Fibonacci integers - Dartmouth College
... difficult because of allowing denominators. That is, if we just looked at the semigroup generated by the Fibonacci numbers, rather than integers in the group that they generate, life would be simpler. In fact, because of Carmichael’s primitive prime factors, if we throw out Fn for n = 1, 2, 6, 12, t ...
... difficult because of allowing denominators. That is, if we just looked at the semigroup generated by the Fibonacci numbers, rather than integers in the group that they generate, life would be simpler. In fact, because of Carmichael’s primitive prime factors, if we throw out Fn for n = 1, 2, 6, 12, t ...
Lucas-Lehmer criterion for primality of Mersenne numbers
... For example, β = 3 + ρ will do. Note that the norm (3 + ρ)(3 − ρ) is 6. We claim that for a + bρ with a, b ∈ Z, (a + bρ)q = (a + bρ)σ = a − bρ (in K) To see this, note first that the image ρq of ρ under the Frobenius map γ → γ q must be another root of the equation x2 − 3 = 0, and is not equal to ρ ...
... For example, β = 3 + ρ will do. Note that the norm (3 + ρ)(3 − ρ) is 6. We claim that for a + bρ with a, b ∈ Z, (a + bρ)q = (a + bρ)σ = a − bρ (in K) To see this, note first that the image ρq of ρ under the Frobenius map γ → γ q must be another root of the equation x2 − 3 = 0, and is not equal to ρ ...
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.