Showing papers on "Prime number published in 1993"
1 Jan 1993
TL;DR: In this article, the authors considered the problem of the distribution of prime numbers in short intervals and gave an explicit formula for Waring's problem and a new boundary for the Zeros of the Zeta Function.
Abstract: I. Integer Points.- 1. Statement of the Problem, Auxiliary Remarks, and the Simplest Results.- 2. The Connection Between Problems in the Theory of Integer Points and Trigonometric Sums.- 3. Theorems on Trigonometric Sums.- 4. Integer Points in a Circle and Under a Hyperbola.- Exercises.- II. Entire Functions of Finite Order.- 1. Infinite Products. Weierstrass's Formula.- 2. Entire Functions of Finite Order.- Exercises.- III. The Euler Gamma Function.- 1. Definition and Simplest Properties.- 2. Stirling's Formula.- 3. The Euler Beta Function and Dirichlet's Integral.- Exercises.- IV. The Riemann Zeta Function.- 1. Definition and Simplest Properties.- 2. Simplest Theorems on the Zeros.- 3. Approximation by a Finite Sum.- Exercises.- V. The Connection Between the Sum of the Coefficients of a Dirichlet Series and the Function Defined by this Series.- 1. A General Theorem.- 2. The Prime Number Theorem.- 3. Representation of the Chebyshev Functions as Sums Over the Zeros of the Zeta Function.- Exercises.- VI. The Method of I.M. Vinogradov in the Theory of the Zeta Function.- 1. Theorem on the Mean Value of the Modulus of a Trigonometric Sum.- 2. Estimate of a Zeta Sum.- 3. Estimate for the Zeta Function Close to the Line ? = 1.- 4. A Function-Theoretic Lemma.- 5. A New Boundary for the Zeros of the Zeta Function.- 6. A New Remainder Term in the Prime Number Theorem.- Exercises.- VII. The Density of the Zeros of the Zeta Function and the Problem of the Distribution of Prime Numbers in Short Intervals.- 1. The Simplest Density Theorem.- 2. Prime Numbers in Short Intervals.- Exercises.- VIII. Dirichlet L-Functions.- 1. Characters and their Properties.- 2. Definition of L-Functions and their Simplest Properties.- 3. The Functional Equation.- 4. Non-trivial Zeros Expansion of the Logarithmic Derivative as a Series in the Zeros.- 5. Simplest Theorems on the Zeros.- Exercises.- IX. Prime Numbers in Arithmetic Progressions.- 1. An Explicit Formula.- 2. Theorems on the Boundary of the Zeros.- 3. The Prime Number Theorem for Arithmetic Progressions.- Exercises.- X. The Goldbach Conjecture.- 1. Auxiliary Statements.- 2. The Circle Method for Goldbach's Problem.- 3. Linear Trigonometric Sums with Prime Numbers.- 4. An Effective Theorem.- Exercises.- XI. Waring's Problem.- 1. The Circle Method for Waring's Problem.- 2. An Estimate for Weyl Sums and the Asymptotic Formula for Waring's Problem.- 3. An Estimate for G(n).- Exercises.- Hints for the Solution of the Exercises.- Table of Prime Numbers < 4070 and their Smallest Primitive Roots.
276 citations
TL;DR: In this paper, the problem of finding discrete logarithm in a finite field of prime order was studied and the expected running time of the algorithm was shown to be O(1/3; (64/9)$ 1/3 + o(1) + o (1)
Abstract: Let K be a number field and $\scr{O}\_{K}$ its ring of integers. Let l be a prime number and e a positive integer. We give a method to construct l$^{e}$th powers in $\scr{O}\_{K}$ using smooth algebraic integers. This method makes use of approximations of the l-adic logarithm to identify l$^{e}$th powers. One version we give is successful if the class number of K is not divisible by l and if the units in $\scr{O}\_{K}$ which are congruent to 1 modulo l$^{e+1}$ are l$^{e}$th powers. A second version only depends on Leopoldt's conjecture. We use the technique of constructing l$^{e}$th powers to find discrete logarithms in a finite field of prime order. Our method for computing discrete logarithms is closely modelled after Gordon's adaptation of the number field sieve to this problem. We conjecture that the expected running time of our algorithm is L$\_{p}$[1/3; (64/9)$^{1/3}$ + o(1)] for p $\rightarrow \infty $, where L$_{p}$[s;c] = exp (c (log q)$^{s}$ (log log q)$^{1-s}$). This is the same running time as is conjectured for the number field sieve factoring algorithm.
179 citations
TL;DR: In this paper, the authors give a systematic treatment of Newton polygons of exponential sums and show that the Adolphson-Sperber conjecture is false in its full form, but true in a slightly weaker form.
Abstract: In this article we give a systematic treatment of Newton polygons of exponential sums. The Newton polygon is a nice way to describe p-adic values of the zeroes or poles of zeta functions and L functions. Our main objective is to show that the Adolphson-Sperber conjecture 12], which asserts that under a simple condition the generic Newton polygon of L functions coincides with its lower bound, is false in its full form, but true in a slightly weaker form. We also show that the full form is true in various important special cases. For example, we show that for a generic projective hypersurface of degree d, the Newton polygon of the interesting part of the zeta function coincides with its lower bound (the Hodge polygon). This gives a p-adic proof of a recent theorem of Illusie, conjectured by Dwork and Mazur. For more examples, let us consider the family of affine hypersurfaces of degree d or the family of affine hypersurfaces defined by polynomials f(xi, . . ., x7n) of degree di with respect to xi (1 < i < n), where the di are fixed positive integers. Then, for all large prime numbers p, the generic Newton polygon for the zeta functions of each of the two families of hypersurfaces coincides with its lower bound. We obtain our main results, namely several decomposition theorems, using certain maximizing functions from linear programming. Our work suggests a possible connection between Newton polygons and the resolution of singularities of toric varieties. Let p be a prime, q = pa, and let Fq be the finite field of q elements and Fqm its extension of degree m. Fix a nontrivial additive character qP of Fp. For any Laurent polynomial f(xi, . . ., xn) E Fq[xl, xj1,.. . ., x, xi1] we form
105 citations
TL;DR: In this paper, the authors presented the full prime factorization of the 9 Fermat number F9 = 2(512) + 1, which is the product of three prime factors that have 7, 49, and 99 decimal digits.
Abstract: In this paper we exhibit the full prime factorization of the ninth Fermat number F9 = 2(512) + 1. It is the product of three prime factors that have 7, 49, and 99 decimal digits. We found the two largest prime factors by means of the number field sieve, which is a factoring algorithm that depends on arithmetic in an algebraic number field. In the present case, the number field used was Q(fifth-root 2) . The calculations were done on approximately 700 workstations scattered around the world, and in one of the final stages a supercomputer was used. The entire factorization took four months.
103 citations
Patent•
27 Sep 1993TL;DR: In this paper, a method of generating and verifying electronic signatures for signed communication via a public digital network system by using an elliptic curve is described, which is characterized by the step of supplying on the network system public data to each of users from a system provider, wherein an element P whose x-coordinate has 0 is chosen, a single parameter is chosen for the elliptic curve E over a finite field and its base point, and a prime number p is chosen such that one of p = 2t +α and p=2t -α where t is
Abstract: The present invention discloses a method of generating and verifying electronic signatures for signed communication via a public digital network system by using an elliptic curve. The method is characterized by the step of supplying on the network system public data to each of users from a system provider, wherein an element P whose x-coordinate has 0 is chosen, a single parameter is chosen for the elliptic curve E over a finite field and its base point, and a prime number p is chosen such that one of p=2t +α and p=2t -α where t is a positive integer and α is a positive integer. Accordingly, fewer parameters can represent the elliptic curve E, base point P, field of definition GF(p), and order of the base point P, and either the x-coordinate or y-coordinate of the base point P have a small value. As a result, the elliptic curve addition kP can be calculated faster for any k.
88 citations
TL;DR: In this article, the smallest strong pseudoprime to all of the first k primes taken as bases was determined and upper bounds for 5, q6, q7, q8, q9, q10, q11, q12, q13, q14, q15, q16, q17, q18, q19, q20, q21, q22, q23, q24, q25, q26, q27, q28, q29, q30, q31, q32, q33, q
Abstract: With Y'k denoting the smallest strong pseudoprime to all of the first k primes taken as bases we determine the exact values for 5, q6, q7, q8 and give upper bounds for V/9, / W t,' 1 . We discuss the methods and underlying facts for obtaining these results. 1. PRIMALITY TESTS BY MEANS OF STRONG PSEUDOPRIMES Computer algebra systems, as for instance AXIOM [2], use strong pseudoprimes for testing primality of integers. The advantage of such tests is that they are very efficient. The disadvantage is that they are only probabilistic tests when the integers are not restricted to certain intervals. To make such tests deterministic for integers in prescribed intervals, one has to know the exact number of necessary so-called "strong pseudoprimality tests". For this purpose we introduce the numbers V1i, V/2, . .. for which we compute lower and upper bounds. These numbers are defined and discussed in this section; in ?2 we derive some facts which are the basis for finding bounds for the numbers V/k. In ?3 we discuss the methods which led to our results. In view of Fermat's "Little Theorem" we know that n is certainly not a prime when we have bn-1 i 1 mod n for an integer b with 1 0, and when n is a composite number, then n is called a "strong pseudoprime to base b" if either
73 citations
TL;DR: In this article, the simple connectivity of p-subgroup complexes of finite groups was investigated and simple connectivity was shown to be a function of the number of subgroups in the complex.
Abstract: We investigate the simple connectivity ofp-subgroup complexes of finite groups.
51 citations
TL;DR: In this paper, the authors show how the analogues of Jacobi sums, in the context of function fields, introduced and studied in [T1, T2, T3] can be obtained from shtukas introduced in [D2, D3, M].
Abstract: We show how the analogues of Jacobi sums, in the context of function fields, introduced and studied in [T1, T2, T3] can be obtained from shtukas introduced and studied in [D2, D3, M]. We apply this to obtain some results on the prime factorization of analogues of Gauss sums and to prove an analogue of the Gross-Koblitz formula for general function field, generalizing the results in [T2]. For this purpose, we also introduce and interpolate a new analogue of gamma function.
48 citations
TL;DR: In this paper, the authors studied the divisibility of the quadratic fields of discriminant 8p, −8p, and −4p by powers of 2 for p ≡ 1 mod 4 a prime number.
37 citations
Patent•
16 Apr 1993TL;DR: In this article, the authors proposed a method of privacy communication in which an elliptic curve E and an element thereof are notified to all parties who wish to communicate, and data are transmitted from one party to another by using a calculation of the element and coded data made in secret by each party.
Abstract: The present invention provides a method of privacy communication, in which an elliptic curve E and an element thereof are notified to all parties who wish to communicate, and data are transmitted from one party to another by using a calculation of the element and coded data made in secret by each party. The method is characterized by a construction of E(GF(p)) such whose number of elements has exactly p, assuming that p is a prime number and E(GF(p)) is a group of elements of GF(p) on the elliptic curve E. More particularly, E(GF(p)) is constructed by an algorithm: let d be a positive integer such that gives an imaginary quadratic field Q((-d1/2)) with a small class number; then find a prime number p such that 4·p-1=d·square number; and find a solution of a class polynomial Hd (x)=0 modulo p such that is defined by d and given with a j-invariant.
34 citations
TL;DR: In this article, the ternary Goldbach problem and the Goldbach-Waring problem are solved in prime numbers lying in intervals of a special form, and a binary additive problem with numbers in the sequence, 1$ SRC=http://ej.iop.org/images/1468-4810/41/3/A03/tex_im_2271_img2.gif
Abstract: Variants of two classical additive problems--the ternary Goldbach problem and the Goldbach-Waring problem--are solved in prime numbers lying in intervals of a special form. A binary additive problem with numbers in the sequence , 1$ SRC=http://ej.iop.org/images/1468-4810/41/3/A03/tex_im_2271_img2.gif/>, is also solved.
Patent•
13 Sep 1993
TL;DR: In this article, a two-dimensional primitive root diffusor includes a 2D pattern of wells, the depths of which are determined through operation of primitive root sequence theory, and a prime number N is chosen such that N-1 has two coprime factors which are non-divisible into each other.
Abstract: A two-dimensional primitive root diffusor includes a two-dimensional pattern of wells, the depths of which are determined through operation of primitive root sequence theory A prime number N is chosen such that N-1 has two coprime factors which are non-divisible into each other From the prime number, a primitive root is determined and, in the preferred embodiment, an algorithm is used to determine sequence values for each well Each sequence value is proportional to the well depth, with each sequence value being multiplied by the design wavelength and then divided by 2N to arrive at the actual well depth value
Book•
1 Jan 1993TL;DR: The fundamental theorem, GCDs and LCMs, and Miller's test and strong pseudoprimes are explained.
Abstract: Preface 1 The fundamental theorem, GCDs and LCMs 2 Listing primes 3 Congruences 4 Powers and pseudoprimes 5 Miller's test and strong pseudoprimes 6 Euler's theorem, orders and primality testing 7 Cryptography 8 Primitive roots 9 The number of divisors d and the sum of divisors 10 Continued fractions and factoring 11 Quadratic residues References Index
TL;DR: In this paper, the Chebyshev type estimates for Beurling generalized prime numbers in the general case n N(x) = x ∑ ν=1 n A ν log ρν−1 x + O(x log −γ x) were shown to hold even under weaker conditions.
Abstract: Let N(x) be the distribution function of the integers in a Beurling generalized prime system. The Chebyshev type estimates for Beurling generalized prime numbers in the general case n N(x) = x ∑ ν=1 n A ν log ρν−1 x + O(x log −γ x) is a long standing question. In this paper we shall give an affirmative answer to the question by proving that the Chebyshev type estimates ... (formule)... hold even under weaker condition ... (formule)... with ρ n = τ ≥ 1, 0 0. This generalizes a result of Diamond and a result of the present author
1 Jan 1993
TL;DR: In this paper, it was shown that if max (D,p≥10 10193) and max (d,p)¬=(3s 2 + 1,4s 2+1,4 s 2 +1), then the equation x 2 +D=p n has at most one positive integer solution (x, n)
Abstract: Let D be a positive integer, and let p be an odd prime with p|D. In this Note we prove that if max (D,p≥10 10193 and (D,p)¬=(3s 2 +1,4s 2 +1), then the equation x 2 +D=p n has at most one positive integer solution (x, n)
TL;DR: A necessary condition for Menon difference sets in groups of the formZ2p×Z 2p×Gq, whereGq is an abelianq-group andp, q are distinct prime numbers, will be proved and constraints on the magnitudes of p andq are given.
Abstract: Menon difference sets have the parameters (4N2, 2N2±N,N2±N). In this paper, a necessary condition for Menon difference sets in groups of the formZ2p×Z2p×Gq, whereGq is an abelianq-group andp, q are distinct prime numbers, will be proved. We will also focus on the groupsZ2pq×Z2pq and give constraints on the magnitudes ofp andq. Finally, we show that if the groupZ6p×Z6p contains a Menon difference set, thenp=3 or 13 only.
Journal Article•
TL;DR: It is shown that there are recursive sets two-variable simple programs cannot recognize, including the set of prime numbers and the set L e of integers raised to the power for any fixed integer e ⩾ 2.
TL;DR: A necessary and sufficient condition for the polyhedron P(A, b) to have integral extreme points only is given and its special version for the case where k is a prime number is provided.
TL;DR: It is proved that the maximal number of mutually disjoint subgroups in a group G of order p^2n cannot be more than (P^(n-1) - l) (p - 1)^-1 provided that n >= 4 and that G is not elementary abelian.
TL;DR: If one of two independent random variables (possibly both) is uniformly distributed, then so is their sum modulo m; it is shown that the converse is true if m is a prime number and the relevant probabilities are rational numbers.
TL;DR: In this paper, the authors give a short elementary proof of a result of Almkvist and Meurman [1] on an integrality property of the values taken by the Bernoulli polynomials at a rational number.
Abstract: We give a short elementary proof of a result of Almkvist and Meurman [1] on an integrality property of the values taken by the Bernoulli polynomials at a rational number. We use a lemma on the divisibility properties of certain binomial coefficients which seems to be of independent interest. 0. Introduction As is well known, the Bernoulli polynomials Bn(i) defined by Xe Z occur naturally while summing powers of the natural numbers. They also appear in other places, like in the evaluation of the Riemann zeta function at even integers, or while finding out whether or not a prime number is regular. As such, the properties of these polynomials are of some number-theoretic interest. Recently, G. Almkvist and A. Meurman proved the following result in [1]. THEOREM. Writing Bn(t) = BJJ) Bn(0), we have for all h,k,ne N, The purpose of this note is to give a short and completely elementary proof of this theorem. REMARK. AS observed in [1], it is enough to prove the theorem with the assumption that h = 1, since we have the addition formula Bn(x+y)= m 0 From now on, we write an = k Bn I -1 for simplicity. We employ two different \kj recursions for the numbers an and a lemma on the divisibility properties of certain binomial coefficients which seems to be of independent interest. Received 10 February 1992; revised 7 April 1992. 1991 Mathematics Subject Classification 11B68. Bull. London Math. Soc. 25 (1993) 327-329
TL;DR: In this article, the authors consider the problem of representing a positive integer N as the sum of the difference of two norms of integral ideals of a number field of degree k > 1, and show that for N to be represented in these ways the following congruences must be solvable in α β є k, respectively.
Abstract: Let K be a number field of degree k > 1. We would like to know if a positive integer N can be represented as the sum, or the difference, of two norms of integral ideals of K. Suppose K/ℚ is abelian of conductor Δ. Then from the class field theory (Artin's reciprocity law) the norms are fully characterized by the residue classes modulo Δ. Precisely, a prime number p ∤ Δ (unramified in K) is a norm (splits completely in K), if, and only if,where k is a subgroup of (ℤ/Δℤ)* of index k. Accordingly we may ask N to be represented as the sumor the differenceof positive integers a, b each of which splits completely in K. For N to be represented in these ways the following congruencesmust be solvable in α β є k, respectively. Moreover the conditionmust hold. Presumably the above local conditions are sufficient for (−) to have infinitely many solutions and for (+) to have arbitrarily many solutions, provided N is sufficiently large in the latter case.
TL;DR: In this article, the notion of locally everywhere embedded in a ℤl-extension is introduced. But the authors focus on the Galois group of idele and do not address the problem of finding the root of unity in a prime number.
Abstract: For a number fieldF that contains ζl a l
th
root of unity (l is a prime number), we determine thex such thatF
$$(\sqrt[\ell ]{x})$$
can be embedded in a ℤl-extension. We approach the corresponding Kummer radical with the notion of being locally everywhere embedded in a ℤl-extension. An idelic description of Galois group is appropriate especially as we utilize the l-adic group of idele of [15]. The illustration concerns l=3 and biquadratic field ℚ
$$(\zeta _3 ,\sqrt d )$$
. We detail the step of the calculus and fournish numerical tables.
1 Oct 1993
TL;DR: In this paper, it was shown that an extended Schlomilch formula for Stirling-type pairs of numbers and the inversion formula of Lagrange are implied by each other.
Abstract: It is shown that an extended Schlomilch formula for Stirling-type pairs of numbers and the inversion formula of Lagrange are implied by each other. Also proved are some congruence relations modulo a prime number p(>2) associated with generalized Stirling numbers. The third result is concerned with the asymptotic expansions of Stirling-type pairs involving large parameters.
TL;DR: The investigation of a conjecture about essential prime factors of R n leads to a proof that CaseI of Fermat's Last Theorem holds for any prime exponent p > 2 such that np + 1 is prime for some integer n ≤ 500 not divisible by 3.
Abstract: After a brief review of partial results regarding CaseI 1 of Fermat's Last Theorem, we discuss the relationship between the number of points on Fermat's curve modulo a prime and the resultant $R_n$ of the polynomials $X^n - 1$ and $(-1-X)^n - 1$, called Wendt's determinant. The investigation of a conjecture about essential prime factors of $R_n$ (Conjecture 1.3) leads to a proof that Case 1 of Fermat's Last Theorem holds for any prime exponent $p>{}$2 such that $np+{}$1 is prime for some integer $n\le{}$500 not divisible by 3. EDITOR'S NOTE: In addition to providing insight into Wendt's determinant, an object of interest in its own right, this paper belongs to a continuing line of investigations that may prove fruitful in spite of the recent announcement by Wiles of his proof of Fermat's Last Theorem. It is not unreasonable to hope for a more elementary proof than Wiles'.
1 Jan 1993
TL;DR: An alternative formulation of Fermat's Little Theorem states that for any prime n and an arbitrary integer b with gcd(b, n) = 1 the congruence b n −1 ≡ 1 mod n holds as mentioned in this paper.
Abstract: Several mathematical procedures require the provision of large “random” prime numbers. Fermat’s Little Theorem plays an important role in many primality tests and in motivating the concept of pseudoprimes. It tells us that for a prime number n and an integer b with gcd(b, n) = 1 the congruence b n −1 ≡ 1 mod n holds. An alternative formulation of Fermat’s Little Theorem states that for any prime n and an arbitrary integer b one has
$$ {b^n} \equiv b\;\bmod \,n. $$
(1)
17 Jan 1993
TL;DR: To identify systematically all irreducible components of the canonical diagrams for first - order spectral nulls at f, a set of channel symbol sequences specifying all of them is given.
Abstract: Irreducible components of canonical diagrams for spectral null constraints at f = fsk/n are studied, where k and n are relatively prime integers with 0 /spl les/s k < n and fs is the symbol frequency. To identify systematically all irreducible components of the canonical diagrams for first - order spectral nulls at f, we give a set of channel symbol sequences specifying all of them. If n is a prime number, then each sequence in the set corresponds to exactly one irreducible component up to label-preserving graph isomorphism. We also give a set of channel symbol sequences specifying all irreducible components of canonical diagrams for second-order spectral nulls at dc (i.e., f = 0).