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.

Consecutive integers with equally many principal divisors

Abstract: Classifying the positive integers as primes, composites, and the unit, is so familiar that it seems inevitable. However, other classifications can bring interesting relationships to our attention. In that spirit, let us classify positive integers by the number o? principal divisors they possess, where we define a principal divisor of a positive integer n to be any prime-power divisor pa \ n which is maximal (so p is prime, a is a positive integer, and pa+l is not a divisor of n). The standard notation pa \ can be read as "/?fl is a principal divisor of n." The Fundamental Theorem of Arithmetic is usually stated in a form emphasizing how primes enter the structure of the positive integers, such as: Every positive integer is the product of a unique finite multiset of primes. (Recall that a multiset is a collection of elements in which multiple occurrences are permitted.) Alternatively, the Fundamental Theorem of Arithmetic can be stated in a form that focuses on how maximal prime powers enter the structure of the positive integers, such as: Every positive integer is the product of a unique finite set of powers of distinct primes. Consequently every positive integer is the product of its principal divisors, and every finite set of powers of distinct primes is the set of principal divisors of a unique positive integer. Of course, the number of principal divisors of n is equal to the number of distinct prime factors of n, but here the principal divisors are the simple structural components of n, whereas the distinct prime factors are but a shadow of that structure. Readers who find the present paper of interest might find similar interest in [6], where upper bounds on the sum of principal divisors of n are established by elementary means. For each integer n > 0, let Pn be the set of all positive integers with exactly n principal divisors, so Pq = {1}, and

Odd values of the Ramanujan tau function

Abstract: We prove a number of results regarding odd values of the Ramanujan $\tau$-function. For example, we prove the existence of an effectively computable positive constant $\kappa$ such that if $\tau(n)$ is odd and $n \ge 25$ then either \[ P(\tau(n)) \; > \; \kappa \cdot \frac{\log\log\log{n}}{\log\log\log\log{n}} \] or there exists a prime $p \mid n$ with $\tau(p)=0$. Here $P(m)$ denotes the largest prime factor of $m$. We also solve the equation $\tau(n)=\pm 3^{b_1} 5^{b_2} 7^{b_3} 11^{b_4}$ and the equations $\tau(n)=\pm q^b$ where $3\le q < 100$ is prime and the exponents are arbitrary nonnegative integers. We make use of a variety of methods, including the Primitive Divisor Theorem of Bilu, Hanrot and Voutier, bounds for solutions to Thue--Mahler equations due to Bugeaud and Győry, and the modular approach via Galois representations of Frey-Hellegouarch elliptic curves.

Factorization of matrices over elementary divisor domain

Abstract: We propose constructive criteria of divisibility and associativity of matrices over commutative elementary divisor ring without zero divisors. On this base, the explicit form for all non-associated divisors which have prescribed canonical diagonal forms (c.d.f.) is indicated. A relation between c.d.f. for matrix and c.d.f. for its divisors is established. The uniqueness theorem is proved.

Method for achieving fixed point and floating point mixed division in general-purpose digital signal processor (GPDSP)

Abstract: The invention discloses a method for achieving fixed point and floating point mixed division in a general-purpose digital signal processor (GPDSP). The method comprises a first step of inputting a divisor and a dividend, and if the divisor and the dividend are fixed point integers, shifting to execute a second step; if the divisor and the dividend are floating point data, shifting to execute a third step; the second step of enabling the divisor and the dividend to perform shifting according to a precursor zero number, calculating iteration times for executing division iterations, and executing iterations of a one-stage or multi-stage SRT algorithm according to fixed point data types and iteration times; performing shifting on a quotient result and obtaining a final quotient result, and obtaining a final remainder according to the final quotient result; a third step of working out mantissas of the divisor and the dividend, adopting the SRT algorithm to execute division iterative computation of the mantissas, and enabling the iterative computation to undergo one-stage execution or multi-stage execution by truncation; and normalizing the mantissas of the quotient result according to floating point data types and the number of stages executed in iterative computation. The achieving method has the advantages of being complete in division function, simple, short in execution cycle, small in time delay and high in division execution efficiency.

Reduction of divisors for classical superintegrable $GL(3)$ magnetic chain

Abstract: Separated variables for a classical GL(3) magnetic chain are coordinates of a generic positive divisor D of degree n on a genus g non-hyperelliptic algebraic curve. Because n > g, this divisor D has unique representative ρ(D) in the Jacobian, which can be constructed by using dim|D| = n − g steps of Abel’s algorithm. We study the properties of the corresponding chain of divisors and prove that the classical GL(3) magnetic chain is a superintegrable system with dim|D| = 2 superintegrable Hamiltonians.