Nonzero Integer Research Articles

Simultaneous Pell Equations

It is proved that ifaandbare different non-zero rational integers then the “simultaneous Pell equations”[formula]have at most 132 solutions in rational integersx,y,z.

A problem of Hooley in Diophantine approximation

In [5] Professor Hooley announced without proof the following result which is a variant of well-known work by Heilbronn [4]and Danicic [3] (see [1]).Let k≥2 be an integer, b a fixed non-zero integer, and a an irrational real number. Then, for any ɛ> 0, there are infinitely many solutions to the inequalityHere

The Diophantine Equation 3u2−2=v6

The theorem of Delaunay–Nagell states that:If d is a cube-free integer>1,then the equation x3+dy3=1has at most one solution in non-zero integers x, y, and if such a solution exists then[formula]is either the fundamental unit of the field[formula]or its square, the latter occurring for only finitely many values of d. Investigation of these exceptionaldvalues has led to the equation of the title [5, 3.9], which has only finitely many solutions. We prove that the title equation has no integer solution other than |u|=|v|=1, which give the known valuesd=19, 20, 28, therefore there are no otherdvalues.

Vibration and Stability of a Spinning Disk Under Stationary Distributed Edge Loads

The vibration and stability of a spinning disk under conservative distributed edge tractions are studied both numerically and analytically. The edge traction is circumferentially stationary in the space. When the compressive traction is uniform, it is found that no modal interaction occurs and the natural frequencies of all nonreflected waves decrease, while the natural frequencies of the reflected waves increase. When the spinning disk is under distributed traction in the form of cos kθ, where k is a nonzero integer, it is found that the eigenvalue only changes slightly under the edge traction if the natural frequency of interest is well separated from others. When two modes are almost degenerate, however, modal interaction may or may not occur. It is observed that when the difference between the number of nodal diameters of these two modes is equal to ±k, frequency veering occurs when both modes are nonreflected, and merging occurs when one of these two modes is a reflected wave. In applying this rule, the number of nodal diameters of the forward and the reflected wave is considered as negative.

Fermat’s last theorem

After more than three centuries of effort by some of the best mathematicians, Gerhard Frey, J-P Serre, Ken Ribet and Andrew Wiles have finally succeeded in proving Fermat’s assertion that the equationXn + Yn = Zn has no solutions in non-zero integers ifn ≥ 3. Each of the four mathematicians made a decisive contribution with Wiles delivering thecoup de grace. The proof, as it finally came to be, is in some sense a triumph for Fermat.

Polynomials with roots modulo every integer

Given a polynomial with integer coefficients, we calculate the density of the set of primes modulo which the polynomial has a root. We also give a simple criterion to decide whether or not the polynomial has a root modulo every non-zero integer.

A Reduction Theorem on Purely Singular Splittings of Cyclic Groups

A set M of nonzero integers is said to split a finite abelian group G if there is a subset S of G for which $M \cdot S = G\backslash \{ 0\}$. If, moreover, each prime divisor of $|G|$ divides an element of M, we call the splitting purely singular. It is conjectured that the only finite abelian groups which can be split by $\{ 1, \ldots ,k\}$ in a purely singular manner are the cyclic groups of order $1,k + 1$ and $2k + 1$. We show that a proof of this conjecture can be reduced to a verification of the case $\gcd (|G|,6) = 1$.

Modular Elliptic Curves and Fermat's Last Theorem

When Andrew John Wiles was 10 years old, he read Eric Temple Bell’s The Last Problem and was so impressed by it that he decided that he would be the first person to prove Fermat’s Last Theorem. This theorem states that there are no nonzero integers a, b, c, n with n > 2 such that an + bn = cn. The object of this paper is to prove that all semistable elliptic curves over the set of rational numbers are modular. Fermat’s Last Theorem follows as a corollary by virtue of previous work by Frey, Serre and Ribet.

Eigenvalues and the Smith normal form

We compare the Smith normal form (SNF) over the integers of an integral nonsingular matrix with its spectrum when its eigenvalues are integers. Our results include tight bounds on the size of the largest element of the SNF when the matrix is diagonalizable with nonzero integer eigenvalues, with no assumptions on the diagonalizing matrices.

On ranks of twists of elliptic curves and power-free values of binary forms

Let E be an elliptic curve over Q. By the rank of E we shall mean the rank of the group of rational points of E. Mestre [31], improving on the work of Neron [34] (cf. [13], [39] and [46]), has shown that there is an infinite family of elliptic curves over Q with rank at least 12. However, computational work (see [3], [4], [7] and [47]) suggests that a typical elliptic curve will have much smaller rank, with curves of rank 0 or 1 being predominant. Indeed, Brumer [6] has proved, subject to the Birch, Swinnerton-Dyer conjecture, the Shimura, Taniyama, Weil conjecture and the generalized Riemann hypothesis, that the average rank of an elliptic curve, ordered according to its Faltings height, is at most 2.3. In this article we shall study the behaviour of the rank as we run over twists of a given elliptic curve over Q. That is, we shall restrict our attention to families of elliptic curves defined over Q which are isomorphic over C. There are families of quadratic, cubic, quartic and sextic twists (see, for example, Proposition 5.4 of Chapter X of [42]). Let E be an elliptic curve over Q with Weierstrass equation y2 = X + ax + b and for any non-zero integer d let Ed 2 3 denote a quadratic twist of E given by the equation dy2 = X + ax + b. Let r(d) denote the rank of Ed . Note that if d1 and d2 are non-zero integers, then Ed is isomorphic to Ed over Q if and only if d1 /d2 is the square of a rational number. Subject to the conjectures of Birch and Swinnerton-Dyer and of Shimura, Taniyama, and Weil, Goldfeld [14] conjectured in 1979 that

A reduction theorem on purely singular splittings of cyclic groups

A set M of nonzero integers is said to split a finite abelian group G if there is a subset S of G for which M ⋅ S = G ∖ { 0 } M \cdot S = G\backslash \{ 0\} . If, moreover, each prime divisor of | G | |G| divides an element of M, we call the splitting purely singular. It is conjectured that the only finite abelian groups which can be split by { 1 , … , k } \{ 1, \ldots ,k\} in a purely singular manner are the cyclic groups of order 1 , k + 1 1,k + 1 and 2 k + 1 2k + 1 . We show that a proof of this conjecture can be reduced to a verification of the case gcd ( | G | , 6 ) = 1 \gcd (|G|,6) = 1 .

The Primes for Which an Abelian Cubic Polynomial Splits

Let $X^3+AX+B$ be an irreducible abelian cubic polynomial in $Z[X]$. We determine explicitly integers $a_1,\ldots,a_t$, $F$ such that, except for finitely many primes $p$, \[ x^3+Ax+B\equiv 0\pmod{p} \text{ has three solutions} \Leftrightarrow p\equiv a_1,\ldots,a_t\pmod{F}. \]

Thermal contraction at the spin-Peierls transition inCuGeO3

We report the results of an x-ray-scattering study of the [ital b]-axis lattice constant of the quasi-one-dimensional spin-1/2 antiferromagnetic chain system CuGeO[sub 3] as a function of temperature and magnetic field. A spontaneous thermal contraction [Delta][ital b] along the [ital b] axis perpendicular to the chain direction is observed below the spin-Peierls transition temperature [ital T][sub [ital c]] near 14 K. The contraction, [Delta][ital b], is well described by a simple power law, [Delta][ital b][proportional to](1[minus][ital T]/[ital T][sub [ital c]])[sup [ital x]], where the exponent [ital x] is found to be close to 0.5. Below the transition temperature, [Delta][ital b] scales with the intensity of the superlattice reflections with indices ([ital h]/2,[ital k],[ital l]/2)([ital h],[ital l]: odd, [ital k]: nonzero integer) measured by neutron scattering. The shift of the transition temperature, [Delta][ital T][sub [ital c]][equivalent to][ital T][sub [ital c]](0)[minus][ital T][sub [ital c]]([ital H]), is found to scale as [ital H][sup 2] in quantitative agreement with the results of magnetic susceptibility measurements and with theory. A small increase in the [ital a]-axis lattice constant is observed below [ital T][sub [ital c]].

Division by invariant integers using multiplication

Integer division remains expensive on today's processors as the cost of integer multiplication declines. We present code sequences for division by arbitrary nonzero integer constants and run-time invariants using integer multiplication. The algorithms assume a two's complement architecture. Most also require that the upper half of an integer product be quickly accessible. We treat unsigned division, signed division where the quotient rounds towards zero, signed division where the quotient rounds towards -∞, and division where the result is known a priori to be exact. We give some implementation results using the C compiler GCC.

Some Remarks on the abc-Conjecture

Let r(x) be the product of all distinct primes dividing a nonzero integer x . The abc-conjecture says that if a, b, c are nonzero relatively prime integers such that a + b + c = 0, then the biggest limit point of the numbers logmax(lal, ibl, cil) log r(abc) equals 1. We show that in a natural anologue of this conjecture for n > 3 integers, the largest limit point should be replaced by at least 2n 5. We present an algorithm leading to numerous examples of triples a, b, c for which the above quotients strongly deviate from the conjectural value 1.

Comment: New signature scheme with message recovery

Piveteau (see ibid., vol. 29, p. 2185, 1993) proposes a digital signature scheme with message recovery. We have p a large prime, such that computation of discrete logarithms module p is intractable, and a message represented as a nonzero integer m module p Nyberg and Rueppei have found independently and generalised the scheme described by Piveteau. Their message recovery procedure coincides with the method proposed by Pinch. Furthermore, as observed by K. Nyberg, the equivalence between the schemes presented by Piveteau is still an open problem, and has not yet been established, in contrast to what he stated.

Some remarks on the 𝑎𝑏𝑐-conjecture

Let r ( x ) r(x) be the product of all distinct primes dividing a nonzero integer x . The abc-conjecture says that if a, b, c are nonzero relatively prime integers such that a + b + c = 0 a + b + c = 0 , then the biggest limit point of the numbers \[ log ⁡ max ( | a | , | b | , | c | ) log ⁡ r ( a b c ) \frac {{\log \max (|a|,|b|,|c|)}}{{\log r(abc)}} \] equals 1. We show that in a natural anologue of this conjecture for n ≥ 3 n \geq 3 integers, the largest limit point should be replaced by at least 2 n − 5 2n - 5 . We present an algorithm leading to numerous examples of triples a, b, c for which the above quotients strongly deviate from the conjectural value 1.

Two simple applications of the unimodularity property

One of the most important and useful properties of Totally Unimodular (TU) matrices is the integrality of all the extreme points of the polyhedron [ x; Ax ⩽ b, x ⩾0, b integer] where A is TU. In this paper we present two applications of this property to situations where integer optimal solutions are generated from noninteger ones. The first applications relates to a manpower distribution problem where one wishes to maximize the number of nonzero (integer) assignments. The second model is applied to a network problem with flow separation side constraints.

Remark on a relation between magnetic field and topological degree of order parameter in a type-II superconductivity model

It is proved that in the absence of an applied magnetic field, the two space dimensional steady-state Helmholtz free energy, which models the type-II superconductors, admits no minimizer in the following admissible function set: {(u, A) ϵ H 1 (Ω C) × H 1 (Ω, R 2): |u| ∂Ω = 1, - 1 2 i ∫ ∂Ω [ uD τu − uD τ u] dΓ = 2kπ} . Here C is the complex plane, k is a fixed nonzero integer which is the topological degree of u| ∂Ω with respect to its value 0. The case k = 0 is discussed separately under different assumptions.

Scattering of light from random surfaces that are periodic on average

We study the scattering of p-polarized light from a one-dimensional randomly rough metal surface that is periodic on average, rather than planar. We find that the contribution to the mean differential reflection coefficient from the diffuse component of the scattered light displays features, e.g., dips, at scattering angles θs given by sin θs = −sin θ0 + n(λ/d), where θ0 is the angle of incidence, n is a nonzero integer, λ is the wavelength of the incident light, and d is the period of the average surface, in addition to the enhanced backscattering peak at θs = −θ0. These additional features are caused by the interference of each multiply scattered surface-plasmon polariton path with its time-reversed partner.