Small prime solutions of quadratic equations II

Let be non-zero integers and any integer. Suppose that and for . In this paper we prove that (i) if the are not all of the same sign, then the above quadratic equation has prime solutions satisfying and (ii) if all the are positive and , then the quadratic equation is soluble in primes Our previous results are and in place of and above, respectively.

     

Small prime solutions of some additive equations

Essentially sharp bounds for small prime solutionsp _j ,q _i of the following two different types of equations are obtained.

Small prime solutions of a pair of linear equations in five prime variables

Assume an additional congruent condition on the coefficients. We prove that the pair 5 of linear equations ∑j=1^5 αλjpj = bλ （λ= 1, 2） has solutions in primes pj satisfying pj 〈〈 （｜b1｜＋｜b2｜＋1） maxλ,j ｜αλj｜^2318＋ε. This improves the exponent 79680 without assuming the additional condition of the second author＇s.     

Small solutions of quadratic equations with prime variables in arithmetic progressions

A necessary and sufficient solvable condition for diagonal quadratic equation with prime variables in arithmetic progressions is given, and the best qualitative bound for small solutions of the equation is obtained,     

Small solutions of additive cubic congruences

We give a somewhat improved estimate for the smallest non-trivial solution of a cubic congruence of the form a₁ x₁³ + ?+ a_s x_s³ o 0  (mod  m)a_1 x_1^3 + cdots + a_s x_s^3 equiv 0 ,(hbox {mod},, m). This also yields new results about small fractional parts of additive cubic forms, where we restrict ourselves to the case of five variables.     

Explicit and exact solutions to cubic Duffing and double-well Duffing equations

In this paper, a kind of explicit exact solution of nonlinear differential equations is obtained using a new approach applied in this case to look for exact solutions of the Duffing and double-well Duffing equations. The new proposed procedure is applied by using a quotient trigonometric function expansion method. The method can also be easily applied to solve other nonlinear differential equations.     

Conditional bounds for small prime solutions of linear equations

Leta ₁,a ₂,a ₃ be non-zero integers with gcd(a ₁ a ₂,a ₃)=1 and letb be an arbitrary integer satisfying gcd (b, a _i,a _j) =1 fori≠j andb≡a ₁+a ₂+a ₃ (mod 2). In a previous paper [3] which completely settled a problem of A. Baker, the 2nd and 3rd authors proved that ifa ₁,a ₂,a ₃ are not all of the same sign, then the equationa ₁ p ₁+a ₂ p ₂+a ₃ p ₃=b has a solution in primesp _j satisfying $$\mathop {\max }\limits_{1 \leqslant j \leqslant 3} p_j \leqslant 3\left| b \right| + (3\mathop {\max }\limits_{1 \leqslant j \leqslant 3} \left| {a_j } \right|)^A $$ whereA>0 is an absolute constant. In this paper, under the Generalized Riemann Hypothesis, the authors obtain a more precise bound for the solutionsp _j . In particular they obtainA<4+∈ for some ∈>0. An immediate consquence of the main result is that the Linnik's courtant is less than or equal to 2.     

Number of solutions for cubic Thue equations with automorphisms

Let be an irreducible cubic form with positive discriminant, and with non-trivial automorphisms. We show that the Thue equation F(x,y) = 1 has at most three integer solutions except for a few known cases. For the proof, we use an explicitly expressed cubic form which is equivalent to F. To obtain an upper bound for the size of solutions, we use the Padé approximation method developed in our former work. To obtain a lower bound for the size of solutions, we use a result of R. Okazaki on gaps between solutions, which is obtained by geometric consideration. 2000 Mathematics Subject Classification Primary—11D25, 11D59     

Small solutions of linear Diophantine equations

Let Ax = B be a system of m × n linear equations with integer coefficients. Assume the rows of A are linearly independent and denote by X (respectively Y) the maximum of the absolute values of the m × m minors of the matrix A (the augmented matrix (A, B)). If the system has a solution in nonnegative integers, it is proved that the system has a solution X = (x_i) in nonnegative integers wity x_i ⩽ X for n - m variables and x_i ⩽ (m - m + 1)Y for m variables. This improves previous results of the authors and others.     

A numerical bound for small prime solutions of some binary equations

In this paper we consider the linear equation a1p1+ a2p2 = n in prime variables pi and estimate the numerical value of a relevant constant in the upper bound for small prime solutions of the above equation in terms of max ai.     

Small perturbation solutions for nonlocal elliptic equations

We present a small perturbation result for nonlocal elliptic equations, which says that for a class of nonlocal operators, the solutions are in C^σ+α for any α∈(0,1) as long as the solutions are small. This is a nonlocal generalization of a celebrated result of Savin in the case of second order equations.     

On the solutions of a parametric family of cubic Thue equations

In this paper, we solve a family of Diophantine equations associated with families of number fields of degree 3. In fact, we use Baker’s method find all solutions to the Thue equation

Asymptotics of odd solutions for cubic nonlinear Schrödinger equations

We consider the Cauchy problem for a cubic nonlinear Schrödinger equation in the case of an odd initial data from H²∩H0,2. We prove the global existence in time of solutions to the Cauchy problem and construct the modified asymptotics for large values of time.     

Kinklike solutions of fourth-order differential equations with a cubic bistable nonlinearity

For the equation y ⁽⁴⁾+2y(y ²?1) = 0, we suggest an analytic construction of kinklike solutions (solutions bounded on the entire line and having finitely many zeros) in the form of rapidly convergent series in products of exponential and trigonometric functions. We show that, to within sign and shift, kinklike solutions are uniquely characterized by the tuple of integers n ₁, …, n _k (the integer parts of distances, divided by π, between the successive zeros of these solutions). The positivity of the spatial entropy indicates the existence of chaotic solutions of this equation.