Topic
Divisor
About: Divisor is a research topic. Over the lifetime, 2462 publications have been published within this topic receiving 21394 citations. The topic is also known as: factor & submultiple.
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TL;DR: In this paper, the largest square-free divisor of a natural numbern is defined, and an asymptotic formula for γ(n) wherek is an arbitrary integer is given.
Abstract: Let γ(n) be the largest squarefree divisor of a natural numbern. In this note we give an asymptotic formula for\(\sum\limits_{n \leqslant x} {\gamma (n)/n^k } \) wherek is an arbitrary integer.
01 Jan 2008
TL;DR: This paper demonstrates the factorization of large integers using matrices and shows that the method can be generalized on RSA cryptosystem algorithm.
Abstract: Factorization of large integers gives a method to successfully attack on RSA cryptosystem algorithm. Williams 1 p + gives us such algorithm to factorize the integer n ; if there exists a prime divisor p , such that 1 p + will have only a small prime divisors. In this paper we demonstrate this algorithm using matrices and show that the method can be generalized.
TL;DR: In this paper, the complexity of satisfiability in positive integers of simultaneous divisibility of linear polynomials with nonnegative coefficients was studied, and it was shown that the problem is NP-hard.
Abstract: The paper studies the algorithmic complexity of subproblems for satisfiability in positive integers of simultaneous divisibility of linear polynomials with nonnegative coefficients. In the general case, it is not known whether this problem is in the class NP, but that it is in NEXPTIME is known. The NP-completeness of two series of restricted versions of this problem such that a divisor of a linear polynomial is a number in the first case, and a linear polynomial is a divisor of a number in the second case is proved in the paper. The parameters providing the NP-completeness of these problems have been established. Their membership in the class P has been proven for smaller values of these parameters. For the general problem SIMULTANEOUS DIVISIBILITY OF LINEAR POLYNOMIALS, NP-hardness has been proven for its particular case, when the coefficients of the polynomials are only from the set {1, 2} and constant terms are only from the set {1, 5}.
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TL;DR: For a general hyperelliptic curve of genus G, the authors obtained a complete description of the effective cone of a divisor on a general curve with non-equidimensional stable base locus.
Abstract: We obtain new information about divisors on the $d-$th symmetric power $C_{d}$ of a general curve $C$ of genus $g \geq 4.$ This includes a complete description of the effective cone of $C_{g-1}$ and a partial computation of the volume function on one of its non-nef subcones, as well as new bounds for the effective and movable cones of $C_{d}$ in the range $\frac{g+1}{2} \leq d \leq g-2.$ We also obtain, for each $g \geq 5,$ a divisor on $C_{g-1}$ with non-equidimensional stable base locus. For a general hyperelliptic curve $C$ of genus $g,$ we obtain a complete description of the effective cone of $C_{d}$ for $2 \leq d \leq g$ and an integral divisor on $C_{g-1}$ which has non-integral volume whenever $g$ is not a power of 2.
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TL;DR: In this article, the authors compute the divisor of the modular equation on the modular curve and then find recurrence relations satisfied by the modular traces of the Hauptmodul for any congruence subgroup of genus zero.
Abstract: We compute the divisor of the modular equation on the modular curve $\Gamma_0(N) \backslash \mathbb H^*$ and then find recurrence relations satisfied by the modular traces of the Hauptmodul for any congruence subgroup $\Gamma_0(N)$ of genus zero. We also introduce the notions and properties of $\Gamma$-equivalence and $\Gamma$-reduced forms about binary quadratic forms. Using these, we can explicitly compute the recurrence relations for $N = 2, 3, 4, 5$.