Prime Modulus Research Articles

Binary Logic for Residue Arithmetic Using Magnitude Index

We consider a residue number system using n pairwise relatively prime moduli m 1 ,⋯,m n to represent any integer X in the range M/ 2≤X>M/2, when M = ∏mi. The moduli m i are chosen to be of the 2-1 type, in order that the residue arithmetic can be implemented by means of binary registers and binary logic. Further, for each residue number X, a magnitude index P x is maintained for all arithmetic operations. We investigate the properties of such a system and derive the addition, subtraction, multiplication, sign determination, and overflow detection algorithms. The proposed organization is found to improve the operation times for sign detection and overflow detection operations, while rendering multiplication to be a difficult operation.

The size of L(1, χ) for real nonprincipal residue characters χ with prime modulus

In this paper it is shown that, as q runs through the odd primes in an arithmetic progression, the sum ∑ n−1 ∞ n q 1 q has considerable variation in size. The proof uses the uniform prime-number theorem and a recent version of the large sieve.

Elementary method in the theory of congruences for a prime modulus

Elementary method in the theory of congruences for a prime modulus

ON THE NUMBER OF POINTS OF A HYPERELLIPTIC CURVE OVER A FINITE PRIME FIELD

A new method is proposed in this paper for investigating algebraic congruences with prime modulus, leading in the case of hyperelliptic curves to estimates of the same order of strength as the classical estimates of Hasse and Weil.

A comparison of the correlational behavior of random number generators for the IBM 360

Hutchinson states that the “new” (prime modulo) multiplicative congruential pseudorandom generator, attributed to D.H. Lehmer, has passed the usual statistical tests for random number generators. It is here empirically shown that generators of this type can produce sequences whose autocorrelation functions up to lag 50 exhibit evidence of nonrandomness for many multiplicative constants. An alternative generator proposed by Tausworthe, which uses irreducible polynomials over the field of characteristic two, is shown to be free from this defect. The applicability of these two generators to the IBM 360 is then discussed. Since computer word size can affect a generator's statistical behavior, the older mixed and simple congruential generators, although extensively tested on computers having 36 or more bits per word, may not be optimum generators for the IBM 360.

The Difference between Consecutive Prime Numbers V

Let pn denote the nth prime and let ε be any positive number. In 1938 (3) Ishowed that, for an infinity of values of n,where, for k≧1, logk+1x = log (logk x) and log1x = log x. In a recent paper (4) Schönhage has shown that the constant ⅓ may be replaced by the larger number ½eγ, where γ is Euler's constant; this is achieved by means of a more efficient selection of the prime moduli used. Schönhage uses an estimate of mine for the number B1 of positive integers n≦u that consist entirely of prime factors p≦y, whereHerer x is large and α and δ are positive constants to be chosen suitably.

A Problem on Partitions With a Prime Modulus p ≧3

A Problem on Partitions With a Prime Modulus p ≧3

A problem on partitions with a prime modulus 𝑝≥3

A problem on partitions with a prime modulus 𝑝≥3

On the distribution with respect to a prime modulus of the products of primes with a given value of a character

On the distribution with respect to a prime modulus of the products of primes with a given value of a character

The solubility of certain Diophantine inequalities

The method devised by Hardy and Littlewood for the solution of Waring's Problem, and further developed by Vinogradov, applies quite generally to Diophantine equations of an additive type. One particular result that can be proved by this method is that if a 1 , …, a s are integers not all of the same sign, and if s is greater than a certain number depending only on k , the Diophantine equation has infinitely many solutions in positive integers x 1 , …, x s , provided that the corresponding congruence has a non-zero solution to every prime power modulus. A result of a similar kind holds if on the right of (1) there stands an arbitrary integer in place of 0.

A Partition Function with the Prime Modulus P > 3

1. Recently Lehner [2]1 developed a convergent series for pi(n) and p2(n), the number of partitions of a positive integer n into summands of the form 51 ? 1 and 51 + 2 respectively. The purpose of this paper is to develop a similar series for p1(n), p2 (n), * (,-1)/2(n), the number of partitions of a positive integer n into summands ofthe form pl ? 1, pl ? 2, pl ? (p 1)72, p being a prime greater than 3. Here, as in the previous paper, we follow the method of Rademacher [5]. We consider the generating functions

On the Factorization of Polynomials to a Prime Modulus

These numbers have been studied recently by D. H. Lehmer in another connection.2 2. Let W denote the residue class of all polynomials of degree N which are congruent to A(x) modulo p, and consider for each polynomial A'(x) of a the highest power of p which divides A(n) (A') = Res { xpn x, A'(x) }. For a given value of n, this power is either zero for every such polynomial, or else a positive integer, which may be thought of as arbitrarily large if the resultant happens to vanish. If the power is not zero there clearly exist polynomials of 21 for which it assumes a minimum value. We denote this minimum by pqm, so that we shall have for some polynomial A'(x) of degree N,

Tables of Irreducible Polynomials for the First Four Prime Moduli

There seem to be no complete lists giving irreducible polynomials for prime moduli. Dickson' gives an irreducible polynomial for each degree within certain limits for the moduli 2, 3, 5, 7, 11. Arnoux2 lists the decomposition of all polynomials of degree less than or equal to three for the modulus 5. More complete and more systematic lists do not seem to exist. The following tables were constructed to satisfy this need. The notation of detached coefficients is used throughout; that is, the polynomial

Rozdělení n-tých potenčních zbytků pro prvočíselný modul

Rozdělení n-tých potenčních zbytků pro prvočíselný modul

Anharmonic polynomial generalizations of the numbers of Bernoulli and Euler

We consider twelve infinite systems of polynomials in z which for z = 1 degenerate either to the numbers of Bernoulli or Euler, or to others simply dependent upon these. The first part proceeds from the definition of anharmonic polynomials to the specific twelve systems discussed; the second presents an adaptation of the symbolic calculus of Blissard and Lucas in sufficient detail for rapidly developing a simple isomorphism between the algebra of the polynomials and that of the twelve elliptic functions sn, cn, ns, nc, sc, ... of Glaisher, and the third contains a short selection from the simpler algebraic and congruential relations between the polynomials. Incidentally there is pointed out in the second part a new interpretation of Kronecker's work on certain symmetric functions and their connections with Bernoulli's nlumbers. Owing to the length of the paper the development stops short of the quadratic transformation of the polynomials which corresponds to the transformation of the second order in elliptic functions, but the material given is a necessary foundation for all higher transformations. For the same reason only prime moduli are considered in the congruences, although the case in which the modulus is a power of a prime can be treated in essentially the same way, but at greater length. All references are at the end of the paper.

Fundamental Systems of Formal Modular Seminvariants of the Binary Cubic

The present paper owes its origin to an attempt to solve the problem of Hurwitz in the case of the binary cubic form, viz., to find a set of formal modular invariants of the binary cubic form such that all others could be expressed as polynomials with integral coefficients with these as arguments. In this attempt I have not yet been successful, but methods have been developed which will aid in the solution of the problem and which have resulted in the solution of the similar, though less difficult problem of the determination of a fundamental system of formal modular seminvariants of the binary cubic form, modulis 5 and 7. These methods are so general in their nature that they could easily be used in the determination of a like system with respect to any prime modulus greater than three. The problem of a fundamental system of the seminvariants of the cubic here solved in the case of the moduli 5 and 7 has already been solved by Dickson in the case of the modulus 5 and the methods which he used in that case could no doubt have been used for its solution in other cases. Whatever interest may attach to the present paper does not then lie in the fact that a solution exists nor yet in the fundamental system exhibited, but in whatever power or generality the methods used in the solution may have, and in the fact, hitherto unknown, that in the case of the cubic the number of members of a fundamental system is a function of the modulus instead of being a constant as in the case of the quadratic. The method of annihilators, fundamental in the classical theory of invariantive concomitants, seems at first to be almost useless in the formal modular case. This is due to the fact that not only are these annihilators not linear but they are of an order which depends upon the modulus. Their complexity does indeed make the computation of seminvariants and invariants through their use very laborious in any except the simplest cases, but they furnish simple proofs of general theorems of far reaching significance. * Presented to the Society, October 30, 1920. 56

The numbers of sums of quadratic residues and of non-residues respectively taken n at a time and congruent to any given integer to an odd prime modulus p.

Article The numbers of sums of quadratic residues and of non-residues respectively taken n at a time and congruent to any given integer to an odd prime modulus p. was published on January 1, 1893 in the journal Journal für die reine und angewandte Mathematik (volume 1893, issue 112).