Odd Positive Integer Research Articles

The Spectrum of Room Cubes

A Room cube of side n is an n by n by n cube such that each 2-dimensional projection is a Room square. We show that there exists a Room cube of side n if and only if n is an odd positive integer other than 3 or 5.

Some Howell Designs of prime side II

In part I of this paper we showed that for any pair ( m, r) of positive integers, r odd, there is a positive integer N 1( m, r) such that if p is a prime, p = 2 m rs+1> N 1( m>, r), r, s odd positive integers, then Howell Designs of all types H ∗(p, 2n) , p+1⩽2 n⩽2 p−2 s exist. We now verify that under the same hypotheses (except that ( m>, r) = (1, 1) is not allowed), there is a positive integer N 2( m>, r) such that if p = 2 m rs+1> N 2( m, r), then Howell Designs of all types H ∗(p, 2n) . 2 p−2 s⩽2 n⩽2 p−6 exist. Since designs of type H ∗(p>, 2p−4) and H ∗(p, 2p) are known to exist for p⩾7, it follows that if p = 2 m rs+1> N( m, r) = max{ N 1( m, r). N 2( m, r)}, then Howell Designs of all types H ∗(p, 2n) , except possibly type H ∗(p, 2p−2) , exist. When this is the case, we say that almost all H ∗(p, 2n) exist. As in part I, the method of construction appears to be much better than the general bounds obtained. We are able to show that N(2, 1) = 5. Thus, if 5< p = 4 s+1, s odd, then almost all H ∗(p, 2n) exist. Evidence is presented to show that N(1, 3) is almost certainly 1. In particular, if p = 6 s+1<1000, s odd, then almost all H ∗(p, 2n) exist.

Periodic orbits of continuous mappings of the circle

Let f be a continuous map of the circle into itself and let P ( f ) P(f) denote the set of positive integers n such that f has a periodic point of period n. It is shown that if 1 ∈ P ( f ) 1\, \in \,P(f) and n ∈ P ( f ) n\, \in \,P(f) for some odd positive integer n then for every integer m &gt; n m\, &gt; \,n , m ∈ P ( f ) m\, \in \,P(f) . Furthermore, if P ( f ) P(f) is finite then there are integers m and n (with m ⩾ 1 m\, \geqslant \,1 and n ⩾ 0 n\, \geqslant \,0 ) such that P ( f ) = { m , 2 m , 4 m , 8 m , … , 2 n m } P(f)\, = \,\{ m,\,2\,m,\,4\,m,\,8\,m,\,\ldots ,\,{2^n}\,m\} .

Some Howell designs of prime side

This paper deals with the existence question for types of Howell Designs of odd side. One of the difficulties that must be surmounted before complete knowledge can be obtained is the problem of constructing designs of prime side. Previously, relatively little information has been available about these types of designs. In this paper, we show that for every pair ( m, r) of positive integers, r odd, there is a positive integer P( m, r) such that if p is a prime, p = 2 m rs + 1> P( m, r), r, s odd positive integers, then at least (2 m−1r−1) (2 m−1r) of the types of Howell Designs of side p other than the Room Square type can be constructed. The method of construction appears to be much better than the general bounds that are obtained. We are able to show by ad hoc procedures that P(2,1) = 1, Thus if p = 4 t + 1, t odd, then at least 1 2 of the types of designs of side p other than the Room Square type can be constructed. We offer compelling evidence in favor of the conjecture that P(1,3) = 1 also. In particular, if p = 6 t + 1< 1000, t odd, then at least 3 3 of the types of designs of side p other than the Room Square type can be constructed.

Rivest-Shamir-Adleman public key cryptosystems do not always conceal messages

To every pair p, q of distinct primes there correspond 9 positive integers x no larger than pq such that χ c  χ mod ( pq) for every odd positive integer c. Therefore these 9 messages x are unconcealable in any Rivest-Shamir-Adleman public key cryptosystem which has the product pq for its encoding modulus. There are, in fact, such cryptosystems which do not conceal any messages, and others which conceal less than 50% of all messages possible within them. These examples point up the need for effective criteria for evaluating the concealing power of any such cryptosystem. This paper treats these questions for a general square free encoding modulus m. It contains easy to use necessary and sufficient conditions that the function f( x) = x c be: 1. a permutation of the residue classes module m; 2. a derangement (i.e. a permutation without fixed points) of the collection of all the residue classes modulo m other than the unavoidably unconcealable ones. It also provides a construction of all such “deranging” exponents.

On multiple node Gaussian quadrature formulae

Let μ 1 , … , μ k {\mu _1}, \ldots ,{\mu _k} be odd positive integers and n = Σ i = 1 k ( μ i + 1 ) n = \Sigma _{i = 1}^k({\mu _i} + 1) . Let { μ i } i = 1 n \{ {\mu _i}\} _{i = 1}^n be an extended Tchebycheff system on [ a , b ] [a,b] . Let L be a positive linear functional on U ≡ span ( { μ i } ) U \equiv {\operatorname {span}}(\{ {\mu _i}\} ) . We prove that L has a unique representation in the form \[ L ( p ) = ∑ i = 1 k ∑ j = 0 μ i − 1 a i j p ( j ) ( t i ) , a &gt; t 1 &gt; ⋯ &gt; t k &gt; b , L(p) = \sum \limits _{i = 1}^k {\sum \limits _{j = 0}^{{\mu _i} - 1} {{a_{ij}}{p^{(j)}}({t_i}),\quad a &gt; {t_1} &gt; \cdots &gt; {t_k} &gt; b,} } \] for all p ∈ U p \in U . The proof uses the topological degree of a mapping F : D ¯ ⊂ R k → R k F:\overline D \subset {R^k} \to {R^k} . The result is proved by showing that the equation F ( t _ ) = 0 F(\underline {t}) = 0 has a unique solution, which in turn is proved by showing that F has degree 1 and that for any solution t _ \underline {t} to the equation F ( t _ ) = 0 F(\underline {t}) = 0 , det F ′ ( t _ ) &gt; 0 \det F\prime (\underline {t}) &gt; 0 . We also give extensions to the cases when the { u i } \{ {u_i}\} are a periodic extended Tchebycheff system and when L is a nonnegative linear functional.

On the “3𝑥+1” problem

It is an open conjecture that for any positive odd integer m the function \[ C ( m ) = ( 3 m + 1 ) / 2 e ( m ) , C(m) = (3m + 1)/{2^{e(m)}}, \] where e ( m ) e(m) is chosen so that C ( m ) C(m) is again an odd integer, satisfies C h ( m ) = 1 {C^h}(m) = 1 for some h. Here we show that the number of m ⩽ x m \leqslant x which satisfy the conjecture is at least x c {x^c} for a positive constant c. A connection between the validity of the conjecture and the diophantine equation 2 x − 3 y = p {2^x} - {3^y} = p is established. It is shown that if the conjecture fails due to an occurrence m = C k ( m ) m = {C^k}(m) , then k is greater than 17985. Finally, an analogous " q x + r qx + r " problem is settled for certain pairs ( q , r ) ≠ ( 3 , 1 ) (q,r) \ne (3,1) .

Oscillation, nonoscillation, and asymptotic behavior for third order nonlinear differential equations

This paper discusses the behavior of real-valued solutions to the equation $$y''' + p(t)y'' + q(t)y' + r(t)y^\mu = 0$$ (1.1) where p, q, r are continuous and real-valued on some half-line (t0, + ∞) and μ is the quotient of odd positive integers. Oriteria are obtained for the existence of nonoscillatory solutions, several stability theorems are proved, and the existence of oscillatory solutions is shown. Of primary concern are the two cases q(t) ≤ 0, r(t) > 0, and q(t)>0, r(t)>0. Some of the main techniques used involve comparison theorems for linear equations and results in the theory of second order nonlinear oscillations.

On Rational Triangles

(1976). On Rational Triangles. The American Mathematical Monthly: Vol. 83, No. 7, pp. 517-521.

On Derivatives of Characteristic Functions

If $k$ is a positive odd integer, it is shown that it is possible to construct a characteristic function $f(t)$ such that $f^{(k)}(0)$ exists but $f^{(k)}(t_m)$ does not exist for a sequence of numbers $\{t_m\}$ where $t_m \rightarrow 0$ as $m \rightarrow \infty$.

Solution of the Room square existence problem

We prove that if there is a Room square of side r with a subsquare of side s, then there is a Room square of side 5( r − s) + s. This enables us to construct a Room square of side 257, thus proving that Room squares exist for all odd positive integer sides except 3 and 5.

Behavior of the Solutions of the Differential Equation y&amp;129;&amp;338;+qyp=r

Some results concerning boundedness and asymptotic properties of the solutions of the differential equation (I)y+q(x)yp ˭ r(x),p an odd positive integer, are obtained, Particular emphasis is given to the investigation of the behavior of the solutions of equation (I) when the quotient ruq is monotonic.

Divisor matrices in the Cayley ring

Suppose that m is an odd positive integer and that A is an integral Cayley number which is primitive mod m. The principal result of Section 1 is the following theorem.

Concerning the order approximation of periodic continuous functions by trigonometric interpolation polynomials II

Letxkn=2θk/n,k=0,1 …n−1 (n odd positive integer). LetRn(x) be the unique trigonometric polynomial of order 2n satisfying the interpolatory conditions:Rn(xkn)=f(xkn),Rn(j)(xkn)=0,j=1,2,4,k=0,1…,n−1. We setw2(t,f) as the second modulus of continuity off(x). Then we prove that |Rn(x)-f(x)|=0(nw2(1/nf)). We also examine the question of lower estimate of ‖Rn-f‖. This generalizes an earlier work of the author.

On Ramanujan's formula for values of Riemann zeta-function at positive odd integers

On Ramanujan's formula for values of Riemann zeta-function at positive odd integers

Concerning the order of approximation of periodic continuous functions by trigonometric interpolation polynomials

Letx kn=2θk/n,k=0,1 …n−1 (n odd positive integer). LetR n(x) be the unique trigonometric polynomial of order 2n satisfying the interpolatory conditions:R n(xkn)=f(xkn),R n (j)(xkn)=0,j=1,2,4,k=0,1…,n−1. We setw 2(t,f) as the second modulus of continuity off(x). Then we prove that |R n(x)-f(x)|=0(nw2(1/nf)). We also examine the question of lower estimate of ‖R n-f‖. This generalizes an earlier work of the author.

First order and critical point phase transitions in cooperative assemblies with many-body interactions: some exact results

The complete thermodynamics of a model hamiltonian with general-spin infinite range Ising 2-point, 4-point,... interaction terms is found exactly in two different statistical mechanical ensembles which prove to be essentially equivalent. The coefficients of the above terms may be chosen to give critical behaviour which is largely consistent with the idea of scaling and which has delta equal to any odd positive integer and gamma =1; alternatively a first order phase transition may be obtained, a finite spontaneous magnetization appearing as the temperature is decreased through that of the transition. The Heisenberg case is also considered and found to be closely similar to the Ising case.

On the sum of a prime and of two powers of two

It has been shown by different methods that there is an infinity of positive odd integers not representable as the sum of a prime and a (positive) power of 2, thus disproving a conjecture to the contrary which had been made in the nineteenth century. The question then arises as to whether or not all sufficiently large positive odd integers can be represented as the sum of a prime and of two positive powers of 2; that is, as p + 2α -f 2 6, where α, b > 0 and p is prime. (The corresponding question has been discussed for bases other than 2 but is really quite trivial.) Theorem I gives a negative answer to this question. THEOREM I. There is an infinity of distinct, positive odd integers not representable as the sum of a prime and of two positive powers of 2.

Preassigned character values in the Gaussian integers

Let γ 1, γ 2,…, γ t be distinct prime Gaussian integers. With k an odd positive rational integer, take ω 1, ω 2,…, ω t to be k-th roots of unity, not necessarily distinct nor primitive. We show there exist infinitely many prime Gaussian integers π having a k-th-power character χ such that χ( γ i ) = ω i , 1 ≤ i ≤ t.

On the Leaf Curves

The equation pn = an cos p0, where p is a positive integer, (the results remain valid if we replace cos pG by sin p0) n is a positive odd integer, p is the principal root of pn, are generalizations of the equation of rose curves to which they reduce if n = 1. We wish to consider the area enclosed by curves of this type. If p is odd there are p leaves and if p is even there are 2p leaves. THEOREM 1. For every k the area enclosed by pn = an cos (2k + 1)0 is /n + / 2n) a2. 2 (n+1)