Complete solution of the polynomial version of a problem of Diophantus

In this paper, we prove that if {a,b,c,d} is a set of four non-zero polynomials with integer coefficients, not all constant, such that the product of any two of its distinct elements plus 1 is a square of a polynomial with integer coefficients, then

(a+b−c−d)2=4(ab+1)(cd+1).     

On correspondence between solutions of a family of cubic Thue equations and isomorphism classes of the simplest cubic fields

Let m?−1 be an integer. We give a correspondence between integer solutions to the parametric family of cubic Thue equations

X³−mX²Y−(m+3)XY²−Y³=λ     

Heights of algebraic numbers modulo multiplicative group actions

Given a number field K and a subgroup G⊂K^∗ of the multiplicative group of K, Silverman defined the G-height H(θ;G) of an algebraic number θ as

     

Simplifying method for algebraic approximation of certain algebraic numbers

A new simple method for approximating certain algebraic numbers is developed. By applying this method, an effective upper bound is derived for the integral solutions of the quartic Thue equation with two parameters $tx^4 - 4sx^3 y - 6tx^2 y^2 + 4sxy^3 + ty^4 = N$ , where s > 32t ³. As an application, Ljunggren’s equation is solved in an elementary way.     

Fekete-like polynomials

In 2001, Borwein, Choi, and Yazdani looked at an extremal property of a class of polynomial with ±1 coefficients. Their key result was:

Theorem. (See Borwein, Choi, Yazdani, 2001.) Letf(z)=±z±z²±?±zN−1, and ζ a primitive Nth root of unity. If N is an odd positive integer then

     

A Cassels-Schmidt theorem for non-homogeneous Markov chains

Let r be an integer not less than 2. Suppose that we have a (not necessarily homogeneous) Markov chain with state space {0,1,…,r−1} given by the sequence of r×r transition matrices

     

Stability of the quadratic functional equation in Lipschitz spaces

Let G be an Abelian group with a metric d and E a normed space. For any f:G→E we define the quadratic difference of the function f by the formula

Qf(x,y):=2f(x)+2f(y)−f(x+y)−f(x−y)     

On the spacings between C-nomial coefficients

Let (Cn)_n?0 be the Lucas sequence Cn+2=aCn+1+bCn for all n?0, where C₀=0 and C₁=1. For 1?k?m−1 let

     

On certain exponential sums over primes

Let f(x) be a real valued polynomial in x of degree k?4 with leading coefficient α. In this paper, we prove a non-trivial upper bound for the quantity

     

The sixth and eighth moments of Fourier coefficients of cusp forms

Let λ(n) be the nth normalized Fourier coefficient of a holomorphic Hecke eigencuspform f(z) of even integral weight k for the full modular group. In this paper we are able to prove the following results.

(i)

For any ε>0, we have

     

On the Waring-Goldbach Problem for Six Cubes and Two Biquadrates

Let P_r denote an almost prime with at most r prime factors, counted according to multiplicity. In the present paper, it is proved that for any sufficiently large even integer n, the equation

$$n = {x^3} + p_1^3 + p_2^3 + p_3^3 + p_4^3 + p_5^3 + p_6^4 + p_7^4$$

has solutions in primes p_i with x being a P₆. This result constitutes a refinement upon that of Hooley C.

     

On the number of solutions of a diophantine equation with symmetric entries

For a class of strictly increasing real valued functions f(n) we obtain an upper bound for the number of solutions of the equation

     

On the distribution of zeros of solutions of first order delay differential equations

This paper contains new estimates for the distance between adjacent zeros of solutions of the first order delay differential equation

x^′(t)+p(t)x(t−τ)=0     

The exceptional set for Diophantine inequality with unlike powers of prime variables

Suppose that λ₁, λ₂, λ₃, λ₄ are nonzero real numbers, not all negative, δ > 0, V is a well-spaced set, and the ratio λ₁/λ₂ is algebraic and irrational. Denote by E(V,N, δ) the number of v ∈ V with v ≤ N such that the inequality

$$\left| {{\lambda _1}p_1^2 + {\lambda _2}p_2^3 + {\lambda _3}p_3^4 + {\lambda _4}p_4^5 - \upsilon } \right| < {\upsilon ^{ - \delta }}$$

has no solution in primes p₁, p₂, p₃, p₄. We show that

$$E\left( {\upsilon ,N,\delta } \right) \ll {N^{1 + 2\delta - 1/72 + \varepsilon }}$$

for any ? > 0.

     

On character sums over flat numbers

Let q?2 be an integer, χ be any non-principal character mod q, and H=H(q)?q. In this paper the authors prove some estimates for character sums of the form

     

Hyperbolic prime number theorem

We count the number S(x) of quadruples \( {\left( {x_{1} ,x_{2} ,x_{3} ,x_{4} } \right)} \in \mathbb{Z}^{4} \) for which

$ p = x^{2}_{1} + x^{2}_{2} + x^{2}_{3} + x^{2}_{4} \leqslant x $

is a prime number and satisfying the determinant condition: x ₁ x ₄???x ₂ x ₃?=?1. By means of the sieve, one shows easily the upper bound S(x)???x/log x. Under a hypothesis about prime numbers, which is stronger than the Bombieri–Vinogradov theorem but is weaker than the Elliott–Halberstam conjecture, we prove that this order is correct, that is S(x)???x/log x.

     

Some identities involving theta functions

Let (|q|<1). For k∈N it is shown that there exist k rational numbers A(k,0),…,A(k,k−1) such that

     

Simple arguments on consecutive power residues

By some extremely simple arguments, we point out the following:

(i)

If n is the least positive kth power non-residue modulo a positive integer m, then the greatest number of consecutive kth power residues mod m is smaller than m/n.

(ii)

Let OK be the ring of algebraic integers in a quadratic field with d∈{−1,−2,−3,−7,−11}. Then, for any irreducible π∈OK and positive integer k not relatively prime to , there exists a kth power non-residue ω∈OK modulo π such that .

     

Note on integer powers in sumsets

Let k,m,n?2 be integers. Let A be a subset of {0,1,…,n} with 0∈A and the greatest common divisor of all elements of A is 1. Suppose that