Nonzero Integer Research Articles

On the congruent number problem over integers of cyclic number fields

Abstract Given a cyclic field extension K/ℚ of degree d and a nonzero rational integer m, we show that the equation mp2 = x4 - y 2 has no nontrivial solutions in K / Q ${K/{\mathbb Q}}$ when p belongs to a subset of rational prime numbers of relative density at least φ(d)/(2d).

Small prime solutions to cubic equations

Let $a_1,\dots,a_9$ be nonzero integers not of the same sign, and let $b$ be an integer. Suppose that $a_1,\dots,a_9$ are pairwise coprime and $a_1+\cdots+a_9\equiv b\pmod{2}$. We apply the $p$-adic method of Davenport to find an explicit $P=P(a_1,\ldots,a_9,n)$ such that the cubic equation $a_1p_1^3+\cdots+a_9p_9^3=b$ is solvable with $p_j\ll P$ for all $1\le j\le 9$. It is proved that one can take $P= \max\{|a_1|,\ldots,|a_9|\}^c+|b|^{1/3}$ with $c=2$. This improves upon the earlier result with $c=14$ due to Liu (2013).

On the class number divisibility of pairs of quadratic fields obtained from points on elliptic curves

Let l be the prime 3;5 or 7 and let m be a nonzero integer. We give a method for constructing an innite family of pairs of quadratic elds Q ( p D) and Q ( p mD) with both class numbers divisible by l. Such quadratic elds are parametrized by rational points on a specied elliptic curve.

Diagonal diophantine equations with small prime variables

Let (ai,aj)=1, 1≤i<j≤s and s=2k+1, where a1,⋯,as,s and k≥4 are nonzero integers. In this paper, we show that if the diagonal diophantine equation a1p1k+⋯+aspsk=n is satisfying some necessary conditions, then we have the following results: For any ϵ>0, we have(i)if a1,⋯,as are not all of the same sign, then the above equation has solutions in primes pj satisfying pj≪|n|1/k+A3⋅2k−1+ϵ,(ii)if a1,⋯,as are all positive, then the above equation is solvable in prime pj whenever n≫A3k⋅2k−1+1+ϵ. This result is the general case of the Diophantine equations with Small Prime Variables.

The Square Terms in Generalized Lucas Sequence with Parameters $P$ And $Q$

Let $P$ and $Q$ be nonzero integers. Generalized Lucas sequence is defined as follows: $V_{0}=2$, $V_{1}=P$, and $V_{n+1}=PV_{n}+QV_{n-1}$ for $n\geq 1$. We assume that $P$ and $Q$ are odd relatively prime integers. Firstly, we determine all indices $n$ such that $V_{n}=kx^{2}$ and $V_{n}=2kx^{2}$ when $k|P$. Then, as an application of our these results, we find all solutions of the equations $V_{n}=3x^{2}$ and $V_{n}=6x^{2}$. Moreover, we find integer solutions of some Diophantine equations.

Construction of a Special Integer Triplet-I

This paper is concerned with an interesting Diophantine problem on triplet. A search is made on finding three non-zero distinct integers namely a,b,c such that each of the expressions a + 2b, a + 2c is a perfect square and b − c is twice a cubical integer.Infinitely may such triplets are obtained.

On the k-free values of the polynomial $${xy^k + C}$$ x y k + C

Consider the polynomial \({f(x, y) = xy^k + C}\) for \({k \geq 2}\) and any nonzero integer constant C. We derive an asymptotic formula for the k-free values of \({f(x, y)}\) when \({x, y \leq H}\). We also prove a similar result for the k-free values of \({f(p, q)}\) when \({p, q \leq H}\) are primes, thus extending Erdős’ conjecture for our specific polynomial. The strongest tool we use is a recent generalization of the determinant method due to Reuss.

Diphoton signatures from heavy axion decays at the CERN Large Hadron Collider

Recently, the LHC collaborations, ATLAS and CMS, have announced an excess in the diphoton channel with local significance of about $3\,\sigma$ around an invariant mass distribution of $\sim 750$ GeV, after analyzing new data collected at centre-of-mass energies of $\sqrt{s} = 13~{\rm TeV}$. We present a possible physical interpretation of such a signature, within the framework of a minimal UV-complete model with a massive singlet pseudo-scalar state $a$ that couples to a new TeV-scale coloured vector-like fermion $F$, whose hypercharge quantum number is a non-zero integer. The pseudo-scalar state $a$ might be a heavy pseudo-Goldstone boson, such as a heavy axion, which decays into two photons and whose mass lies around the excess region. The mass of the CP-odd state $a$ and its coupling to $F$ may be due to non-perturbative effects, which can break the original Goldstone shift symmetry dynamically. The possible role that the heavy axion $a$ can play in the radiative generation of a seesaw Majorana scale and in the solution to the so-called strong CP problem is briefly discussed.

On The Lucas Sequence Equations $$V_{n}=7\square $$ V n = 7 □ and $$V_{n}=7V_{m}\square $$ V n = 7 V m □

Let P be a nonzero integer and let \((U_{n})\) and \((V_{n})\) denote Lucas sequences of first and second kind defined by \(U_{0}=0\), \(U_{1}=1; V_{0}=2, V_{1}=P;\) and \(U_{n+1}=PU_{n}+U_{n-1}\), \(V_{n+1}=PV_{n}+V_{n-1}\) for \(n\ge 1\). In this study, when P is odd, we show that the equation \(U_{n}(P,1)=7\square \) has only the solution \((n,P)=(2,7\square )\) when 7|P and the equation \(V_{n}(P,1)=7\square \) has only the solution \((n,P)=(1,7\square )\) when 7|P or \((n,P)=(4,1)\) when \(P^{2}\equiv 1({\mathrm{mod }}\, 7)\). In addition, we show that the equation \(V_{n}(P,1)=7V_{m}(P,1) \square \) has a solution if and only if \(P^{2}=-3+7\square \) and \((n,m)=(3,1)\). Moreover, we show that the equation \(U_{n}(P,1)=7U_{m}(P,1) \square \) has only the solution \((n,m,P,\square )=(8,4,1,1)\) when P is odd.

A Systematic Method for Constructing Sparse Gaussian Integer Sequences With Ideal Periodic Autocorrelation Functions

A Gaussian integer is a complex number whose real and imaginary parts are both integers. Meanwhile, a sequence is defined as perfect if and only if it has an ideal periodic auto-correlation function. This paper proposes a method for constructing sparse perfect Gaussian integer sequences (SPGISs) in which most of the sequence elements are zero. The proposed SPGISs are obtained by linearly combining four base sequences or their cyclic-shift equivalents using nonzero Gaussian integer coefficients of equal magnitudes. Each base sequence contains four nonzero elements belonging to the set $\{\pm 1, \pm j\}$ . The number of nonzero elements of the constructed SPGISs depends on the choice of complex coefficients and cyclic shifts. However, each SPGIS has at most 16 nonzero elements, irrespective of the sequence length. A systematic investigation is performed into the properties of the SPGISs and their Fourier dual equivalents. Finally, a general expression is derived for a perfect Gaussian integer sequence (PGIS) of length $4n$ , where $n$ is any positive integer and most of the sequence elements are nonzero.

On the OCTIC Equation with Six Unknowns 4(x6−y6) − (x2−y2)(x4+y4+10x2y2) = (z4−w4) p2−4T4p2(x2−y2)

We obtain infinitely many non-zero integer sextuples (x, y, z, w, p, T) satisfying the non-homogeneous equation of degree eight with six unknowns given by 4(x6−y6) − (x2−y2)(x4+y4+10x2y2) = (z4−w4) p2−4T4p2(x2−y2). Various interesting relations between the solutions and special numbers, namely, polygonal numbers, Star numbers, Stella Octangular numbers, Pronic number, Jacobsthal number, Jacobsthal-Lucas number, keynea number are exhibited.

SMALL SOLUTIONS OF QUADRATIC CONGRUENCES, AND CHARACTER SUMS WITH BINARY QUADRATIC FORMS

Let $Q(x,y,z)$ be an integral quadratic form with determinant coprime to some modulus $q$. We show that $q\mid Q$ for some non-zero integer vector $(x,y,z)$ of length $O(q^{5/8+\varepsilon})$, for any fixed $\varepsilon>0$. Without the coprimality condition on the determinant one could not achieve an exponent below $2/3$. The proof uses a bound for short character sums involving binary quadratic forms, which extends a result of Chang.

Digital semigroups

The well-known expansion of rational integers in an arbitrary integer base different from 0,1, − 1 is exploited to study relations between numerical monoids and certain subsemigroups of the multiplicative semigroup of nonzero integers.

Fragile surface zero-energy flat bands in three-dimensional chiral superconductors

We study surface zero-energy flat bands in three-dimensional chiral superconductors with $p_z (p_x +ip_y)^{\nu}$-wave pairing symmetry ($\nu$ is a nonzero integer), based on topological numbers and tunneling conductance. It is shown that the surface flat bands are fragile against (i) the misorientation angle and (ii) surface Rashba spin-orbit interaction. The fragility of (i) is specific to chiral SCs, whereas that of (ii) happens for general odd-parity SCs. We demonstrate that these flat band instabilities vanish or suppress a zero-bias conductance peak in a normal/insulator/superconductor junction, which behavior is clearly different from high-$T_c$ cuprates and noncentrosymmetric superconductors. By calculating the angle resolved conductance, we also discuss a topological surface state associated with the coexistence of line and point nodes.

On Ternary Quadratic Diophantine Equation 8x2 + 8y2 -15xy = 40z2

The ternary quadratic Diophantine equation 5(x2 +y2) - 6xy = 4z2 is analyzed for its non-zero distinct integer solutions. Five different patterns of integral solutions are obtained. Few interesting relations among the solutions and some special polygonal numbers are presented.

Set of uniqueness of shifted Gaussian primes

In this paper, we show that any additive complex valued function over non-zero Gaussian integers which vanishes on the shifted Gaussian primes is necessarily identically zero.

Denominators and differences of boundary slopes for (1,1)-knots

We show that every nonzero integer occurs as the denominator of a boundary slope for infinitely many (1,1)-knots and that infinitely many (1,1)-knots have boundary slopes of arbitrarily small difference. Specifically, we prove that for any integers $$m,n>1$$ with n odd the exterior of the Montesinos knot $$K(-1/2, m/(2m\pm 1),1/n)$$ in $$S^3$$ contains an essential surface with boundary slope $$r = 2(n-1)^2/n$$ if m is even and $$2(n+1)^2/n$$ if m is odd. If $$n \ge 4m + 1$$ , we prove that $$K(-1/2, m/(2m+1),1/n)$$ also has a boundary slope whose difference with r is $$(8m-2)/(n^2-4mn+n)$$ , which decreases to 0 as n increases. All of these knots are (1,1)-knots.

Instability of point defects in a two-dimensional nematic liquid crystal model

We study a class of symmetric critical points in a variational 2D Landau–de Gennes model where the state of nematic liquid crystals is described by symmetric traceless 3×3 matrices. These critical points play the role of topological point defects carrying a degree k2 for a nonzero integer k. We prove existence and study the qualitative behavior of these symmetric solutions. Our main result is the instability of critical points when |k|≥2.

On square classes in generalized Lucas sequences

Let P and Q be nonzero integers. Generalized Fibonacci and Lucas sequences are defined as follows: U0= 0, U1= 1, and Un+1= PUn+ QUn-1for n ≥ 1 and V0= 2, V1= P, and Vn+1= PVn+ QVn-1for n ≥ 1, respectively. For all odd relatively prime values of P and Q such that P ≥ 1, we determine all indices n and m such that Vn= wVmx2or VnVm= wx2with w = 1, 2, 3 or 6 under the assumptions P2+ 4Q &gt; 0 and Vm≠ 1 for all positive integers m.

Mordell’s equation: a classical approach

We solve the Diophantine equation$Y^{2}=X^{3}+k$for all nonzero integers$k$with$|k|\leqslant 10^{7}$. Our approach uses a classical connection between these equations and cubic Thue equations. The latter can be treated algorithmically via lower bounds for linear forms in logarithms in conjunction with lattice-basis reduction.