D. H. Lehmer

Mathematical Tables and Other Aids to Computation.

where y(x) denotes the solution of the differential equation. The idea is to use a quadrature formula to estimate the integral of (1). This requires knowledge of the integrand at specified arguments x¿ in (xo, -To + h)—hence we require the values of y(x) at these arguments. A numerical integration method may be used to estimate y(x) for the required arguments. In this way a numerical integration method is combined with a quadrature formula to obtain another numerical integration method. A large number of methods may be devised, depending on which combination of quadrature formula and integration method is used. In particular, the Gauss two-point quadrature formula combined with the Runge-Kutta fourth order method appears to give excellent results [1]. We propose here the combination of the Radau three-point quadrature formula with the Runge-Kutta fourth order method. The resulting method seems to give greater accuracy with the same amount of work. 2. The Method. The Radau quadrature formula [2] gives

A Ternary Analogue of Abelian Groups

Introduction. A class K with an operation called multiplication applied to pairs of elements of K is an abstract group provided certain postulates are satisfied. Unfortunately the name group does not suggest the binary character of multiplication. The entities with which this note is concerned are similar to groups, the class K being subjected to a ternary operation however. For want of a more descriptive name we have called them triplexes. The need for their consideration arose in an attempt to obtain solutions of a pair of functional equations, but their investigation would seem justified by intrinsic interest especially when compared with Abelian groups. The system of postulates on which we base our investigation is modelled after Hurwitzs t system for Abelian groups, and accordingly differs somewhat from the definitions of a group found in most treatises on group theory. In this way we reduce the proofs of several theorems to a minimum and have at the same time a more perfect system from a strictly logical. point of view. The role that Abelian groups play in this theory is described in ? 3. The rest of the paper deals with finite triplexes and concepts analogous to the fundamental notions of the theory of Abelian groups such as order and inverse of an element, sub-group, cyclic group, quotient group, etc. Two notions however are conspicuous by their absence, the unit and the basis. Other facts such as the existence of triplexes with no subtriplex stand out as being different from what one might expect from the theory of Abelian groups.

A Machine Method for Solving Polynomial Equations

The problem of finding numerical approximations to the roots of a polynomial has a long and interesting history. The various methods have always been proposed in terms of the state of the art of computataon then current. Since the advent of high-speed computing systems there has been, the writer feels, an unusual lag m the development of new techniques for deahng with this timehonored problem, techmques bet ter stated to the capabihties and inadequacies of automatic computers. A recent survey of available methods [1] indicates tha t no one method is desirable for automatic computers. I t is true that methods such as those of Newton or Bernoulli for real roots and Graeff~ or Bairstow for complex roots have been programmed for automatm computers. Each of these classic methods requires a good deal of judgment in connection with the isolation or separation of roots. These judicial decasions are relatively easy for a human being to make when operating a desk calculator but are more difficult to anticipate and furnish to the machines program. On the other hand, the machine is prepared to undertake thousands of times more arithmetical activity than was ever contemplated by the inventors of the classmal methods. Hence it is tame to consider methods of more uniform applmcabihty, ~dth possibly slower convergence rates, that may be far too laborious to carry out by hand but which nevertheless are sufficiently easy for an automatic computer. Such a method should be applicable to polynomials with complex coefficients whose roots are therefore any arbitrary finite set of points an the complex plane, distinct or not. I t thus becomes a problem of searching the complex plane for roots. One such method has already been tried by J. A. Ward [1]. I t seeks to minimize

On Euler's totient function

where k is an integer, and <t>{n) is Eulers totient function, giving the number of integers <n and prime to n. Our main purpose is to show that if n is a solution of (1), then n is a prime or the product of seven or more distinct primes. One is tempted to believe the stronger statement that (1) has no composite solutions or, in other words, the integer n is a prime if (and only if) cj>(n) divides n — 1. We have not been able to establish this, however. The proof of the nonexistence of composite solutions of (1) seems about as remote as the proof of the nonexistence of odd perfect numbers and the two problems though not equivalent are not dissimilar. Let w b e a composite solution of (1) and let a be any number prime to n\ then

A new approach to Bernoulli polynomials

AbstractBeginning with Jacob Bernoullis discovery before 1705 of the polynomials that bear his name, there have been five approaches to the theory of Bernoulli polynomials. These can be associated...

Integer sequences having prescribed quadratic character

For the odd primes p, = 3, p2 = 5, ■ • -, we determine integer sequences Np such that the Legendre symbol (N/p,) = +1 for all pi S P for a prescribed array of signs ± 1; (i.e., for a prescribed quadratic character). We examine six quadratic characters having special interest and applications. We present tables of these Np and examine some applications, particularly to questions concerning extreme values for the smallest primitive root (of a prime N), the class number of the quadratic field R(J — N), the real Dirichlet L functions, and quadratic character sums.

On Fermat’s quotient, base two

This paper extends the search for solutions of the congruence 2P1 =_ 0 (modp2) to the limit p < 6 * 109. No solution, except the well-known p = 1093 and p = 3511, was found. In 1969 Brillhart, Tonascia, and Weinberger [1] reported on the search for solutions of the congruence a 1 (modp2), a = 2(1)199. For a = 2, a special effort was made to consider all p < 3 109. Only the two known solutions p = 1093 and p = 3511 were found. Having the occasional opportunity to use the Illiac IV, one of the projects decided upon was the recalculation and extension of the above result for a = 2 by a somewhat different method. The calculation was pushed to twice the above limit, that is top < 6 * 109, without finding any further solutions. The parallel construction of the Illiac IV makes it possible to look for 64 different values of p at the same time. The speed of Illiac IV is such that a range of 100000 numbers can be searched in one second. Thus, the range for p < 6 * 109 was broken up into 60 runs of 1000 seconds each. The program was run as one of a few backlog problems over the past two years. Let n belong to one of the 64 residue classes that are prime to 240 and let 2m _ Am + nBm (mod n2), where 0 < Am < n, 0 < Bm < n. For numbers n as large as 109, the number n2 is a doubly precise integer. Nevertheless, the calculation of Am and Bm can be accomplished by single precision arithmetic in only O(log m) operations. In fact, one uses one or the other of the following two recurrences: If m = 2h + 1, then Am -2A2h and Bm 2B2h (mod n). If m = 2h, and if A 2 = Rm + nQm (0 < Rm < n) and if 2AhBh Dm (mod n), then Am= Rm and Bm = Qm + Dm. Received March 30, 1980. 1980 Mathematics Subject Classification. Primary 10-04. ( 1981 American Mathematical Society 0025-5718/81 /0000-0027/

Extended computation of the Riemann zeta-function

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On the roots of the Riemann zeta-function

In 1955 a programme of study of the first 10000 zeros of the Riemann Zeta-function was completed. Use was made of the high-speed digital computer SWAC and a report of this programme has appeared recently [1]. More recently still, the programme has been extended to the first 25000 zeros. All these zeros have σ= ½ The purpose of this paper is to summarize the methods needed for this (and possibly future) work from the highspeed computer point of view.

On the series for the partition function

on the critical line a=l/s, t > 0 . These results confirm those made previously by Gram [1], Hutchinson [2], Titchmarsh [3] and Turing [4] and extend these to the first 10,000 zeros of