Diophantine inequality involving binary forms

Let r =2^d-1 ＋ 1. We investigate the diophantine inequality
｜∑i=1^r λiФi（xi,yi）＋η｜〈（max 1≤i≤r{｜xi｜,｜yi｜}）^-δ,
where Фi（x,y）∈X[x,y]（1≤i≤r） are nondegenerate forms of degree d = 3 or 4.     

On a binary Diophantine inequality involving prime numbers

The Ramanujan Journal - Let $$1< c < frac{59}{44},, cne frac{4}{3}$$ . In this paper it is proved that for any sufficiently large real N, for almost all real $$Tin (N,2N]$$ (in...     

On a Diophantine inequality involving prime numbers

The Ramanujan Journal - Let $$2< c < \frac{52}{25}$$. In this paper, it is proved that for any sufficiently large real number N, the Diophantine inequality $$|p_1^{c} + p_2^{c} +...     

On a Diophantine inequality involving prime powers

Suppose that N is a sufficiently large real number. In this paper it is proved that if 1 < c < 108/53, c ≠ 2, then the Diophantine inequality ${| p^c_1 + p^c_2+ p^c_3 +p^c_4+ p^c_5 - N| < \log^{-1}N}$ is solvable in primes p ₁, p ₂, p ₃, p ₄, p ₅. This result constitutes an improvement upon that of Zhai and Cao for the range 1 < c < 81/40, c ≠ 2.     

On a ternary Diophantine inequality

Let \(\Omega (n)\) be the total number of prime factors of n, and let \(\lambda _j\) be the real numbers satisfying suitable conditions. Let \(J_k(N)\) denote the number of solutions to the inequality

$$\begin{aligned} \left\{ \begin{array}{ll} |\lambda _1n_1+\lambda _2n_2+\lambda _3n_3+\eta |<\varepsilon ,&{} \\ \Omega (n_j)= k,&{} \\ 2\le n_j\le N \quad (j=1,2,3).&{} \end{array} \right. \end{aligned}$$

In this note, we investigate the properties of \(J_k(N)\) for any integer \(k\ge 1\), which is allowed to tend to infinity with respect to N. Using an asymptotic formula for the weighted exponential sums, we obtain a sharper lower bound for it and also discuss an application of the main result.

     

Diophantine inequality by unlike powers of primes

The Ramanujan Journal - Suppose that $$\lambda _1,\ldots ,\lambda _5$$ are nonzero real numbers, not all of the same sign, satisfying that $$\frac{\lambda _1}{\lambda _2}$$ is irrational. Then for...     

On a Diophantine equation involving primes

The Ramanujan Journal - For all positive integers k and N, we prove that there are infinitely many totally real multiquadratic fields K of degree $$2^k$$ over $$mathbb {Q}$$ such that each...     

Diophantine inequalities involving several power sums

Let denote the ring of power sums, i.e. complex functions of the form for some and _iA, where is a multiplicative semigroup. Moreover, let We consider Diophantine inequalities of the form where >1 is a quantity depending on the dominant roots of the power sums appearing as coefficients in F(n,y), and show that all its solutions have y parametrized by some power sums from a finite set. This is a continuation of the work of Corvaja and Zannier [4–6] and of the authors [10, 18] on such problems.Mathematics Subject Classification (2000):11D45,11D61Revised version: 6 May 2004     

Note on a Diophantine inequality in several variables

We establish estimates for the number of points that belong to an aligned box in in terms of certain exponential sums. These generalize previous results that were known only in case .

     

Modular forms and effective Diophantine approximation

After the work of G. Frey, it is known that an appropriate bound for the Faltings height of elliptic curves in terms of the conductor (Frey?s height conjecture) would give a version of the ABC conjecture. In this paper we prove a partial result towards Frey?s height conjecture which applies to all elliptic curves over $\mathbb{Q}$, not only Frey curves. Our bound is completely effective and the technique is based in the theory of modular forms. As a consequence, we prove effective explicit bounds towards the ABC conjecture of similar strength to what can be obtained by linear forms in logarithms, without using the latter technique. The main application is a new effective proof of the finiteness of solutions to the S-unit equation (that is, S-integral points of ${\mathbb{P}}^1?\{0,1,\infty \}$), with a completely explicit and effective bound, without using any variant of Baker?s theory or the Thue–Bombieri method.     

Diophantine inequalities for forms of odd degree

If $\text{F}$ is a form of odd degree k with real coefficients in s variables where s ≥ c₁(k), then there are integers x₁,… x_s not all zero, with |$\text{F}$(x₁,… x_s)| < 1.     

On the Aleksandrov-Fenchel inequality involving zonoids

The equality case in the general quadratic inequality V(K, L, K ₁, ..., K _n–2)² V(K, K, K ₁, ..., K _n–2) V(L, L, K ₁, ..., K _n–2) for mixed volumes is settled under the assumption that K and L are centrally symmetric and K ₁, ..., K _n–2 are zonoids. This result partly confirms a conjecture on the general case made in an earlier paper.     

The set of solutions of a proportionally modular Diophantine inequality

The set of solutions of the inequality is a numerical semigroup. We present in this paper a tool for finding the set of minimal generators of this set, and thus the set of solutions to such an inequality. This tool will also enable us to give characterizations of those numerical semigroups that are the set of integer solutions of inequalities of this form. Finally, we give a deeper study of the embedding dimension three case.     

混合幂为2,3,a和b的丢番图不等式

运用Davenport-Heilbronn方法证明了如果η是实数,λ1,μ1,μ2,μ3,μ4,θ1,θ2是非零实数,并且不同一符号,且至少一个λ1/μi(i=1,2,3,4)是无理教,假设(i)a=3,3≤b≤11,或者(ii)a=4,4≤b≤5,那么对某些σ=σ(a,b)>0,混合幂为2,3,a和b的丢番图不等式有无穷多正整数解x1,...,x7.     

An inequality in binary vector spaces

We prove that if an n-dimensional vector space over GF(2) is the irredundant union of k subspaces, and this collection of subspaces has zero intersection, then n>k. This settles a conjecture by G. Bruns.     

Universal binary Hermitian forms

We will determine (up to equivalence) all of the integral positive definite Hermitian lattices in imaginary quadratic fields of class number 1 that represent all positive integers.

     

Unsolvability of binary forms

A property of a binary unit is obtained which is invariant in all number fields of fixed degree n, n ≥ 3. This property is then used to formulate a necessary condition for the solvability of a binary form.