Showing papers on "Prime number published in 1981"
TL;DR: All methods that have been proposed thus far are either restricted to very special cases or are so laborious and prolix that even for numbers that do not exceed the limits of tables constructed by estimable men, they try the patience of even the practiced calculator.
Abstract: Is there an algorithm that can decide whethern is prime or composite and that runs in polynomial time? The Adleman-Rumely algorithm and the Lenstra variations come so close, that it would seem that almost any improvement would give the final breakthrough. Is factoring inherently more difficult than distinguishing between primes and composites? Most people feel that this is so, but perhaps this problem too will soon yield. In his “Disquisitiones Arithmeticae” Gauss [9], p. 396, wrote “The problem of distinguishing prime numbers from composite numbers and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be superfluous to discuss the problem at length. Nevertheless we must confess that all methods that have been proposed thus far are either restricted to very special cases or are so laborious and prolix that even for numbers that do not exceed the limits of tables constructed by estimable men, i.e., for numbers that do not yield to artificial methods, they try the patience of even the practiced calculator. And these methods do not apply at all to larger numbers.” The struggle continues!
68 citations
TL;DR: In this article, conditions sufficient for a matrix to be prime were provided, and a characterization of primes in terms of a nation of rank was given, and an order property of primate factors was used to obtain a result on prime factors.
32 citations
TL;DR: In this article, the authors give affirmative answers to the conjecture under various conditions, including the assumption that the complex projective line is a polynomial of degree p with p a prime number.
Abstract: Here, P1 is the complex projective line. The purpose of this paper is to give affirmative answers to the conjecture under various conditions. It is separated into 3 parts. In Part 1, we assume that n=p is a prime number and obtain a result. Recently, Kato [2] has improved this result extensively. In Part 2, we assume that f(x) is a polynomial of degree p with p a prime number, and obtain a result. In Part 3, we assume:
29 citations
TL;DR: In this paper, a simple, straightforward procedure, which requires no special tables or generators, is presented for constructing resolvable incomplete block designs for v =pk, v=p2k, …, treatments, for k≥p, in incomplete blocks of size k.
Abstract: A simple, straightforward procedure, which requires no special tables or generators, is presented for constructing resolvable incomplete block designs for v=pk, v=p2k, …, treatments, for k≥p, in incomplete blocks of size k. Also, it is shown, how to obtain incomplete block designs for any v in blocks of size k and k+1. The procedure allows construction of balanced incomplete block designs for p = k a prime number. For p = n not a prime number, incomplete block designs can be obtained by the procedure, but are not balanced. However, for ps being the smallest prime factor of n, ps + 1 for v = n2, ps2+ ps + 1 for v = n3, …, arrangements can be obtained for which the occurrence of any treatment pair in the blocks is either zero or one. This is called a zero-one concurrence design. Procedures are described for obtaining additional zero-one concurrence arrangements. It is shown that the efficiency of these designs is maximum. Both intra-block and inter-block analyses are described.
29 citations
TL;DR: In this article, it was shown that if p = (2qn)2 + 1 is a prime, where q is an odd prime and n ≧ 1 an integer, then H(p) > 1.
Abstract: Let p be an integer and let H(p) be the class-number of the field where ζ p is a primitive p-th root of unity and Q is the field of rational numbers. It has been proved in [1] that if p = (2qn)2 + 1 is a prime, where q is a prime and n > 1 an integer, then H(p) > 1. Later, S. D. Lang [2] proved the same result for the prime number p = ((2n + 1)q)2 + 4, where q is an odd prime and n ≧ 1 an integer. Both results have been obtained in the case p ≡ 1 (mod 4). In this paper we shall prove the similar results for a certain prime number p ≡ 3 (mod 4). We designate by h(p) the class-number of the real quadratic field
12 citations
TL;DR: In fact, the only known prime values of En occur for n = 1, 2, 3, 4, 5, 6, 7, 8, 14, 16, 17, and 11 as discussed by the authors.
Abstract: Euclid's proof that there are infinitely many primes is based on the observation that if En = Pn + 1 is not itself prime (where Pn =P1P2 p Pn is the product of the first n primes), it must still contain a prime factor larger than pn. Little is known about the values of n for which En is prime. In fact, the only known prime values of En occur for n = 1, 2, 3, 4, 5, and 11. When asked by a student whether E,, is prime for infinitely many values of n, George P6lya is reported to have replied, "There are many questions which fools can ask that wise men cannot answer." The anthropologist Reo Fortune (once married to Margaret Mead) conjectured that if Qn is the smallest prime number strictly greater than En, then the difference Fn = QnP is always prime. (This conjecture first appeared in print [2] in 1980, and is discussed further in [3].) To illustrate, E = (2 X 3 X5X .** X 17) + 1= 51051 1, and the next larger prime is Q7= 510529. Sure enough, the difference is F7 = Q7P7 = 510529 510510 = 19, a prime. The sequence (Fn} of "fortunate numbers" begins: 3, 5, 7, 13, 23 ,17, 19, 23, 37, 61, 67, 61,71,47,107,59,61,109,89,103,79,..., and indeed, all the listed numbers are prime. Is this merely a remarkable coincidence? At first glance, most mathematicians are tempted to dismiss this conjecture as "almost certainly false." However, a closer inspection reveals that it is quite likely to be true. Since Q,1 is known to be prime, Fn = Pn cannot be divisible by p 1, p2,. . .,pn. Thus Fn ,> p + I for all n, with equality observed at n= 1,2,3,6,7,8,14,16,17,.... On the other hand, so long as F,
10 citations
TL;DR: The existence of hyperperfect numbers with more than two different prime factors is shown by five examples as mentioned in this paper, each of which has the form paq, where p and q are prime numbers, p < q and a E N.
Abstract: The existence of hyperperfect numbers with more than two different prime factors is shown by five examples. Recently, Minoli [2] has defined n-hyperperfect numbers as positive integers m such that there is some positive integer n with (1) m = 1 + n[a(m)-m-1]. 1-hyperperfect numbers are the classical perfect numbers. Minoli gives a list of all n-hyperperfect numbers 1, and these numbers have the form paq, where p and q are prime numbers, p < q and a E N. Minoli wonders whether all hyperperfect numbers might have this form. By using a well-known technique, which was used, for instance, by Euler [1] to compute amicable number pairs, we have computed five hyperperfect numbers, each with three different prime factors. Let m = pqr, p < q < r prime numbers, be an n-hyperperfect number. By (1) we have pqr = 1 + n(pq + pr + qr + p + q + r). Now, if we assume that p and n are given, this is a quadratic equation in q and r. We write it as (p n)qr n(p + 1)q n(p + 1)r = 1 + np. Multiplying by (p n), and adding n2(p + 1)2 to both sides yields [(p n)qn(p + 1)][(p n)rn(p + 1)] = (p -n)(1 + np) + n2(p + 1)2. If AB, A < B, is a factorization of the known right-hand side, then we can write q =[n(p + 1) + A]/ (pn), r = [n(p + 1) + B]/ (pn). If now both q and r are integers and prime, then pqr is an n-hyperperfect number. Clearly, a small value of (p n) will facilitate finding integers q and r. The simplest choice isp n = 1; this gives (2) q = p -1 + A, r = p2 _ 1 +B, AB = p4 p2 -p + 2, A < B. If A = 1, then q = p 2, not a prime. If p _ 2 (mod 3), then p2 1 0 (mod 3) and AB = p4 p2-p +2 (mod 3), so that 3 1 A or 3 1 B; hence, at least one of q Received June 5, 1980. 1980 Mathematics Subject Classification. Primary IOA20.
10 citations
TL;DR: The primitive elements of a finite field are those elements of the field that generate the multiplicative group of k as discussed by the authors, and if a polynomial over k of small degree compared to the size of k represents at least one primitive element of k, then f(x) represents an lth power at a primitive element if l is also small.
9 citations
TL;DR: In this paper, a class of theorems of the type "q is a prime number iff R(q) is a divisor of the binomial coefficient" were presented.
Abstract: We find a class of theorems of the type “q is a prime number iff R(q) is a divisor of the binomial coefficient\(\left( {\begin{array}{*{20}c} {S(q)} \\ {T(q)} \\ \end{array} } \right)\) ”; here R, S, T are certain fully significant functions that are superpositions of addition, subtraction, multiplication, division, and raising to a power. Similar criteria were also obtained for prime Mersenne numbers, prime Fermat numbers, and twin-prime numbers.
9 citations
TL;DR: In this paper, some theorems on n × n circulant matrices were introduced under the hypothesis n a prime number and extended to the case n = 2⋅ r where r is a prime greater than 2.
4 citations
Patent•
18 Dec 1981
TL;DR: In this paper, an input/output device (28) is connected to an L/O device for converting the input data from the constant radix number system to the prime or relatively prime number system.
Abstract: This system is configured to operate in a number system in which the radix of each digit is a different prime number, or a system in which all of the radices are relatively prime, hereafter a prime or relatively prime radix number system. The system includes an input/output device (28) which inputs data in a constant radix number system and outputs results of operations carried out in the system in a constant radix number system. A means (32) is connected to the l/O device (28) for converting the input data from the constant radix number system to the prime or relatively prime radix number system. A processing means (12, 14-1 to 14-K) is connected to the converting means (32) for carrying out operations on the input data in prime or relatively prime radix form. A memory means (18-1 to 18-K) connected to the processing means (12, 14-1 to 14-K) stores the data and results of operations thereon in prime or relatively prime radix form. Since there is no carry required to perform arithmetic operations except divide in the prime or relatively prime radix number system, such operations are substantially simplified in comparison with corresponding operations with a constant radix number system.
TL;DR: In this paper, an asymptotic formula for the sum of the sums of the elements of the continued fractions for numbers of the forma/p (p is a prime number).
Abstract: One obtains an asymptotic formula for the sum of the sums of the elements of the continued fractions for numbers of the forma/p (p is a prime number).
TL;DR: In this paper, an application of Tauberian theorems to the problem of connection between the mean values of multiplicative functions for the cases when the argument runs through all natural numbers and all prime numbers, respectively, is given.
Abstract: One gives an application of Tauberian theorems to the problem of connection between the mean values of multiplicative functions for the cases when the argument runs through all natural numbers and all prime numbers, respectively.
TL;DR: Using I. M. Vinogradov's method, one obtains an asymptotic formula regarding the distribution of the fractional parts as mentioned in this paper, where P runs through the prime numbers.
Abstract: Using I. M. Vinogradov's method, one obtains an asymptotic formula regarding the distribution of the fractional parts
where
is a constant and P runs through the prime numbers.
Journal Article•
TL;DR: In this article, the relationship between memory contention and (a,M), the greatest common divisor of a and M, is investigated, where M is the number of parallel memory modules, and a is a constant increment of address.
Abstract: In this paper the relationship between memory contention and (a,M), the greatest common divisor of a and M, is investigated, where M is the number of parallel memory modules, and a is a constant increment of address. A quantitative representation of average efficiency against M is derived. It shows that if M is a prime number, the efficiency is higher than that when M equals 2~n.
1 Jan 1981
TL;DR: In this article, the Riemann hypothesis implies that dn = O(pn112logpn), and there has been much interest recently in estimating f(l /2) (see [2], [11], [12], [16] and [20]).
Abstract: fl ){0 for, p> 1/2, 1/2. Since the Riemann hypothesis implies that dn = O(pn112logpn), Erdos asked whether f(l /2) < 1 and there has been much interest recently in estimating f(l /2) (see [2], [11], [12], [16] and [20]). Heath-Brown [7] has given an unconditional proof that
TL;DR: By listing the positions of the l's in the binary representation of Gandhi's summation, the prime p and all primes succeeding it which are less than p can be found from the primes preceding it, and a large number of primes can be calculated from relatively few.
Abstract: Gandhi's formula, which allows the odd prime p to be calculated, by summation, from the primes preceding it, is not an immediately suitable problem for computer solution since for p > 7 the terms of the summation become smaller than 10~ and cannot be added to unity, and for;? > 11 the terms are smaller than the limits of the floating-point system in use (10~). By representing Gandhi's formula in the binary system however, I am here going to show that the formula gives rise to a rather interesting algorithm for the computation of the prime p, and (more importantly), by listing the positions of the l's in the binary representation of Gandhi's summation, the prime p and all primes succeeding it which are less than p can be found from the primes preceding it. Thus a large number of primes can be calculated from relatively few. Referring to the proof of Gandhi's formula given by C. V. Eynden (Amer. Math. Monthly, 1972, 79, p. 625), Gandhi's inequalities
TL;DR: In this article, it was proved that the equation xp + yp + Zp = q, (xyz, p)=1 has no solutions in rational integers x, y, z for all odd prime numbers p for which q =pk + 1 is a prime number, k ⩽ 82, k ≠ 0 (mod 3).
Abstract: It is proved that the equation xp + yp + Zp=Q, (xyz, p)=1 has no solutions in rational integers x, y, z for all odd prime numbers p for which q=pk + 1 is a prime number, k ⩽ 82, k ≠ 0 (mod 3).
1 Jan 1981
TL;DR: In this paper, conditions sufficient for a matrix to be prime are provided, and a characterization of primes in terms of a notion of rank is given, and an order property of prime factors is used to obtain a result on prime factors.
Abstract: Let A, B, C be R X n matrices of zeros and ones. Using Boolean addition and multiplication, we say that A is prime if it is not a permutation matrix and if A =BC implies that B or C must be a permutation matrix. Conditions sufficient for a matrix to be prime are provided, and a characterization of primes in terms of a notion of rank is given. Finally, an order property of primes is used to obtain a result on prime factors.
Dissertation•
1 Jan 1981
TL;DR: In this article, the authors considered the problem of finding an upper bound on the number of sign changes of a strongly additive arithmetical function and showed that the upper bound can be computed using a convolutional identity.
Abstract: The thesis is divided into five sections:
(a) Trigonometric sums involving prime numbers and applications,
(b) Mean-values and Sign-changes of S(t)-- related to Riemann's Zeta function,
(c) Mean-values of strongly additive arithmetical functions,
(d) Combinatorial identities and sieves,
(e) A Goldbach-type problem.
Parts (b) and (c) are related by means of the techniques used but otherwise the sections are disjoint.
(a) We consider the question of finding upper bounds for sums like
∑_PSN▒〖e(ap2)〗
and using a method of Vaughan, we get estimates which are much better than those obtained by Vinogradov. We then consider two applications of these, namely, the distribution of the sequence (αp2) modulo one.
Of course we could have used the improved results to get improvements in estimates in various other problems involving p[superscript]2 but we do not do so.
We also obtain an estimate for the sum
∑_PSN▒〖(ap3)〗
and get improved estimates by the same method.
(b) Let N(T) denote the number of zeros of ς(s) - Riemann's Zeta function. It is well known that
N(T) = L(T) + S(T),
where
L(T) = 1/2π Tlog(T/2π)-T⁄(2π+7⁄(8+0 ((1)⁄(T))))but the finer behaviour of S(T) is not known. It is known that
S(t) ≪ log t ; ∫_o^t▒〖Slu)du〗 ≪ log t
so that S(T) has many changes of sign. In 1942, A. Selberg showed that the number of sign changes of Set) for t ∈ (O,T) exceeds
T (log T)1/3 exp(-A loglog T), (1)
but stated to Professor Halberstam in 1979 that one can improve the constant 1/3 in (1) to 1 – ∈. It can be shown easily that the upper bound for the number of changes of sign is log T.
We give a proof of Selberg's statement in (b), but in the process we do much more. Selberg showed that if k is a positive integer then
∫_T^(T+H)▒〖ls(t)l〖2k〗_dt 〗 = C CkH(loglog T) k ,{1+0( (loglogT)(-1)/2) } (2)
where TT 1⁄2< H ≤ T[superscript]2 and C[subscript]k is some explicit constant in k. We have found a simple technique which gives (2) with the constant k replaced by any non-negative real number. Using this type of result, I prove Selberg's statement, with
(log T)-∈ replaced by
Exp (-A√loglogT (logloglogT) -□(1/2)).
(c) I use the" method for finding mean-values above to answer similar questions for a class of strongly additive arithemetical functions.
We say that f is strongly additive if
(1) f(mn) = f(m) = f(n), if m and n are coprime,
(2) f(p[superscript]a) = f(p) for all primes p and positive integer a.
(d) This section contains joint work with Professor Halberstam and is still in its infancy. We have found a general identity and a type of convolution which serves to be the starting point of most investigations in Prime Number Theory involving the local and the global sieves. The term global refers to sieve methods of Brun, Selberg, Rosser and many more. The term local refers to things like Selberg's formula in the elementary proof of the prime number theorem, Vaughan's identity and so on. We have shown that both methods stem from the same source and so leads to a unified approach to such research.
(e) I considered the question of solving the representation of an integer N in the form
N = P_(1^2 )+ P _(2^2 ) +K P[subscript] 3,
where the Pi’s are prime numbers. This problem was motivated by Goldbach's Problem and is exceedingly difficult. So I looked into getting partial answers.
Let E(x) denote the numbers less than x not representable in the required form. Then there is a computable constant δ > 0
such that
E(x) ≪ X i- δ
To do this we use a method of Montgomery and Vaughan but the proof is long and technical, and we do not give it here.
We show by sieve methods that the following result holds true:
N = P_(1^2 ) + P _(2^2 ) +kP3P4P5.
We have been unable to replace the product of three primes by two.
Note: k is a constant depending on the residue class of N modulo 12.
TL;DR: The main results of the present paper are as follows: as mentioned in this paper showed that the distance of x from the nearest integer can be expressed in terms of a positive constant depending at most on an arbitrary positive number e and on an integer k. The letter p always denotes a prime number.
Abstract: We write e(x) for e2πix and let ‖x‖ denote the distance of x from the nearest integer. The notation A ≪ B will mean |A| ≤ C|B| where C is a positive constant depending at most on an arbitrary positive number e, and on an integer k. The letter p always denotes a prime number. The main results of the present paper are as follows.
TL;DR: A new algorithm is presented for the problem of finding all primes between 2 and N that improves on Mairson's sieve algorithm by using a dynamic sieve technique that avoids most of the nonprimes in the range 2 to N, and byUsing a tabulation method to simulate multiplications.
Abstract: A new algorithm is presented for the problem of finding all primes between 2 and N. It is based on Mairson's sieve algorithm which uses θ(N) additions and multiplications. The new algorithm improves on this algorithm by using a dynamic sieve technique that avoids most of the nonprimes in the range 2 to N, and by using a tabulation method to simulate multiplications. It is shown to require θ(N/log log N) additions. A related algorithm is outlined that has the same complexity but a storage requirement of only θ(N/log log N) bits.
1 Jan 1981
TL;DR: In this paper, Morita proved that for each prime number p, one can define a p-adic continuous function from p to p, interpolating the sequence where m runs through the integers m prime to p with 1 ≤ m < n.
Abstract: Y. Morita proved that, for each prime number p, one can define a p-adic continuous function Γp(x) from p to p, interpolating the sequencewhere m runs through the integers m prime to p with 1 ≤ m < n. Our aim is to show how this result is related to Dwork's result on the radius of convergence of