Families of elliptic curves of high rank with nontrivial torsion group
... Moreover, Q3they pass through the points whose x-coordinates are the roots of F2 (x) i=1 (F2 (x) − g2 (xi )). If we apply this method in the case where x1 = 2, x2 = 4, and x3 = 6, then we obtain the points P1 , . . . , P6 of x-coordinates −7, −7/9, −19/9, Q3 −19/25, −39/4, −39/49 (6 of the 9 roots o ...
... Moreover, Q3they pass through the points whose x-coordinates are the roots of F2 (x) i=1 (F2 (x) − g2 (xi )). If we apply this method in the case where x1 = 2, x2 = 4, and x3 = 6, then we obtain the points P1 , . . . , P6 of x-coordinates −7, −7/9, −19/9, Q3 −19/25, −39/4, −39/49 (6 of the 9 roots o ...
32(2)
... m = 4; Note that x4 = aif^, x = d^, and x3 = i^. Therefore, applying concatenation to the alignments cdz)d;c and P4 ZDP2;P3 implies that x4 IDX;CX3. Consequently, by Lemma 1,(1) cannot hold for m = 4, since x2 begins with a rf. Similar reasoning shows that (1) is false for m = 9,12,... . To generali ...
... m = 4; Note that x4 = aif^, x = d^, and x3 = i^. Therefore, applying concatenation to the alignments cdz)d;c and P4 ZDP2;P3 implies that x4 IDX;CX3. Consequently, by Lemma 1,(1) cannot hold for m = 4, since x2 begins with a rf. Similar reasoning shows that (1) is false for m = 9,12,... . To generali ...
Algorithmic Number Theory
... the number n = i=1 pi + 1. It is easy to see that none of the primes p1 , . . . , pk is a divisor of n and n is larger than any of them. Hence n must be a prime, contradicting the assummption. ...
... the number n = i=1 pi + 1. It is easy to see that none of the primes p1 , . . . , pk is a divisor of n and n is larger than any of them. Hence n must be a prime, contradicting the assummption. ...
34(5)
... Krattenthaler [5] showed that the preceding formulas are a consequence of his bivariate version of Lagrange inversion; furthermore, he has generalized (1.3) and (1.4). One must note that we do not need to use Lagrange inversion in two variables to prove these types of identities, as Krattenthaler di ...
... Krattenthaler [5] showed that the preceding formulas are a consequence of his bivariate version of Lagrange inversion; furthermore, he has generalized (1.3) and (1.4). One must note that we do not need to use Lagrange inversion in two variables to prove these types of identities, as Krattenthaler di ...
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.