pressure region and so with the inverse condition with the decrease in speed.
Bernoulli’s Principle complies with the principle of conservation of energy. In a steady Flow, at all points of the streamline of a flowing fluid is the sum of all forms of mechanical energy along a streamline. It was first derived by the Swiss Mathematician Daniel Bernoulli; the theorem states that when a fluid flows from one place to another without friction, its total energy (kinetic+ potential+ pressure) remains constant.
Many of schools, academies or universities cannot provide their student an equipment which can help them in understanding fluid dynamics. They don’t have a “hands on” environment which can develop their knowledge and theoretical concepts.
Our Bernoulli’s Apparatus which is an instructional material purposes will provide for those interested viewer and learners a demonstration of related Bernoulli’s Theorem takes into effect.
Our research...