Showing papers on "Prime number published in 1976"
TL;DR: In this article, it was shown that the number of primes in an interval (n, n + h), averaged over n ≤ N, tends to the limit λ, when n and h tend to infinity in such a way that h ∼ λ log N, with λ a positive constant.
Abstract: One of the formulations of the prime number theorem is the statement that the number of primes in an interval (n, n + h], averaged over n ≤ N, tends to the limit λ, when N and h tend to infinity in such a way that h ∼ λ log N, with λ a positive constant.
173 citations
TL;DR: The first few irregulär primes are 37, 59, 67, 101, 103, 131, 149 (see as discussed by the authors for a short list) and are known to be infinite; a short proof for this can be found in [1], pp. 381-382.
Abstract: The first few irregulär primes are 37, 59, 67, 101, 103, 131, 149 (for longer lists, see [7], or [1], pp. 430—431, and [4]). The number of irregulär primes is known to be infinite; a short proof for this, due to Carlitz [2], can be found in [1], pp. 381—382. Jensen [3] proved in 1915 that there exist infinitely many irregulär primes l (mod 4). Montgomery [6] generalized this by replacing 4 by any integer > 2. In this note we shall give a further generalization, by proving the following theorem.
11 citations
1 Jan 1976
TL;DR: In this article, the indecomposability of positive definite binary quadratic lattices was studied and shown to be equivalent to positive definite quadrastic lattices with indecomposition.
Abstract: THEOREM 1. Let Li9Mi be indecomposable positive definite binary quadratic lattices with Lt = Lί9 Mt = Mi9 m(Li) = miMi) = 1. For any ίsometry a: (x)?=i Lt = (x)?=i Mi9 we have a — (x)?=i at where Gt is an isometry from Li on Mu changing the suffix if necessary. THEOREM 2. Let Li9 Mt be positive definite quadratic lattices with [Lii LJ < oo, [Mi: Mi\ < oo. Assume that ( i ) Li (resp. Mt) is of E-type except at most one, (ii) sLi = sMt = Z, and m(Li),m(Mi) are prime numbers, and (iii) Lί9Mi are indecomposable. Then for any isometry σ: (x)?=i Lt = (x)?Li Mt we have n = m and σ = (x) σί9 where at is an isometry from Li on Mi9 changing the suffix if necessary.
9 citations
Patent•
IBM1
TL;DR: In this paper, a digital filter is described, which includes a circular convolution device using the complex Mersenne transform to convert a sequence of values A n into another sequence A k in which P is prime number and J is square root of zero.
Abstract: A digital filter is disclosed which includes a circular convolution device using the Complex Mersenne transform to convert a sequence of values A n into another sequence A k in which ##EQU1## WHERE P IS PRIME NUMBER AND J IS THE SQUARE ROOT OF MINUS ONE. The convolutor is provided with an input for applying fixed length data blocks made up of input samples appended with an equal number of zeros; circuits for recirculating and accumulating said data; a register for storing said accumulated data; switches for selectively connecting the output of the storage to the inputs of an adder-subtractor; a product device for term-by-term multiplying of the output of the adder-subtractor with the Complex Mersenne transforms of the filter coefficients set appended with zeros; and an inverse transform device for performing the inverse Complex Mersenne transform on the multiplier output blocks of data. The filtered output samples are provided by adding the inverse Mersenne transforms performed on two consecutive data blocks.
9 citations
8 citations
1 Jan 1976
TL;DR: In this article, a more elegant martingale approach was used to obtain sharper results for the Hawkins random sieve, which is a probabilistic analogue of the sieve of Eratosthenes.
Abstract: This paper is concerned with the Hawkins random sieve which is a probabilistic analogue of the sieve of Eratosthenes. Analogues of the prime number theorem and Mertens' theorem have previously been obtained for this sieve by classical probabilistic methods. In the present paper, sharper results akin to the Riemann hypothesis are obtained by a more elegant martingale approach. The following random sieve procedure introduced by Hawkins [1], [2] is a stochastic analogue of the sieve of Eratosthenes: Let A1 = {2, 3, 4, 5, 6, . . . } . Stage 1. Put X1 = min A1. From the set Aj\{X1} each number in turn is (independently of the others) deleted with probability XI-' or not deleted with probability 1 X-'. The set of elements of Al\{X1} which remain is denoted by A2. Stage n. Put Xn = min An. From the set An\{XJ} each number in turn is (independently of the others) deleted with probability Xn-1 or not deleted with probability 1 -Xn. The set of elements of An\{Xn} which remain is denoted by An+1 Define Yn = rI (1 -Xk 1f1 1?n? Wunderlich [5], [6] has obtained the results lim (n log n) IXn = 1 a.s., and lim (log n) 1 Yn = 1 a.s. n-*oo n-*oo which are analogues of the prime number theorem and Mertens' theorem respectively. These results have been obtained by classical probabilistic methods involving detailed estimation of moments. In this paper we shall obtain sharper results, akin to the Riemann hypothesis, using a more elegant martingale approach. These results are given in the following theorem and are different in character from those obtained for a diffusion analogue by Williams [4]. THEOREM. (i) limnc (log log n) (n-'Xn log n) = 1 a.s., (ii) limn(log log n) (1 Ynlog n) = 1 a.s. Received by the editors March 11, 1975. AMS (MOS) subject classifications (1970). Primary 60F15, 1OH30; Secondary 60J05.
8 citations
8 citations
TL;DR: In this paper, it was shown that it is sufficient that P be met-Abelian and have nilpotence class
Abstract: Thompson [1] showed that if p is an odd prime number, A is a p-group of operators of the finite group P in which the Frattini subgroup Φ(P) is elementary and central, and P/Φ(P) is a free ZpA-module, then Cp(A) covers CP/Φ(P)(A). Then he proposed the question of whether it is possible in this theorem to weaken the hypothesis that Φ(P) be elementary and central. In the work it is shown that this hypothesis may be replaced by a much weaker one; it is sufficient that P be met-Abelian and have nilpotence class
8 citations
1 Jan 1976
TL;DR: The prime number theorem is equivalent to the statement as discussed by the authors, where ψ(x) is Chebyshev's function and ψ (x) = √ √ n \leqslant x.
Abstract: The prime number theorem is equivalent to the statement
$$\psi (x) \sim xasx \to \infty ,$$
(1)
where ψ(x) is Chebyshev’s function,
$$\psi (x) = \sum\limits_{n \leqslant x} {\Lambda (n)} .$$
5 citations
1 Feb 1976
TL;DR: In this article, it was shown that the order of the divisor classes of the field of 5th roots of unity is bounded independent of n. Theorem 1.1.
Abstract: Let h~ be the relative class number of the field of 5"th roots of unity. If / is any prime number, then the /-part of h~ is bounded independent of n. Let A: be a number field, K/k the cyclotomic Z^-extension of k, and kn the unique intermediate field of degree p" over k. In [4] it was conjectured that if / ^ p is any prime number then the /-part of the class number of kn is bounded independent of n. This conjecture arose from analogy with the case of function fields over finite fields, where Zp-extensions can be obtained by extending the field of constants. In this case it is not difficult to show that the /-part of the order of the group of divisor classes of degree zero is bounded independent of n [4]. For number fields, the conjecture has been proved when k/Q is abelian and p = 2 or 3. The main obstacle for larger primes p is the existence of p-adic (p — l)st roots of unity, which are, of course, harder to handle when p > 5. In this note we attack the case of the simplest Z5-extension, namely the one obtained by adjoining all 5"th roots of unity, for all n > 1, to the field of 5th roots of unity. Recall that for any imaginary abelian number field K there is a maximal real subfield K +, and since K/K+ is totally ramified (at oo) the class number h+ of K+ divides the class number h of K. The quotient h/h+ is called the relative class number h ~. Theorem. Let h~ be the relative class number of Q(f5n), where f5„ is a primitive 5"th root of unity. Let I be any prime number and let le\\h~. Then e~ is bounded as n -^ oo. In fact, if m, is determined by 5m'||/4 — 1 and n > 2m, + 2, then l\h~/h~_v (la\\b means la\b, la+iJfb). Proof. Since 5 is a regular prime, 5\h~ for any n [2], Therefore, we assume / ¥= 5. The set of odd Dirichlet characters of Q(iV) is obtained as follows: Let Xp X2 be the odd characters for Q(f5)/Q and let xpn be a character of conductor 5" such that xpn(a) depends only on a4 mod 5" (therefore \\>n generates the Received by the editors March 1, 1976. AMS (MOS) subject classifications (1970). Primary 12A50, 12A35. 'Partially supported by NSF Grant MPS74-07491A01. © American Mathematical Society 1977 205 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 206 L. C. WASHINGTON characters of the subfield of Q(£5«) of degree 5"_1 over Q). The odd characters of Q(f5~) are then {X/U^}' where / = 1, 2 and 0 2m, let Tr be the trace function from Fn to Fm. We shall calculate Tr( j 5, ^), where x = Xi or XiNote that Tr(fy) = 0 if c > m. Therefore, Tr(Ua))^0^U"f= l^U"5")= 1 fl4'5" = 1 mod 5" ** a4 = 1 mod 5"'"', in which case Ar(xpn(a)) = 5"~mxpn(a). Therefore,
TL;DR: In this article, the coefficients ξi(rm),i= 1,2, as functions of the powers of the prime numbers ofm and of g. c. d. (r, m) are determined.
Abstract: Letd(n) denote the number of divisors ofn, then the asymptotic formula
$$\sum\limits_{\mathop {n< x}\limits_{n = r(\bmod m)} } {d(n) = \xi _1 (r,m)} x\log x + \xi _2 (r,m)x + O(x^{1/2} )$$
is derived and, as the main result of the paper, the coefficients ξi(rm),i= 1,2, as functions of the powers of the prime numbers ofm and of g. c. d. (r, m) are determined.
TL;DR: The Prime Number Theorem was proved in this paper using properties of the Dirichlet series Σ n = 1 ∞ n −8 in its half plane of convergence, and simple facts of harmonic analysis.
1 Jan 1976
TL;DR: The behavior of π(x) as a function of x has been the object of intense study by many celebrated mathematicians ever since the ighteenth century as discussed by the authors, which led Gauss (1792) and Legendre (1798) to conjecture that π (x) is asymptotic to x/log x.
Abstract: If x > 0 let π(x) denote the number of primes not exceeding x. Then π(x) → ∞ as x → ∞ since there are infinitely many primes. The behavior of π(x)as a function of x has been the object of intense study by many celebrated mathematicians ever since the ighteenth century. Inspection of tables of primes led Gauss (1792) and Legendre (1798) to conjecture that π(x) is asymptotic to x/log x, that is
$$\mathop {\lim }\limits_{x \to \infty } \frac{{\pi \left( x \right)\log x}}{x} = 1. $$
TL;DR: In this article, the authors show that for any well-formed formulas u and v of commutative sentential general recursive calculus, the following inference rules are valid in network J f f E. (1) (Translation Rule of FE-Deductiori) If v is derivable in
Abstract: (H 3) For any well-formed formulas u and v of commutative sentential general recursive calculus ifFE, any FE-translation a[«], any FE-consorts ξ[ι/] and ξ[ι?], and either any FEtranslation or FE-consort denoted s p[t?], the following two inference rules are valid in network J f f E . (1) (Translation Rule of FE-Deductiori) If v is derivable in ^FE from a[w], then we infer that the implication a[w] => p[i>] is a theorem of £fE. (2) (Consortion Rule of FE-Deductiori) If v is derivable in
TL;DR: The first even perfect numbers were given by Euclid in the form of 2P-1(2P 1), where is a prime and 2P? 1 is also a prime as mentioned in this paper.
Abstract: Mersenne numbers are prime or composite. A number is called perfect if it is equal to the sum of its divisors, not including itself. As an example, 6 is a perfect number, since its only divisors are 1, 2, and 3 (besides 6) and these sum to 6. Similarly, the divisors of 28, excluding 28 itself, are 1, 2, 4, 7, and 14, and these sum to 28. These numbers were considered perfect, substantiated by the creation of the universe in 6 days and of the lunar cycle in 28 days. Over 2 000 years ago, Euclid proved that every number of the form 2P-1(2P 1), where is a prime and 2P ? 1 is also a prime, must be an even perfect number. As an example, take = 2. Since 22 ? 1 = 4 ? 1 = 3 is a prime, = 2 may be used in the formula to yield a perfect number. Sub stituting, we obtain 22_1(22 1) = 2 X 3 = 6, the first even perfect number. In the eighteenth century, Euler proved that Euclid's formula gives all even perfect numbers. The first five perfect numbers given by Euclid's formula are 6, 28, 496, 8 128, and 33 550 336. According to Euclid's formula, we are only allowed to substitute primes such that 2P 1 is also prime. Let Mp stand for the number 2P 1; thus Mz = 23 1 = 7, M5 = 2s 1 = 31, and so on. If Mp is a prime, we call that particular Mp a Mer senne prime, named after the seventeenth century mathematician Marin Mersenne. As an example, Mz = 23 1 = 7, a prime; hence 7 is a Mersenne prime. Up to the present time the only known values of the prime that yield Mersenne primes are 2,3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1 279, 2 203, 2 281, 3 217, 4 253, 4 423, 9 689, 9 941, 11 213, 19 937 (24 values). It is to be understood that the omitted primes between these do not yield Mersenne primes; for example, for = 11, Mn = 211 1 = 2 047 = 89 23. In addi tion, if is composite, it is clear that Mn is composite; in fact, 2ab 1 (a and b greater than 1) is divisible by Ia 1. In particular then, if is even and larger than 2, Mn is composite. In 1772, Euler proved that Mzl = 2 147 483 647 is a prime, and this number became the largest known prime for more than 100 years. E. V. Lucas (1877, pp. 184-239, 289-321) developed a beautiful test to determine, for a given prime /?, whether Mp is prime or not. The result was later modified and the proof simplified by D. H. Lehmer (1930, pp. 419-48). Consider the following sequence of numbers, which is called the Lucas-Lehmer sequence: 4, 14, 194, 37 634, 1 416 317 954, and so on. This sequence satisfies the recurrence relation Si = 4, Sk+1 = Sk2 2, k = 1, 2, 3,. ... Thus 14 = 42 2, 194 = 142 2, and so on. The following statement can be proved:
1 Jan 1976
TL;DR: In this article, it was shown that if F is a p-saturated formation of solvable groups, then every solvable group G has an F-projector K with (P, [G:K]) = 1.
Abstract: Publisher Summary This chapter presents the proof of the theorem as per which if F be a p-saturated formation of solvable groups, F can be p-locally defined. For formations of solvable groups, it is also possible to prove that if F is a p-saturated formation, then every solvable group G has an F-projector K with (P. [G:K]) =1. A theorem presented in the chapter states that if G be a solvable group, a subgroup K of G is a PN-projector of G if and only if K is a self-normalizing p–n subgroup with (p, [G:K])=1. In light of this theorem, the only formation of solvable groups that is p-saturated for every prime number p is the class of all solvable groups.
1 Jan 1976
1 Jan 1976
TL;DR: In this paper, a sharpening of Moore's theorem is proposed, which implies that the unique root of unity in a global field which modulo the maximal ideal is congruent to a given function field over a finite field is a prime divisor of F which is not complex archimedean.
Abstract: 1 A sharpening of Moore 's theorem, Let F be a global field, i e, a finite extension of Q or a function field in one variable over a finite field The multiplicative group of F is denoted by F , the group of roots of unity in F by y, and its finite order by m By a prime v of F we shall always mean a prime divisor of F which is not complex archimedean · If v is non-archimedean, then we also use the symbol v to denote the associated normalized exponential valuation For a prime v of F, let FV be the completion of F at v The group of roots of unity in F is'called μ, and its finite order m(v) The m(v)-th power norm residue symbol F* x FV -»μν is denoted by ( , )v> For all but finitely many v this map is given by the so-ealled "tarne formula", cf [l, sec 1], This formula implies that, for those v, and for all a, b e F · with v(a) «= 0, the symbol (a, b) is the unique root /*!· \ of unity in F which modulo the maximal ideal is congruent to a It follows