JRF IN MATHEMATICS 2011
... (22) If a and b are integers such that 9 divides a2 + ab + b2 then show that 3 divides both a and b. (23) Let c be a 3n digit number whose digits are all equal. Show that 3n divides c. (24) Prove that x4 − 10x2 + 1 is reducible modulo p for every prime p. (25) Does there exist an integer x satisfyin ...
... (22) If a and b are integers such that 9 divides a2 + ab + b2 then show that 3 divides both a and b. (23) Let c be a 3n digit number whose digits are all equal. Show that 3n divides c. (24) Prove that x4 − 10x2 + 1 is reducible modulo p for every prime p. (25) Does there exist an integer x satisfyin ...
Day 1 - Intro to the Number System Vocabulary Review
... 14. The numbers 2/3 and 2 are called “exact values”, while 0.66666 and 1.41459 are called “approximate values” for these same numbers. Explain the difference between these two terms. 15. What is the additive identity? ______ What is the multiplicative identity?_______ 16. What is the additive invers ...
... 14. The numbers 2/3 and 2 are called “exact values”, while 0.66666 and 1.41459 are called “approximate values” for these same numbers. Explain the difference between these two terms. 15. What is the additive identity? ______ What is the multiplicative identity?_______ 16. What is the additive invers ...
LP.092314
... The sum and difference of any two integers is still an integer. The product of any two integers is still an integer. The quotient of any two may not be an integer. The set of integers is not closed under the operation division. ...
... The sum and difference of any two integers is still an integer. The product of any two integers is still an integer. The quotient of any two may not be an integer. The set of integers is not closed under the operation division. ...
NON-CONVERGING CONTINUED FRACTIONS RELATED TO THE
... transcendence of both F and G is derived. Then, combining Christol’s theorem and a classical theorem of Cobham (see [2, Theorem 11.2.1]), we obtain that Fp and Gp are transcendental over Fp (X) for every prime number p 6= 2. Note that the idea to combine together Christol’s and Cobham’s theorems in ...
... transcendence of both F and G is derived. Then, combining Christol’s theorem and a classical theorem of Cobham (see [2, Theorem 11.2.1]), we obtain that Fp and Gp are transcendental over Fp (X) for every prime number p 6= 2. Note that the idea to combine together Christol’s and Cobham’s theorems in ...
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.