Showing papers on "Prime number published in 1996"
1 Jan 1996
TL;DR: In this paper, Carmichael-Lucas and Carmichael showed that a natural number is a prime if and only if it can be represented by a pairwise relative prime Integrator.
Abstract: 1 How Many Prime Numbers Are There?.- I. Euclid's Proof.- II. Goldbach Did It Too!.- III. Euler's Proof.- IV. Thue's Proof.- V. Three Forgotten Proofs.- A. Perott's Proof.- B. Auric's Proof.- C. Metrod's Proof.- VI. Washington's Proof.- VII. Furstenberg's Proof.- VIII. Euclidean Sequences.- IX. Generation of Infinite Sequences of Pairwise Relatively Prime Integers.- 2 How to Recognize Whether a Natural Number Is a Prime.- I. The Sieve of Eratosthenes.- II. Some Fundamental Theorems on Congruences.- A. Fermat's Little Theorem and Primitive Roots Modulo a Prime.- B. The Theorem of Wilson.- C. The Properties of Giuga, Wolstenholme, and Mann and Shanks.- D. The Power of a Prime Dividing a Factorial.- E. The Chinese Remainder Theorem.- F. Euler's Function.- G. Sequences of Binomials.- H. Quadratic Residues.- III. Classical Primality Tests Based on Congruences.- IV. Lucas Sequences.- V. Primality Tests Based on Lucas Sequences.- VI. Fermat Numbers.- VII. Mersenne Numbers.- VIII. Pseudoprimes.- A. Pseudoprimes in Base 2 (psp).- B. Pseudoprimes in Base a (psp(a)).- C. Euler Pseudoprimes in Base a (epsp(a)).- D. Strong Pseudoprimes in Base a (spsp(a)).- E. Somer Pseudoprimes.- IX. Carmichael Numbers.- X. Lucas Pseudoprimes.- A. Fibonacci Pseudoprimes.- B. Lucas Pseudoprimes (lpsp(P, Q)).- C. Euler-Lucas Pseudoprimes (elpsp(P, Q)) and Strong Lucas Pseudoprimes (slpsp(P, Q)).- D. Somer-Lucas Pseudoprimes.- E. Carmichael-Lucas Numbers.- XL Primality Testing and Large Primes.- A. The Cost of Testing.- B. More Primality Tests.- C. Primality Certification.- D. Fast Generation of Large Primes.- E. Titanic Primes.- F. Curious Primes.- XII. Factorization and Public Key Cryptography.- A. Factorization of Large Composite Integers.- B. Public Key Cryptography.- 3 Are There Functions Defining Prime Numbers?.- I. Functions Satisfying Condition (a).- II. Functions Satisfying Condition (b).- III. Functions Satisfying Condition (c).- IV. Prime-Producing Polynomials.- A. Surveying the Problems.- B. Polynomials with Many Initial Prime Absolute Values.- C. The Prime-Producing Polynomials Races.- D. Primes of the Form m2 + 1.- 4 How Are the Prime Numbers Distributed?.- I. The Growth of ?(x).- A. History Unfolding.- B. Sums Involving the Mobius Function.- C. Tables of Primes.- D. The Exact Value of ?(x) and Comparison with x/(log x), Li(x), and R(x).- E. The Nontrivial Zeros of ?(s).- F. Zero-Free Regions for ?(s) and the Error Term in the Prime Number Theorem.- G. The Growth of ?(s).- H. Some Properties of ?(x).- II. The n th Prime and Gaps.- A. The n th Prime.- B. Gaps Between Primes.- Interlude.- III. Twin Primes.- Addendum on k-Tuples of Primes.- IV. Primes in Arithmetic Progression.- A. There Are Infinitely Many!.- B. The Smallest Prime in an Arithmetic Progression.- C. Strings of Primes in Arithmetic Progression.- V. Primes in Special Sequences.- VI. Goldbach's Famous Conjecture.- VII. The Waring-Goldbach Problem.- A. Waring's Problem.- B. The Waring-Goldbach Problem.- VIII. The Distribution of Pseudoprimes, Carmichael Numbers, and Values of Euler's Function.- A. Distribution of Pseudoprimes.- B. Distribution of Carmichael Numbers.- C. Distribution of Lucas Pseudoprimes.- D. Distribution of Elliptic Pseudoprimes.- E. Distribution of Values of Euler's Function.- 5 Which Special Kinds of Primes Have Been Considered?.- I. Regular Primes.- II. Sophie Germain Primes.- III. Wieferich Primes.- IV. Wilson Primes.- V. Repunits and Similar Numbers.- VI. Primes with Given Initial and Final Digits.- VII. Numbers kx2n+-1.- VIII. Primes and Second-Order Linear Recurrence Sequences.- IX. The NSW Primes.- 6 Heuristic and Probabilistic Results about Prime Numbers.- I. Prime Values of Linear Polynomials.- II. Prime Values of Polynomials of Arbitrary Degree.- III. Polynomials with Many Successive Composite Values.- IV. Partitio Numerorum.- V. Some Probabilistic Estimates.- A. Distribution of Mersenne Primes.- B. The log log Philosophy.- VI. The Density of the Set of Regular Primes.- Conclusion.- The Pages That Couldn't Wait.- Primes up to 10,000.- Index of Tables.- Index of Names.
516 citations
Book•
1 Jan 1996
TL;DR: In this article, the arithmetical function d(n), its generalizations and its analogues are discussed, and the sequence of prime numbers is considered. But the authors do not consider the problem of estimating the number of elements in a graph.
Abstract: Preface. Basic Symbols. Basic Notations. I. Euler's phi-function. II. The arithmetical function d(n), its generalizations and its analogues. III. Sum-of-divisors function, generalizations, analogues Perfect numbers and related problems. IV. P, p, B, beta and related functions. V. omega(n), Omega(n) and related functions. VI. Function mu k-free and k-full numbers. VII. Functions pi(x), psi(x), theta(x), and the sequence of prime numbers. VIII. Primes in arithmetic progressions and other sequences. IX. Additive and diophantine problems involving primes. X. Exponential sums. XI. Character sums. XII. Binomial coefficients, consecutive integers and related problems. XIII. Estimates involving finite groups and semi-simple rings. XIV. Partitions. XV. Congruences, residues and primitive roots. XVI. Additive and multiplicative functions. Index of authors.
295 citations
Posted Content•
TL;DR: In this article, it was shown that the Hecke algebra acting on the space of weight two forms of level $M$ is reduced if weight at least two cannot be a scalar.
Abstract: Let $p$ be a prime number and $N$ an integer prime to $p$. We show that the operator $U_p$ on the space of cuspidal modular forms of level $pN$ and weight two is semi-simple. It follows from this that the Hecke algebra acting on the space of weight two forms of level $M$ is reduced if $M$ is cube free. Assuming Tate's conjecture for cycles on smooth projective varieties over finite fields, we generalize these results to higher weights. The main point in the proof is that the crystalline Frobenius of the reduction mod $p$ of the motive associated to a newform of level prime to $p$ and weight at least two cannot be a scalar. Assuming Tate's conjecture, it follows that Ramanujan's inequality is strict. For $N$ prime, we relate the discriminant of the weight two Hecke algebra to the height of the modular curve $X_0(N)$, for which we get an upper bound.
60 citations
55 citations
TL;DR: In this paper, the integral coefficients of the modular, function fields with respect to the principal congruence subgroup of SL2 (ℤ) of prime level were determined.
Abstract: We determine the equation with integral coefficients of the modular, function fields with respect to the principal congruence subgroup ofSL2 (ℤ) of prime level.
32 citations
TL;DR: A Hundred Years of Prime Numbers as mentioned in this paper is a collection of prime numbers from the early nineties to the present day, with a focus on prime numbers and their applications in mathematics.
Abstract: (1996). A Hundred Years of Prime Numbers. The American Mathematical Monthly: Vol. 103, No. 9, pp. 729-741.
29 citations
Patent•
3 Oct 1996
TL;DR: In this paper, the authors proposed a method of public key cryptography based on the discrete logarithm that makes use of the computation of the variable r=gk modp where p is a prime number called a modulus, the exponent k is a random number usually with a length of N bits and g is an integer called a base, wherein an entity E carries out operations of authentication and/or of signature, including exchanges of signals with another entity in which this variable comes into play.
Abstract: Method of public key cryptography based on the discrete logarithm that makes use of the computation of the variable r=gk modp where p is a prime number called a modulus, the exponent k is a random number usually with a length of N bits and g is an integer called a base, wherein an entity E carries out operations of authentication and/or of signature, including exchanges of signals with another entity in which this variable comes into play.
19 citations
1 Jan 1996
TL;DR: In this article, it was shown that AssR (M/I an(1) 1... I an(g) g N ) is independent of n for all large n.
Abstract: Let I1, . . . , Ig be ideals of the commutative ring R, let M be a Noetherian R-module and let N be a submodule of M ; also let A be an Artinian R-module and let B be a submodule of A. It is shown that, whenever (am (1) , . . . , am (g))m∈N is a sequence of g-tuples of non-negative integers which is non-decreasing in the sense that ai (j) ≤ ai+1 (j) for all j = 1, . . . , g and all i ∈ N, then AssR ( M/I an(1) 1 . . . I an(g) g N ) is independent of n for all large n, and also AttR ( B :A I an(1) 1 . . . I an(g) g ) is independent of n for all large n. These results are proved without any regularity conditions on the ideals I1, . . . , Ig, and so (a special case of) the first answers in the affirmative a question raised by S. McAdam.
18 citations
15 citations
TL;DR: It is shown that the minimum period modulo p of the Bell exponential integers is (p p − 1)/(p − 1) for all primes p < 102 and several larger p, and the proof of this result requires the prime factorization of these periods.
Abstract: We show that the minimum period modulo p of the Bell exponential integers is (p p − 1)/(p − 1) for all primes p < 102 and several larger p. Our proof of this result requires the prime factorization of these periods. For some primes p the factoring is aided by an algebraic formula called an Aurifeuillian factorization. We explain how the coefficients of the factors in these formulas may be computed.
15 citations
TL;DR: The density of primes dividing at least one term of the Lucas sequence defined by L 0(P) = 2, L 1(P)= P and Ln n-1(P + Ln-2(P)) for n ≥ 2, with P an arbitrary integer, is determined in this paper.
Abstract: The density of primes dividing at least one term of the Lucas sequence defined by L0(P) = 2, L1(P)= P and Ln(P) = PLn-1(P) + Ln-2(P) for n ~ 2, with P an arbitrary integer, is determined.
TL;DR: This paper shows that the procedure described, in which a k-bit odd number is chosen at random and subjected to t random strong probable prime tests, is in fact valid for all k ≥ 2 and t ≥ 1.
Abstract: Recently, Damgard, Landrock and Pomerance described a procedure in which a k-bit odd number is chosen at random and subjected to t random strong probable prime tests. If the chosen number passes all t tests, then the procedure will return that number ; otherwise, another k-bit odd integer is selected and then tested. The procedure ends when a number that passes all t tests is found. Let p kt denote the probability that such a number is composite. The authors above have shown that p k,t ≤ 4 −t when k ≥ 51 and t ≥ 1. In this paper we will show that this is in fact valid for all k ≥ 2 and t ≥ 1.
TL;DR: An algorithm is described which takes as input α and the minimal polynomial of α over Q, and determines if α is a norm of an element of L, and it is shown that, if the authors ignore the time needed to obtain a complete factorization of α and a completefactorization of the discriminant of α, then the algorithm runs in timePolynomial in the size of the input.
Abstract: Let L = Q[α] be an abelian number field of prime degree q, and let α be a nonzero rational number. We describe an algorithm which takes as input α and the minimal polynomial of α over Q, and determines if α is a norm of an element of L. We show that, if we ignore the time needed to obtain a complete factorization of α and a complete factorization of the discriminant of α, then the algorithm runs in time polynomial in the size of the input. As an application, we give an algorithm to test if a cyclic algebra A = (E, σ, σ) over Q is a division algebra.
TL;DR: A problem in number theory and a problem in physics may turn out to be two sides of the same numerical coin this paper, and the connection with physics could break a longtime logjam in pure mathematics by leading to a proof of a century-old problem.
Abstract: A problem in number theory and a problem in physics may turn out to be two sides of the same numerical coin. Quantum physicists think a mathematical beast known as the Riemann zeta function, which encodes information about prime numbers, could provide a key to understanding the behavior of complex atomic systems. At the same time, the connection with physics could break a longtime logjam in pure mathematics by leading to a proof of a century-old problem.
TL;DR: In this paper, it was shown that a full range map Psi: Xk -> X exists which is invariant under the action of S, only if, for all i>1, the elements of the homotopy group IL(X) have orders relatively prime with k. Theorem 3 proves necessary and sufficient condition for a parafinite CW complex X to admit full range invariant maps for any prime number k:X must be contractible.
Abstract: This paper studies maps which are invariant under the action of the symmetry group S,. The problem originates in social choice theory: there arc k individuals each with a space of preferences X, and a social choice map phi:Xk - X which is anonymous i.e. invariant under the action of a group of symmetries. Theorem 1 proves that a full range map Psi: Xk -> X exists which is invariant under the action of S, only if, for all i>1, the elements of the homotopy group IL(X) have orders relatively prime with k. Theorem 2 derives a similar results for actions of subgroups of the group Sk. Theorem 3 proves necessary and sufficient condition for a parafinite CW complex X to admit full range invariant maps for any prime number k:X must be contractible.
Patent•
12 Apr 1996TL;DR: In this paper, a polynomial in the form P(x)=xk +ak-1 xk-1 +... +a1 x+a0, whose coefficients are formed from the message, taking into account a random number.
Abstract: A computerized method for signing a message, where a secret key is used for signing and the signature can be tested with the help of a public key, provides for the public key to be a number n that is the product of two large prime numbers p and q; the secret key includes at least one of the two prime numbers; a polynomial is created in the form P(x)=xk +ak-1 xk-1 + . . . +a1 x+a0, whose coefficients ak-1 . . . a0 are formed from the message, taking into account a random number. This polynomial is used to derive additional polynomials P(x) mod p and P(x) mod q whose zeros in the respective finite fields GF(p) and GF(q) are defined. The zeros are combined into one or more solutions z of the equation P(x) mod n=0, and the random number and the solution z or selected solutions z are added as the signature to the message.
TL;DR: Euclid's elegant proof that there are infinitely many prime numbers is well known as discussed by the authors, and Elder proved the same result, in fact a stronger one, byanalytical methods.
Abstract: Euclid’s elegant proof that there are infinitely many prime numbers is well known. Elder proved the same result, in fact a stronger one, byanalytical methods. This article gives an exposition of Euler’s proof introducing the necessary concepts along the way.
TL;DR: In this paper, the Iwasawa invariant λ p (k) of the cyclotomic Z p -extension of a real quadratic field was studied.
Abstract: Let k be a real quadratic field and p an odd prime number which splits in k. In a previous work, the author gave a sufficient condition for the Iwasawa invariant λ p (k) of the cyclotomic Z p -extension of k to be zero. The purpose of this paper is to study the case p = 3 of this result and give new examples of k with λ 3 (k) = 0, by using information on the initial layer of the cyclotomic Z 3 -extension of k.
1 Jan 1996
TL;DR: The answer to the question of how many prime numbers exist is given by the fundamental theorem as mentioned in this paper, which states that there exist infinitely many numbers of a given number of prime number.
Abstract: The answer to the question of how many prime numbers exist is given by the fundamental theorem:
There exist infinitely many prime numbers.
1 Jan 1996
TL;DR: Ankeny and Artin-Chowla as discussed by the authors obtained several congruences for the class number h k of a quadratic field K, some of which were also obtained by Kiselev.
Abstract: Ankeny–Artin–Chowla obtained several congruences for the class number h k of a quadratic field K , some of which were also obtained by Kiselev. In particular, if the discriminant of K is a prime number p ≡ 1 (mod 4) and e = t + u √ p /2 is the fundamental unit of K , then
TL;DR: In this paper, a geometrical model in the form of a 2-dimensional matrix map of the distribution of divisibles and indivisibles of the natural number sequence was constructed and on that basis, it was possible to derive an analytical expression which describes globally the distributions of primes and non-primes.
Abstract: This paper reports on some interesting properties of the natural number system which may interest mathematicians, scientists and engineers In a recently published paper, a geometrical model in the form of a 2‐dimensional matrix map of the distribution of divisibles and indivisibles of the natural number sequence was constructed and on that basis, it was possible to derive an analytical expression which describes globally the distribution of primes and nonprimes This equation is called the equation of divisiblesBy predicting the presence of divisibles against each integer position, this equation also at the same time predicts the absence of indivisibles in the remaining positions This approach is rather unusual but it works Many interesting geometrical properties of the natural number system were discovered using the matrix map model which would be difficult to unravel by focussing on the prime number series alone Two mathematical problems are posed which at this stage the author finds no proofs or s
TL;DR: It is argued that frequency and irregularity in the occurrence of prime numbers, distribution of the primes gaps, and the behaviour of the prime counting function π(x) are more easily represented on a spreadsheet than in a programming language.
Abstract: This paper presents the use of a spreadsheet in generating prime numbers through the Sieve of Eratosthenes. The approach is applied to visualizing frequency and irregularity in the occurrence of prime numbers, distribution of the primes gaps, and the behaviour of the prime counting function π(x).It is argued that these and other aspects of the theory of numbers are more easily represented on a spreadsheet than in a programming language. The spreadsheet used is Excel 4.0 for Macintosh computers.
[...]
TL;DR: In this paper, it was shown that every sufficiently large even integer can be written as the sum of an odd prime and a number that is either prime or the product of two primes.
Abstract: In this chapter, we shall prove one of the most famous results in additive prime number theory: Chen’s theorem that every sufficiently large even integer can be written as the sum of an odd prime and a number that is either prime or the product of two primes.
TL;DR: This paper takes up a subdomain of binary images, called the weakly taxicab convex image domain, and shows how the indecomposability problem in that shape domain can be approached in a manner closely analogous to the number theoretic way.
Abstract: An indecomposable shape is like a prime number. It cannot be decomposed further as a Minkowski sum of two simpler shapes. With respect to Minkowski addition (dilation), therefore, the indecomposable shapes are the fundamental building blocks of all geometric shapes. However, just as it is difficult to identify whether a given number is a prime number or not, it is equally or more difficult to say whether a given shape is indecomposable or not. In this paper we take up a subdomain of binary images, called the weakly taxicab convex image domain, and show how the indecomposability problem in that shape domain can be approached in a manner closely analogous to the number theoretic way. Apart from our attempt to show that the indecomposability problem is an extremely interesting mathematical problem, our algorithmic treatment of the problem also leads to an efficient method of computing Minkowski addition and decomposition of binary images.
1 Jan 1996
TL;DR: The distribution of Pjateckii-Sapiro prime numbers in arithmetic progressions was investigated in this paper, and a Bombier i- Vinogr adov typ e mean value t heorem and anot her almost all result concerning this problem were established.
Abstract: The distribution of the Pjateckii-Sapiro prime numbers in arithmetic progressions is investigated, and a Bombier i- Vinogr adov typ e mean- value t heorem and anot her almost all resultconcerning this problem are established.
Journal Article•
TL;DR: In this article, a characterization of normal bases and complete normal bases in GF(qrn) over GF (q), whereq> 1 is any prime power, r = 2 and q? 3 mod 4, for anyn? 1, in terms of certain roots of unity, is given.
1 Jan 1996
Patent•
2 Apr 1996
TL;DR: In this paper, a primitive root sequence theory is used to laterally scatter sounds, suppress specified mirror-shaped reflection and increase indirect sound fields for a listener by providing the well of two-dimensional well and determining the depth of the well.
Abstract: PURPOSE: To laterally and uniformly scatter sounds, to suppress specified mirror- shaped reflection and to increase indirect sound fields for a listener by providing the well of two-dimensional well and determining the depth of the well by operating a primitive root sequence theory. CONSTITUTION: The matrix of 12×13 showing the positions of wells 1-156 on a matrix according to an instruction is calculated. Concerning such a matrix, numbers are obliquely continued at -45 deg. until reaching a final enable spot, a sequence continues from the apex of the next column, and the next row of the 1st column is started when the final column is completed. The numbering sequence is continued until all 156 wells are properly positioned. After the primitive root is multiplied corresponding to the number of wells, the number generated as a result is further divided by a prime number such as '157' in this case, the prime number '157' is multiplied to a numeral less than a decimal point, and the result is defined as a sequence value corresponding to the number of wells. The respective sequence values are multiplied with a designed wavelength and when it is divided by the double prime number, the real value of well can be provided.
TL;DR: The prime numbers are often called the building blocks of number theory, a classic case of a sine qua non as discussed by the authors, and if the corpus of the theory of numbers is looked upon as an architectural pile then the primes will be found amongst its foundations, and amongst its walls and buttresses, and indeed amongst the array of pinnacles and turrets which burst forth from it and stand proud, alone and magnificent.
Abstract: The prime numbers are often called the building blocks of number theory, a classic case of a ‘ sine qua non ’. If the corpus of the theory of numbers is looked upon as an architectural pile then the primes will be found amongst its foundations, and amongst its walls and buttresses, and indeed amongst the array of pinnacles and turrets which burst forth from it and stand proud, alone and magnificent. One such pinnacle had been discussed and planned for a hundred years before it was finally constructed in 1896 by two men working totally independently of each other (collapse of stout analogy). With this achievement, the prime numbers, those familiar yet raw beasts of mathematics, in one sense had been tamed. The long journey, which had begun with the tentative definitions of the ancient Greeks, was completed.