Proper Divisors Research Articles

Short Sequel to “Problem Solving In Mathematics”

One charming aspect of mathematics lies in the truism that the solution to a problem usually poses other equally intriguing problems, an example of which was carefully developed by Truman Botts in this journal for October and November 1965. Another serendipitous dividend in our discipline is the satisfaction derived from relating seemingly disjoint topics. The burden of this note is to observe the relationship existing between perfect numbers—that is, integers which are the sum of their proper divisors—and the problem taken from Volume II of the Hungarian Problem Book (this problem is to find three different positive integers such that the sum of their reciprocals is an integer).

A Remark on Flat and Projective Modules

It is the purpose of this note to give some characterizations of flat and projective modules, partly in ideal theoretical terms, partly in terms of the exterior product of a module (“puissance extérieure“); cf. (1).We shall consider left modules over a ring R with identity element and without proper zero divisors. The left module M is called flat if X ⊗R M is an exact functor on the category of right R-modules X. If M is flat over a commutative domain R, M is necessarily torsion-free. Therefore when looking for flatness of a module M over a commutative domain, one may assume from the start that M is torsion-free.In the following theorem, we shall not restrict ourselves to commutative rings R, but the modules concerned have to be torsion-free, which, of course, should mean that rm = 0 implies r = 0 or m = 0.

Synthesis of Binary Ring Counters of Given Periods

A number of papers have been written from time to time about logical counters of a certain type which have quite simple logic and have been variously referred to as Binary Ring Counters, Shift Register Counters, Johnson Counters, etc. To my knowledge, most of these papers confine themselves to certain special cases and usually leave the subject with some speculation as to the possibility of generating periods of any desired length by the use of these special types. The point of view of this paper is to consider all possible counters of this general type to see how one would obtain a particular period. Special emphasis is placed on determining the least number of bits, n , required to produce a given period, K . The rules for counting are as follows. If an n -bit counter is in state ( a n -1 , a n -2 ···, a 2 , a 1 , a 0 ) at a given time, T , then at T + 1 its state is ( b n -1 , b n -2 , ···, b 1 , b 0 ) where b 0 = a n -1 , b i = a i -1 + c i a n -1 for i = 1, 2, ···, n - 1. The a 's, b 's, and c 's are all 0's or 1's, the c 's being constants, and the indicated operations are carried out using modulo 2 arithmetic. This is equivalent to considering the state of the counter as an ( n - 1)th degree polynomial in X , multiplying said polynomial by X and reducing it modulo m ( X ), where m ( X ) is a polynomial of degree n which is relatively prime to X . At time T the state of the counter corresponds to: A ( X ) = a n -1 X n -1 + a n -2 X n -2 + ··· + a 1 X + a 0 . The polynomial which corresponds to the state of the counter at time T + 1 is obtained by forming X · A ( X ) and reducing, if necessary, modulo m ( X ) = X n + c n -1 X n -1 + c n -2 X n -2 + ··· + c 1 X + 1. Since a n -1 · m ( X ) = 0 mod m ( X ), X · A ( X ) = X · A ( X )+ a n -1 m ( X ) mod m ( X ), so X · A ( X ) = ( a n -2 + c n -1 · a n -1 ) X n -1 + ( a n -3 + c n -2 a n -1 ) X n -2 + ··· + ( a 0 + c 1 a n -1 ) X + a n -1 = b n -1 X n -1 + b n -2 X n -2 + ··· + b 1 X + b 0 . It is well known that more than one possible period may be obtained depending upon the initial state of the counter. Several examples are given by Young [4]. However, starting with X itself will always yield the longest possible period for any given m ( X ) and, furthermore, any other periods possible will always be divisors of the major period (Theorem I below). Since these minor periods can always be obtained with moduli of lower degree they are of no real interest here, and throughout the remainder of this paper the expression “period of the counter” will be assumed to refer to the major period. The set of all polynomials whose coefficients are the integers modulo 2 is the polynomial domain GF (2, X ), which has among other things unique factorization into primes (irreducibles). If m ( X ) is in GF (2, X ), then GF (2, X ) modulo m ( X ) is a commutative ring. Thus it is closed under multiplication, but it may have proper divisors of zero. However, any element which is relatively prime to m ( X ) in GF (2, X ) has an inverse in GF (2, X )/ m ( X ) [1].

A Generalized Ring of Quotients I

According to a well-known theorem of algebra [3, p.47], an integral domain can be embedded in a field, called its field of quotients. Every freshman is familiar with the simplest form of this theorem concerning the integers and the rational numbers. Many generalisations have been given in which a "ring of quotients" is constructed for a given ring [e.g. 6, 7, 11, 12, 13]. Of course, we cannot expect a ring of quotients to be a field, or even a skew field. As Malcev [9] has shown, there even exist rings without proper divisors of zero which cannot be embedded isomorphically in a skew field. The most recent construction, by Utumi [12], gives a ring of quotients for any ring with zero left annihilator. We show in this paper that this construction can be extended to arbitrary rings, in fact, to arbitrary modules. The method used is more abstract: a fundamental relation between rings, defined by Utumi in terms of their elements, is here replaced by a corresponding relation between modules, defined by means of homomorphisms.

A note on divisibility sequences

A sequence of rational integers (u): u₀, u₁, u₂, . . .u_n, . . . is called a divisibility sequence if u_r divides u_s whenever r divides s, and any integer M dividing terms of (u) with positive suffix is called a divisor of (u). The suffix s is called a rank of apparition of M if u_s ≡ 0 (mod M), but u_r /≡ (mod M) if r is a proper divisor of s. It follows from a previous note of mine in this Bulletin (Ward [l]) that a necessary and sufficient condition that every divisor of (u) shall have only one rank of apparition is that (u) have the following property.