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... The Four Square Theorem was first conjectured in Bachet’s 1621 edition of Diophantus and the first proof was given by Lagrange in 1770. Fermat claimed to have come up with a proof, but did not publish it. In fact, there is a similar theorem called Fermat’s Two Square Theorem, that is a corollary of ...
... The Four Square Theorem was first conjectured in Bachet’s 1621 edition of Diophantus and the first proof was given by Lagrange in 1770. Fermat claimed to have come up with a proof, but did not publish it. In fact, there is a similar theorem called Fermat’s Two Square Theorem, that is a corollary of ...
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... of digits, there exists a member of the given sequence beginning with this string of digits 9 and define the sequence to be right-normal if there exists a member of the sequence ending with this string of digits. Show that the sequences whose n ...
... of digits, there exists a member of the given sequence beginning with this string of digits 9 and define the sequence to be right-normal if there exists a member of the sequence ending with this string of digits. Show that the sequences whose n ...
The equivariant spectral sequence and cohomology with local coefficients Alexander I. Suciu
... circle, with ν = p∗ : π Z the homomorphism induced by the projection map, p : X → S 1 . The homology of the resulting infinite cyclic cover was studied by J. Milnor in [7]. This led to another spectral sequence, introduced by M. Farber, and further developed by S.P. Novikov, see [6]. The Farber-No ...
... circle, with ν = p∗ : π Z the homomorphism induced by the projection map, p : X → S 1 . The homology of the resulting infinite cyclic cover was studied by J. Milnor in [7]. This led to another spectral sequence, introduced by M. Farber, and further developed by S.P. Novikov, see [6]. The Farber-No ...
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... In several recent papers L. Bernstein [1], [2] introduced a method of operating with units in cubic algebraic number fields to obtain combinatorial identities. In this paper we construct kth degree (k J> 2) algebraic fields with the special property that certain units have Fibonacci numbers for coef ...
... In several recent papers L. Bernstein [1], [2] introduced a method of operating with units in cubic algebraic number fields to obtain combinatorial identities. In this paper we construct kth degree (k J> 2) algebraic fields with the special property that certain units have Fibonacci numbers for coef ...
Two-Year Algebra 2 B Semester Exam Review 2015–2016
... If Brianna bakes 10 pound cakes in one day, what was her average cost per pound ...
... If Brianna bakes 10 pound cakes in one day, what was her average cost per pound ...
Lesson 2
... We have already talked about plotting integers on the number line. It gives a visual representation of which number is bigger, smaller, etc. It would therefore be helpful to plot non-integer rational numbers (fractions) on the number line also. There are 2 ways to graph rational numbers on the numbe ...
... We have already talked about plotting integers on the number line. It gives a visual representation of which number is bigger, smaller, etc. It would therefore be helpful to plot non-integer rational numbers (fractions) on the number line also. There are 2 ways to graph rational numbers on the numbe ...
Factorization
In mathematics, factorization (also factorisation in some forms of British English) or factoring is the decomposition of an object (for example, a number, a polynomial, or a matrix) into a product of other objects, or factors, which when multiplied together give the original. For example, the number 15 factors into primes as 3 × 5, and the polynomial x2 − 4 factors as (x − 2)(x + 2). In all cases, a product of simpler objects is obtained.The aim of factoring is usually to reduce something to “basic building blocks”, such as numbers to prime numbers, or polynomials to irreducible polynomials. Factoring integers is covered by the fundamental theorem of arithmetic and factoring polynomials by the fundamental theorem of algebra. Viète's formulas relate the coefficients of a polynomial to its roots.The opposite of polynomial factorization is expansion, the multiplying together of polynomial factors to an “expanded” polynomial, written as just a sum of terms.Integer factorization for large integers appears to be a difficult problem. There is no known method to carry it out quickly. Its complexity is the basis of the assumed security of some public key cryptography algorithms, such as RSA.A matrix can also be factorized into a product of matrices of special types, for an application in which that form is convenient. One major example of this uses an orthogonal or unitary matrix, and a triangular matrix. There are different types: QR decomposition, LQ, QL, RQ, RZ.Another example is the factorization of a function as the composition of other functions having certain properties; for example, every function can be viewed as the composition of a surjective function with an injective function. This situation is generalized by factorization systems.