Morgan Ward

The arithmetical theory of linear recurring series

which we call the reduced sequence corresponding to (u) modulo m. It is easily seen that after a finite number of terms, the sequence (a) repeats itself periodically, and that any one of its periods is a multiple of a certain least period which is called the characteristic number of (u) (or (a)) modulo m%. The number of non-repeating terms in (a) is called the numeric of (u) modulo m; if it is zero, (u) is said to be purely periodic^ modulo m. If all the terms of (u) after a certain point are divisible by m, so that the repeating part of (a) consists of the single residue zero, (u) is said to be a null sequence modulo m. Three important problems immediately suggest themselves: first, to determine the characteristic number and numeric of the sequence (m) as

THE INTRINSIC DIVISORS OF LEHMER NUMBERS

41, k = 6. The question of the existence of intrinsic divisors when a and fi are real but not necessarily integers was studied some time ago in these Annals by R. D. Carmichael [3], and again quite recently by C. G. Lekkenkerker [6]. In this paper, I study the intrinsic divisors of D. H. Lehmers generalization of the Lucas numbers [5] in which merely (a + p3)2 and afi are required to be integers, again under the assumption that a and fi are real. The method of attack goes back in principle to Sylvester [7], page 607, and is powerful enough to furnish a complete answer. Nothing appears to be known about the intrinsic divisors of Lucas or Lehmer numbers when a and d are complex. Let L and Ml be integers, with L and K = L - 4M positive and M 5 0. Then the roots a and ,3 of the polynomial

The characteristic number of a sequence of integers satisfying a linear recursion relation

we shall call (A)n the reduced sequence corresponding to (W)n modulo m. If after s terms in the reduced sequence, a cycle of t terms keeps repeating itself indefinitely, (W)n will be said to admit the period t, modulo m. The least period that (W)n admits (modulo m) is called its characteristic number.t In this paper, I give a number of new results on the form of the characteristic number of a sequence. The principal result is the following:

Postulates for the inverse operations in a group

It may happen that the function F is of such a character that when the 5-symbols z, x are given in (1), an c5-symbol y is uniquely determined, and when the25-symbols y, z are given in (1), an25-symbol x is uniquely determined. In this case we may associate with the function F(x, y) two other one-valued functions y = G (z, x), x = H (y, z) defined over the collection (B. These functions are called the first and second inverses of the function F. The primary object of this paper is to state the restrictions which must be imposed upon F in order that one of its inverses may define with (E an abstract group. It is convenient, in developing the properties of the system { ( ;o} consisting of the i-symbols, F, and one or more postulates, to replace (1) by the notationt z = xoy.

The distribution of residues in a sequence satisfying a linear recursion relation

the reduced sequence corresponding to (W)n, modulo m. It is easily shown the (A)n is periodic; following Carmichael,: we shall call its smallest period, r, the characteristic number of (W)n modulo m. The object of this memoir is to attack the following fundamental distribution problem:? Given the numerical values of the integers P, Q, R, WO, W1, W2, m and r, to determine the distribution of the residues 0, 1, 2, m, m -1 among any r terms of the reduced sequence (A)n. There are really two distinct problems involved here: the determination of the particular place a given residue occurs in (A)n and the determination of the number of times a given residue occurs in any r terms of (A)n. Both

An enumerative problem in the arithmetic of linear recurring series

is any sequence of rational integers satisfying (1.1), then after a certain point the sequence becomes periodic when considered modulo m. Its least period is called the characteristic number of the sequence (U) modulo m. In a recent paper in these Transactionst, I have considered the problem of determining this characteristic number given m, C, C2, , * * Ck and the k initial values UO, U1, *, Uk-i of the sequence (U), and I have reduced it to certain basic problems in the theory of higher congruences.

Memoir on Elliptic Divisibility Sequences

In the present paper, I am concerned with the following problem which I shall similarly reduce to a problem in the theory of higher congruences: Given any positive integer s: to find the number of distinct sequences (U) modulo m whose characteristic number is exactly equal to s. 2. I obtain here the following results.