Number Theory The Greatest Common Divisor (GCD) R. Inkulu http

... but the converse is not necessarily true. ...

... but the converse is not necessarily true. ...

Commutative Algebra

... that can be given by a polynomial. We therefore call A(X) the ring of polynomial functions or coordinate ring of X. Often we will simply say that A(X) is the ring of functions on X since functions in algebra are always given by polynomials. Moreover, the class of a polynomial f ∈ K[x1 , . . . , xn ] ...

... that can be given by a polynomial. We therefore call A(X) the ring of polynomial functions or coordinate ring of X. Often we will simply say that A(X) is the ring of functions on X since functions in algebra are always given by polynomials. Moreover, the class of a polynomial f ∈ K[x1 , . . . , xn ] ...

Solutions

... of 3 are additive inverses mod 32, all 16 odd congruence classes mod 32 can be written as either one of the above numbers or its negative. (b) Determine the order of 9 modulo 64. By the above list of powers, 9 has order 4 modulo 32. So lifting to 64, 9 has order divisible by 4. In fact, 94 ≡ 33 (mod ...

... of 3 are additive inverses mod 32, all 16 odd congruence classes mod 32 can be written as either one of the above numbers or its negative. (b) Determine the order of 9 modulo 64. By the above list of powers, 9 has order 4 modulo 32. So lifting to 64, 9 has order divisible by 4. In fact, 94 ≡ 33 (mod ...

Advanced Problems and Solutions

... Limit of series with Fibonacci powers, 2.3(1964)204; Corrected, 3.2(1965)123 [Reference to special case], 3.2(1965)123 Generalized Fibonacci numbers, 2.3(1964)205 A Favorable Response, 3.2(1965)123 Quadruple sums of Fibonacci squares, 2.3(1964)205 Iterated Sums of Squares, 3.2(1965)127 Determinant w ...

... Limit of series with Fibonacci powers, 2.3(1964)204; Corrected, 3.2(1965)123 [Reference to special case], 3.2(1965)123 Generalized Fibonacci numbers, 2.3(1964)205 A Favorable Response, 3.2(1965)123 Quadruple sums of Fibonacci squares, 2.3(1964)205 Iterated Sums of Squares, 3.2(1965)127 Determinant w ...

# 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.