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An interesting way to find the smallest prime factor of a given natural number. A computer program for this procedure is supplied.

LetG andK be finite groups whose orders have a common prime divisor. Then there is a groupK * closely related toK for which there is a non-split extension ofK * byG.

Let A be a positive or negative rational integer such that integers in the field of √1 − 4A have unique prime factorization. An elementary criterion will be obtained for x2 + x + A to be a prime number, where x is a positive integer. The criterion implies that for positive A the polynomial x2 + x + A is prime for x = 0, 1,…, A − 2.

Let H denote the set of all the entire functions f ( z ) f(z) of the form: f ( z ) ≡ h ( z ) e p ( z ) + k ( z ) f(z) \equiv h(z){e^{p(z)}} + k(z) where p ( z ) p(z) is a nonconstant polynomial of degree m, and h ( ≢ 0 ) h(\nequiv \;0) , k ( ≢ k(\nequiv constant) are two entire functions of order less than m. In this paper, a necessary and sufficient condition for a function in H to be a prime is established. Several generalizations of known results follow. Some sufficient conditions for primeness of various subclasses of H are derived. The methods used in the proofs are based on Nevanlinna’s theory of meromorphic functions and some elementary facts about algebraic functions.

Denote by ω ( n ) \omega (n) and Ω ( n ) \Omega (n) the number of distinct prime factors of n n and the total number of prime factors of n n , respectively. Given any positive integer α \alpha , we prove that \[ ∑ 2 ≦ n ≦ x Ω ( n ) / ω ( n ) = x + x ∑ i = 1 α a i / ( log ⁡ log ⁡ x ) i + O ( x / log ⁡ log ⁡ x ) α + 1 ) , \sum \limits _{2 \leqq n \leqq x} {\Omega (n)/\omega } (n) = x + x\sum \limits _{i = 1}^\alpha {{a_i}/{{(\log \log x)}^i} + O{{(x/\log \log x)}^{\alpha + 1}}),} \] where a 1 = ∑ p 1 / p ( p − 1 ) {a_1} = \sum \nolimits _p {1/p(p - 1)} and all the other a i {a_i} ’s are computable constants. This improves a previous result of R. L. Duncan.

JVLost children are aware that some numbers are even and the rest are odd, but they know little else about our number system. The "board-stretcher" model provides a means to discovering the idea of factors and obtaining a feeling for prime and composite numbers. It portrays the inverse relation between multiplication and division and, as a finale, it can be used to introduce the concept of multiplication by a fraction. This model suggests the notion that the sum of the prime factors of a number (except for four) is always less than the product of the factors. The Fundamental Theorem of Arithmetic (or Unique Factorization Theorem) is understood by youngsters when they see that every composite number can be expressed as a product of prime factors that is unique except for the order of the factors. The board-stretcher illustrates an idea expressed by the late Dr. Max Beberman in a talk that I heard him give in Chicago in 1964. The "board-stretcher"

We prove imbedding theorems for block designs balanced with respect to pairs, and with the aid of these theorems we establish the existence of (v, k, λ)-resolvable BIB block designs with parameters v, k, λ such that λ=k−1 [and also such that λ=(k−l)/2 if k is odd], k ¦(p−1) for each prime divisor p of the number v/k; we also establish an imbedding theorem for Kirkman triple systems.

This paper investigates a new procedure for solving the general-variable pure integer linear programming problem. A simple transformation converts the problem to one of constructing nonnegative integer solutions to a system of linear diophantine equations. Rubin's sequential algorithm, an extension of the classic Euclidean algorithm, is used to find an integer solution to this system of equations. Two new theorems are proved on the properties of integer solutions to linear systems. This permits a modified Fourier-Motzkin elimination method to be used to construct a nonnegative integer solution. An experimental computer code was developed for the algorithm to solve some test problems selected from the literature. The computational results, though limited, are encouraging when compared with the Gomory all-integer algorithm.

On the elements of the ring of residues modulo v (zτ v, 3τ v) we construct cyclic PBIB-designs with τ(v)-1 classes of connectedness, where τ(v) is the number of divisors of v. We prove the existence of cyclic BIB-designs with parameters b, v, r, k, and λ such that: 1) λ=k (and also λ=k/2 if k is even), k≥4, and (k-1) ¦ (p-1) for each prime divisor p of the number v; 2) λ=(k−l)/2, k odd, k≥3, k ¦ (p−1) for each prime divisor p of the number v.

In recent years several papers [1; 3; 4; 5; 6; 7; 9; 10; 11] have appeared dealing with the problem of "Factorisatio numerorum," the number f(n) of representations of an integer n as an ordered product of factors greater than 1. As a result, the basic combinatorial properties of f(n) and the asymptotic behavior of its summatory function are well known. In this paper, I determine the asymptotic behavior of f(n) itself for squarefree n and use the result to determine a "normal" order of f(n) for all n. If n is written as a product of powers of distinct primes pi, then f(n) is evidently a symmetric function of the exponents of the pi's. If n is squarefree, all the exponents are 1 and f(n) can be considered a function of the single variable r, the number of distinct prime factors of n. It is therefore convenient to define, for positive integral r

We propose to give a computationally feasible procedure for the generation of all the integer points satisfying a given set of inequalities. Five different systems of inequalities will be considered. In order to generate all of these integer points, one requires a particular set of integer points, called fundamental points, and a set of linearly independent vectors with integer components. The number of these fundamental points is given by a simple formula. We show how to generate the fundamental points and the required vectors. We give an application concerning the localization of the integer optimum of a linear objective function subject to constraints which geometrically define a cone or a parallelotope.

as x ~ ~ , for which as for many other interesting results in the theory of numbers we are indebted to Chebyshev, has a t t racted the interest of several mathematicians. Revealed posthumously as little more than a fragment in one of Chebyshev's manuscripts, the theorem was first published and fully proved in a memoir by Markov in 1895 [6], while later in the same year a generalisation by Ivanov [4] appeared in which the polynomial n2§ 1 was replaced by n2+A for any positive A (an account of both Markov's and Ivanov ' s work is to be found in Paragraphs 147 and 149 of Landau's Primzahlen [5]). In 1921 Nagell [7] improved and further generalised Chebyshev's theorem by shewing that for any e log~ x, X

A testing machine has been designed and developed to investigate the performance of a complete journal bearing under a vertical cyclic load of the form P = P0 + P1 sin (ω1 t)+ P n sin (nω1t + φ). This paper presents the results of the experimental study of the bearing performance for the case where Pn = 0 and for values of P1/ P0 between infinity and zero inclusive. Test results show that the ratio of the rate of load application to that of journal rotation (ω1/ω) is a prime factor in the bearing behaviour. Values of this ratio slightly less than 0.5 give rise to critical changes in the bearing performance, of a character mainly dependent on the ratio of P1 to P0. The results do not, in general, confirm the theory put forward by Swift; they do, however, indicate closer agreement with some recent theoretical studies. Further work on the subject is proceeding.

An algorithm is given for solving an integer program over an additive group. Computation times appear to grow more favorably with increases in the number of variables and group elements than with the dynamic programming approach proposed by Gomory. A new property satisfied by optimal solutions to the group problem is established by reference to the structure of the algorithm. Extension of the algorithm to the general integer programming problem is developed in a sequel.

1. In deriving an expression for the number of representations of a sufficiently large integer N in the formwhere k: is a positive integer, s(k) a suitably large function of k and pi is a prime number, i = 1, 2, …, s(k), by Vinogradov's method it is necessary to obtain estimates for trigonometrical sums of the typewhere ω = l/k and the real number a satisfies 0 ≦ α ≦ 1 and is “near” a rational number a/q, (a, q) = 1, with “large” denominator q. See Estermann (1), Chapter 3, for the case k = 1 or Hua (2), for the general case. The meaning of “near” and “arge” is made clear below—Lemma 4—as it is necessary for us to quote Hua's estimate. In this paper, in Theorem 1, an estimate is obtained for the trigonometrical sumwhere α satisfies the same conditions as above and where π denotes a squarefree number with r prime factors. This estimate enables one to derive expressions for the number of representations of a sufficiently large integer N in the formwhere s(k) has the same meaning as above and where πri, i = 1, 2, …, s(k), denotes a square-free integer with ri prime factors.

Title of program: COEF Catalogue number: ACRE Program obtainable from: CPC Program Library, Queen’s University of Belfast, N. Ireland (see application form in this issue) Computer: ICL 1901A, Installation: University of Port Elizabeth Operating system: ICL automatic operator Programming languages used: FORTRAN IV High speed store required: 14080 words. No. of bits in a word: 24 Is the program overlaid? No No. of magnetic tapes required: None What other peripherals are used? Card Reader; Line Printer No. of cards in combined program and test deck: 625 Card punching code: ICL 1900

1. Let d ( n ) denote the number of divisors of the positive integer n , so that, if n = p 1 a 1 . . . p r ar is the canonical expression of n in prime factors, d ( n ) = (1 + a 1 ) . . . (1 + a r ), and let d ( x ) = 0 if x is not an iteger; then if (1. 1) D ( x ) = Σ' n ≤ x d ( n ) = Σ n ≤ x d ( n ) ─ ½ d ( x ), and (1. 2) Δ ( x ) = D ( x ) ─ x log x ─ (2C ─ 1) x ─ ¼, where C is Euler's constant, it was proved by Dirichlet in 1849 that (1. 21) Δ ( x ) = O (√ x )

This paper shows how the asymptotic structure of the integer programming problem extends to the mixed integer problem. The group-theoretic ideas are then used to provide information about the solutions of the asymptotic mixed integer problem, and it is also shown how useful bounds can be obtained from the corresponding all-integer asymptotic problem. An example shows the application of some of the results.

We consider a Beurling generalized number system for which the counting function of the integers satisfies \[ N ( x ) = c x + O ( x log − γ x ) N(x) = cx + O(x{\log ^{ - \gamma }}x) \] for some positive c c and γ \gamma . Beurling showed that the prime number theorem must hold if γ > 3 2 \gamma > \tfrac {3}{2} , but it can fail to hold if γ ≦ 3 2 \gamma \leqq \tfrac {3}{2} . It was shown by Hall in the preceding article that the Chebyshev prime counting estimates \[ 0 > lim inf x → ∞ ψ ( x ) x ; lim sup x → ∞ ψ ( x ) x > ∞ 0 > \lim \inf \limits _{x \to \infty } \frac {{\psi (x)}}{x};\quad \lim \sup \limits _{x \to \infty } \frac {{\psi (x)}}{x} > \infty \] can fail if γ > 1 \gamma > 1 . Here we shall prove that these estimates hold for any system satisfying γ > 1 \gamma > 1 . The proof uses a convolution approximate inverse of the measure d N dN .

A method to find certain common factors of numbers of the form $$A_n = a^{2^n } + 1$$ , when theA n are lacking algebraic factors, is discussed. Using known factors of the Fermat numbers $$F_n = 2^{2^n } + 1$$ , otherA n , containing these factors, are deduced.

Given a polynomial f(x) with rational integral coefficients, find the factorization of f(x) into irreducible factors for a given characteristic, a natural prime p or zero. In the latter case we use the factorization of f(x) for a special natural prime p.

CONTENTSBasic notationIntroductionChapter I. Auxiliary lemmas § 1. The functions , , , and their properties § 2. Estimates for the solutions of some approximating integro-difference equations § 3. Asymptotic behaviour of the solutions of some differential-difference equationsChapter II. An estimate for and its applications § 1. An estimate for . Examples § 2. Local distribution laws for additive arithmetic functions § 3. Integral theorems § 4. Equidistribution of the fractional parts of some additive functionsChapter III. Generalized problem of the numbers with small and large prime factors, and applications § 1. Statement of the problem and some special cases § 2. Solution of the general problem § 3. Upper bounds for sums extended over numbers with small prime factors § 4. Numbers with small prime factors § 5. Applications to the probabilistic theory of numbersChapter IV. The case of arithmetic progressions with increasing differences § 1. Solution of the general problem for arithmetic progressions § 2. Some applicationsReferences

This paper discusses point-symmetric graphs which have a prime number of points. Isomorphism classes of these graphs are enumerated. It is shown that these graphs are characterized by the eigenvalues of their adjacency matrices. Finally it is shown hat point-symmetric graphs with a prime number of points are group-graphs.

where all the prime factors of each hi are of a given form. A search of the literature seemed to indicate that various theorems had been conjectured but none actually proved.t For example, L. Euler stated without proof that every integer of the form 4j+2 is a sum of two primes each of the form 4j+ 1. Even the weaker statement that every integer of the form 4j+2 is a sum of two integers which have all their prime factors of the form 4j+ 1 has not yet been proved. In view of the absence of any definite results in the literature it seems worthwhile to point out that some very interesting theorems can be obtained in an elementary way. This is done in Part I of this paper and the results are summarized in Theorems 1, 2, and 3 below. In Part II we use the method of Viggo Brunt to prove a general theorem and from this we deduce Theorems 4 and 5 below.

In a recently published papert I investigated the number P(x, x') of primes X in a real quadratic field satisfying the inequalities 0

Let u > 3 and β > √e/(√e−1) be real numbers and let β0=β−√e/(√e−1). Let a and q be relatively prime positive integers. Let Ψa(X, Y) denote the number of positive integers ⩽ X and ≡ a (mod q), whose largest prime factor is ⩽ Y. There exists a computable q0(u, β) such that, for q > q0, ψa(quβ,qβ)>quβ−1exp {−2uββ−1)√e(log u+6+t)} where T = max {log log u, 4 + log (ββ0)}.