Odd Positive Integer Research Articles

Variations on twists of tuples of hyperelliptic curves and related results

Let f∈Q[x] be a square-free polynomial of degree ⩾3 and m⩾3 be an odd positive integer. Based on our earlier investigations we prove that there exists a function D1∈Q(u,v,w) such that the Jacobians of the curvesC1:D1y2=f(x),C2:y2=D1xm+b,C3:y2=D1xm+c, have all positive ranks over Q(u,v,w). Similarly, we prove that there exists a function D2∈Q(u,v,w) such that the Jacobians of the curvesC1:D2y2=f(x),C2:y2=D2xm+b,C3:y2=xm+cD2, have all positive ranks over Q(u,v,w). Moreover, if f(x)=xm+a for some a∈Z∖{0}, we prove the existence of a function D3∈Q(u,v,w) such that the Jacobians of the curvesC1:y2=D3xm+a,C2:y2=D3xm+b,C3:y2=xm+cD3, have all positive ranks over Q(u,v,w). We present also some applications of these results. Finally, we present some results concerning the torsion parts of the Jacobians of the superelliptic curves yp=xm(x+a) and yp=xm(a−x)k for a prime p and 0<m<p−2 and k<p and apply our result in order to prove the existence of a function D∈Q(u,v,w,t) such that the Jacobians of the curvesC1:Dyp=xm(x+a),Dyp=xm(x+b) have both positive rank over Q(u,v,w,t).

Weierstrass semigroup and codes over the curve $y^q + y = x^{q^r + 1}$

We compute the Weierstrass semigroup at a pair of rational points on the curve defined by the affine equation$y^q + y = x^{q^r + 1}$ over $\mathbb{F}_{q^{2r}}$, where $r$ is a positive odd integer and $q$ is a prime power. We then construct a two-point AG code on the curve whose relative parameters are better than comparable one-point AG code.

Self-intersection of the relative dualizing sheaf on modular curves X_1(N)

Let $N$ be an odd and squarefree positive integer divisible by at least two relative prime integers bigger or equal than 4. Our main theorem is an asymptotic formula solely in terms of $N$ for the stable arithmetic self-intersection number of the relative dualizing sheaf for modular curves $X_1(N)/ \mathbb{Q}$. From our main theorem we obtain an asymptotic formula for the stable Faltings height of the Jacobian $J_1(N) / \mathbb{Q}$ of $X_1(N)/ \mathbb{Q}$, and, for sufficiently large N, an effective version of Bogomolov's conjecture for $X_1(N) / \mathbb{Q}$.

Oscillation Behavior for a Class of Differential Equation with Fractional-Order Derivatives

By using a generalized Riccati transformation technique and an inequality, we establish some oscillation theorems for the fractional differential equation[atpt+qtD-αxt)γ′ − b(t)f∫t∞‍(s-t)-αx(s)ds = 0, fort⩾t0&gt;0, whereD-αxis the Liouville right-sided fractional derivative of orderα∈(0,1)ofxandγis a quotient of odd positive integers. The results in this paper extend and improve the results given in the literatures (Chen, 2012).

The Relationships Between the Bernoulli Numbers B2n &amp; an And the Euler Numbers E2n &amp; bn

Traditionally, the values of Riemann zeta function at the even positive integers have been formulated in terms of Bernoulli numbers n B2 and the sums of the alternating series of odd powers of the reciprocal of odd positive integers have been calculated in terms of Euler numbers . 2n E However, the present author reproduced the sum of the same series using different procedures in terms of two rational numbers n a and . n b In this paper, the relationships between n B2 & n a and n E2 & n b have been established. Consequently, some of the theorems and corollaries in [5] have been restated in terms of n B2 and E2n.

The exponential Diophantine equation 2 x + b y = c z.

Let b and c be fixed coprime odd positive integers with min{b, c} > 1. In this paper, a classification of all positive integer solutions (x, y, z) of the equation 2x + b y = c z is given. Further, by an elementary approach, we prove that if c = b + 2, then the equation has only the positive integer solution (x, y, z) = (1,1, 1), except for (b, x, y, z) = (89,13,1, 2) and (2r − 1, r + 2,2, 2), where r is a positive integer with r ≥ 2.

Oscillation behavior of third-order nonlinear neutral dynamic equations on time scales with distributed deviating arguments

The aim of this paper is to give oscillation criteria for the third-order neutral dynamic equations with distributed deviating arguments of the form [r(t)([x(t)+p(t)x(?(t))]??)?]? + ?dc f(t,x[?(t,?)])?? = 0; where ?&gt;0 is the quotient of odd positive integers with r(t) and p(t) real-valued rd-continuous positive functions defined on T. By using a generalized Riccati transformation and integral averaging technique, we establish some new sufficient conditions which ensure that every solution of this equation oscillates or converges to zero.

Chen’s conjecture and its generalization

Let l1, l2, ..., lg be even integers and x be a sufficiently large number. In this paper, the authors prove that the number of positive odd integers k ≤ x such that (k + l1)2, (k + l2)2, ..., (k + lg)2 can not be expressed as 2n + pα is at least c(g)x, where p is an odd prime and the constant c(g) depends only on g.

Oscillatory properties of third-order quasilinear difference equations

Some new oscillation criteria are obtained for the third-order quasilinear difference equation $\Delta^2\left(p_n\left(\Delta x_n\right)^\alpha\right)-$ $q_n\left(\Delta x_n\right)^\alpha+r_n f\left(x_n\right)=0, n=0,1,2, \ldots$, where $\alpha&gt;0$ is the ratio of odd positive integers. The method uses techniques based on Schwarz's inequality. Example is inserted to illustrate the result.

CONVOLUTION SUMS OF ODD AND EVEN DIVISOR FUNCTIONS

Let <TEX>${\sigma}_s(N)$</TEX> denote the sum of the s-th power of the positive divisors of N and <TEX>${\sigma}_{s,r}(N;m)={\sum_{d{\mid}N\\d{\equiv}r\;mod\;m}}\;d^s$</TEX> with <TEX>$N,m,r,s,d{\in}\mathbb{Z}$</TEX>, <TEX>$d,s$</TEX> > 0 and <TEX>$r{\geq}0$</TEX>. In a celebrated paper [33], Ramanuja proved <TEX>$\sum_{k=1}^{N-1}{\sigma}_1(k){\sigma}_1(N-k)=\frac{5}{12}{\sigma}_3(N)+\frac{1}{12}{\sigma}_1(N)-\frac{6}{12}N{\sigma}_1(N)$</TEX> using elementary arguments. The coefficients' relation in this identity (<TEX>$\frac{5}{12}+\frac{1}{12}-\frac{6}{12}=0$</TEX>) motivated us to write this article. In this article, we found the convolution sums <TEX>$\sum_{k</TEX><TEX>&</TEX><TEX>lt;N/m}{\sigma}_{1,i}(dk;2){\sigma}_{1,j}(N-mk;2)$</TEX> for odd and even divisor functions with <TEX>$i,j=0,1$</TEX>, <TEX>$m=1,2,4$</TEX>, and <TEX>$d{\mid}m$</TEX>. If N is an odd positive integer, <TEX>$i,j=0,1$</TEX>, <TEX>$m=1,2,4$</TEX>, <TEX>$s=0,1,2$</TEX>, and <TEX>$d{\mid}m{\mid}2^s$</TEX>, then there exist <TEX>$u,a,b,c{\in}\mathbb{Z}$</TEX> satisfying <TEX>$\sum_{k&</TEX> <TEX>lt;2^sN/m}{\sigma}_{1,i}(dk;2){\sigma}_{1,j}(2^sN-mk;2)=\frac{1}{u}[a{\sigma}_3(N)+bN{\sigma}_1(N)+c{\sigma}_1(N)]$</TEX> with <TEX>$a+b+c=0$</TEX> and (<TEX>$u,a,b,c$</TEX>) = 1(Theorem 1.1). We also give an elementary problem (O) and solve special cases of them in (O) (Corollary 3.27).

The Gauss–Wilson theorem for quarter-intervals

We define a Gauss factorial N n ! to be the product of all positive integers up to N that are relatively prime to n. It is the purpose of this paper to study the multiplicative orders of the Gauss factorials $\left\lfloor\frac{n-1}{4}\right\rfloor_{n}!$ for odd positive integers n. The case where n has exactly one prime factor of the form p≡1(mod4) is of particular interest, as will be explained in the introduction. A fundamental role is played by p with the property that the order of $\frac{p-1}{4}!$ modulo p is a power of 2; because of their connection to two different results of Gauss we call them Gauss primes. Our main result is a complete characterization in terms of Gauss primes of those n of the above form that satisfy $\left\lfloor\frac{n-1}{4}\right\rfloor_{n}!\equiv 1\pmod{n}$ . We also report on computations that were required in the process.

ON PAIRS OF ONE PRIME, TWO PRIME SQUARES AND POWERS OF 2

In this short paper, it is proved that every pair of large positive odd integers satisfying some necessary conditions can be represented in the form of a pair of one prime, two prime squares and k powers of 2. In particular, we have k = 332.

Kamenev-type oscillation criteria for higher-order nonlinear dynamic equations on time scales

Abstract In this paper, we investigate the oscillation of the following higher-order dynamic equation: { r n ( t ) [ ( r n − 1 ( t ) ( ⋯ ( r 1 ( t ) x △ ( t ) ) △ ⋯ ) △ ) △ ] γ } △ + F ( t , x ( τ ( t ) ) ) = 0 on an arbitrary time scale T, where n ≥ 2 , 1 r k ( t ) ( 1 ≤ k ≤ n ) are positive rd-continuous functions on T, and γ is the quotient of two odd positive integers, τ : T → T with τ ( t ) &gt; t and F ∈ C ( T × R , R ) . We give sufficient conditions under which every solution of this equation is either oscillatory or tends to zero. MSC:34K11, 39A10, 39A99.

A New Construction of Zero-Difference Balanced Functions and Its Applications

In this paper, a new construction of zero-difference balanced functions defined on is given, where is an odd positive integer. Based on the generic constructions proposed by Ding, optimal constant composition codes and perfect difference systems of sets with new parameters can be generated from the zero-difference balanced functions constructed in this paper.

Fundamental domains for properly discontinuous affine groups

We construct a fundamental region for the action on the $2d+1$-dimensional affine space of some free, discrete, properly discontinuous groups of affine transformations preserving a quadratic form of signature $(d+1, d)$, where $d$ is any odd positive integer.

The integral points on elliptic curves y 2 = x 3 + (36n 2 − 9)x − 2(36n 2 − 5)

Let n be a positive odd integer. In this paper, combining some properties of quadratic and quartic diophantine equations with elementary analysis, we prove that if n > 1 and both 6n 2 − 1 and 12n 2 + 1 are odd primes, then the general elliptic curve y 2 = x 3+(36n 2−9)x−2(36n 2−5) has only the integral point (x, y) = (2, 0). By this result we can get that the above elliptic curve has only the trivial integral point for n = 3, 13, 17 etc. Thus it can be seen that the elliptic curve y 2 = x 3 + 27x − 62 really is an unusual elliptic curve which has large integral points.

An oscillation theorem for second order superlinear dynamic equations on time scales

In this paper, the oscillatory behavior of the second order superlinear dynamic equation (0.1)(r(t)xΔ(t))Δ+p(t)xα(σ(t))=0,α>1,is studied under the assumption∫∞Δtr(t)<∞,where r,p∈Crd(T,R),r(t)>0,T in our main theorem is assumed to be a regular time scale, α is the quotient of odd positive integers. When the coefficient function p(t) is allowed to be negative for arbitrarily large values of t, we establish a sufficient condition for oscillation of all solutions of Eq. (0.1). As special cases, we get that the superlinear differential equation(r(t)x′(t))′+p(t)xα(t)=0,α>1,is oscillatory, if∫∞Rα(t)p(t)dt=∞,R(t)=∫t∞dsr(s),and the superlinear difference equationΔ(r(n)Δx(n))+p(n)xα(n+1)=0,α>1,is oscillatory, if∑∞Rα(n+1)p(n)=∞,R(n)=∑k=n∞1r(k).

Optimal Frequency Hopping Sequences of Odd Length

In this paper, a new generalized cyclotomy with respect to a positive odd integer is introduced, and a construction of frequency hopping sequence sets and two constructions of frequency hopping sequences are proposed as its applications. The frequency hopping sequence sets and frequency hopping sequences obtained in this paper can be optimal with respect to the Peng-Fan bound and Lempel-Greenberger bound, respectively. Further, the length of sequences in the optimal frequency hopping sequence sets can be any odd integer larger than 3. Some of them have new parameters.

A note on the Diophantine equation $$y^2=px(A{x^{2}}+2)$$

Recently, Li Min Chen (Acta Math Sinica Chin Ser 53:83–86, 2010) considered a variant \(y^2=px(x^2+2)\) of Cassels’ equation \(y^2=3x(x^2+2)\). He proved that the equation has at most two solutions in positive integers \((x, y)\). Therefore, he improved a result obtained Luca and Walsh (Glasgow Math J 47:303–307, 2005). In this note, we consider another variant of Cassels’ equation and we show that for any prime \(p\) and any odd positive integer \(A\), the Diophantine equation \(y^2=px(Ax^2+2)\) has at most seven solutions in positive integers \((x, y)\).

Oscillation criteria for a class of nonlinear fourth order neutral differential equations

Abstract In this paper, sufficient conditions are obtained for oscillation of a class of nonlinear fourth order mixed neutral differential equations of the form (E)$$\left( {\frac{1} {{a\left( t \right)}}\left( {\left( {y\left( t \right) + p\left( t \right)y\left( {t - \tau } \right)} \right)^{\prime \prime } } \right)^\alpha } \right)^{\prime \prime } = q\left( t \right)f\left( {y\left( {t - \sigma _1 } \right)} \right) + r\left( t \right)g\left( {y\left( {t + \sigma _2 } \right)} \right)$$ under the assumption $$\int\limits_0^\infty {\left( {a\left( t \right)} \right)^{\tfrac{1} {\alpha }} dt} = \infty .$$ where α is a ratio of odd positive integers. (E) is studied for various ranges of p(t).