The complete solution in integers of x3 + 3y3 = 2n

Some general remarks are made concerning the equation f(x, y) = qⁿ in the integral unknowns x, y, n, where f is an integral form and q > 1 is a given integer. It is proved that the only integral triads (x, y, n) satisfying x³ + 3y³ = 2ⁿ are (x, y, n) = (?1, 1, 1), (1, 1, 2), (?7, 5, 5,), (5, 1, 7).     

Singular (k,n − k) boundary value problems between conjugate and right focal

For 1 ⩽k⩽n − 1 and 0 ⩽q⩽k − 1, solutions are obtained for the boundary value problem, (−1)^n−k = f(x,y), y⁽ⁱ⁾=0, 0⩽i⩽k − 1, and y⁽ⁱ⁾ = 0, q⩽j⩽n − k + q − 1, where f(x,y) is singular at y = 0. An application is made of a fixed point theorem for operators that are decreasing with respect to a cone.     

The exponential Diophantine equation x2 + (3n 2 + 1) y = (4n 2 + 1) z

Let n be a positive integer. In this paper, using the results on the existence of primitive divisors of Lucas numbers and some properties of quadratic and exponential diophantine equations, we prove that if n ≡ 3 (mod 6), then the equation x ² + (3n ² + 1) ^y = (4n ² + 1) ^z has only the positive integer solutions (x, y, z) = (n, 1, 1) and (8n ³ + 3n, 1, 3).     

An application of L-functions of imaginary quadratic fields

Let F₁(x, y),…, F_2h+1(x, y) be the representatives of equivalent classes of positive definite binary quadratic forms of discriminant ?q (q is a prime such that q ≡ 3 mod 4) with integer coefficients, then the number of integer solutions of F_i(x, y) = n (i = 1,…, 2h + 1) can be calculated for each natural number n using L-functions of imaginary quadratic field $\text{Q}$((?q)^1/2).     

Enumerating pairs of permutations with the same up-down form

For each sequence q = {q_i} = ± 1, i = 1, …, n−1 let N_q = the number of permutations σ of 1, 2, …, n with up-down sequence sgn(σ_i+1 − σ_i) = q_i, i = 1,…, n−1. Clearly Σ_q (N_q/n!) =1 but what is the probability p_n = Σ_q (N_q/n!)² that two random permutations have the same up-down sequence? We show that p_n = (Kⁿ⁻¹1,1) where 1 = 1(x, y) ≡ 1 and Kⁿ⁻¹ is the iterated integral operator with Kφ(x, y) = ∫₀¹∫₀¹ K(x, y; x′, y′)φ(x′, y′) dx′ dy′ on L²[0, 1] × [0, 1] where K(x, y; x′, y′) is 1 if (x− x′)(y − y′) > 0 otherwise, and (f, g) = ∫₀¹∫₀¹fg. The eigenexpression of K yeilds p_n ∼ cαⁿ as n → ∞, where c ≈ 1.6, α ≈ 0.55.We also give a recursion formula for a polynomial whose coefficients are the frequencies of all the possible forms.     

Obrechkoff methods having additional parameters for general second-order differential equations

A class of two-step implicit methods involving higher-order derivatives of y for initial value problems of the form y″ = f(t, y, y′) is developed. The methods involve arbitrary parameters p and q, which are determined so that the methods become absolutely stable when applied to the test equation y″ + λy′ + μy = 0. Numerical results for Bessel's and general second-order differential equations are presented to illustrate that the methods are absolutely stable and are of order O(h⁴), O(h⁶) and O(h⁸).     

Short cubic exponential sums over primes

For y ≥ x ^4/5 L ^8B+151 (where L = log(xq) and B is an absolute constant), a nontrivial estimate is obtained for short cubic exponential sums over primes of the form S ₃(α; x, y) = ∑ _x?y<n≤x Λ(n)e(αn ³), where α = a/q + θ/q ², (a, q) = 1, L ^32(B+20) < q ≤ y ⁵ x ^?2 L ^?32(B+20), |θ| ≤ 1, Λ is the von Mangoldt function, and e(t) = e ^2πit.     

The diophantine equation x3 + 3y3 = 2n

It is proved that the equation of the title has a finite number of integral solutions (x, y, n) and necessary conditions are given for (x, y, n) in order that it can be a solution (Theorem 2). It is also proved that for a given odd x₀ there is at most one integral solution (y, n), n ≥ 3, to x₀³ + 3y³ = 2ⁿ and for a given odd y₀ there is at most one integral solution (x, n), n ≥ 3, to x³ + 3y₀³ = 2ⁿ.     

On the nonvanishing of homogeneous product sums

For integer n ≥ 1 let H_n = H_n(x, y, z) = Σ_{p + q + r = n}x^py^qz^r be the homogeneous product sum of weight n on three letters x, y, z. Morgan Ward conjectured that H_n ≠ 0 for all integers n, x, y, z with n > 1 and xyz ≠ 0. In support of this conjecture he proved that H_n ≠ 0 if n is even or if n + 2 is a prime number greater than 3. This paper adds considerably more evidence in support of Ward's conjecture by showing that in many cases H_n(a, b, c)¬=0 modulo 2, 4, or 16. The parity of H_n(a, b, c) is determined in all cases and, when H_n(a, b, c) is even, further congruences are given modulo 4 or 16.     

Uniqueness and existence for nth-order right focal boundary value problems

Assuming f is bounded and solutions to the linearized equation are unique, the uniqueness and existence of solutions is established for solutions of the equation y⁽ⁿ⁾ = f(t,y,y′,…,y⁽ⁿ⁻¹⁾) subject to the right focal boundary conditions.     

A second-order nonlinear difference equation: Oscillation and asymptotic behavior

Discrete analogues are investigated for well-known results on oscillation, growth, and asymptotic behavior of solutions of y″ + q(t) y^γ = 0, for q(t) ? 0 and for q(t) ? 0. The analogue of Atkinson's oscillation criterion is shown to be true for Δ²y_{n ? 1} + q_ny_n^γ = 0, but the analogue for Atkinson's nonoscillation criterion is shown to be false.     

Error bounds for least squares approximation by polynomials

Let f ε Cⁿ⁺¹[−1, 1] and let H[f](x) be the nth degree weighted least squares polynomial approximation to f with respect to the orthonormal polynomials q_k associated with a distribution dα on [−1, 1]. It is shown that if q_n+1/q_n max(q_n+1(1)/q_n(1), −q_n+1(−1)/q_n(−1)), then f − H[f] f^{n + 1} · q_n+1/q_{n + 1}^{(n + 1)}, where · denotes the supremum norm. Furthermore, it is shown that in the case of Jacobi polynomials with distribution (1 − t)^α (1 + t)^β dt, α, β > −1, the condition on q_n+1/q_n is satisfied when either max(α,β) −1/2 or −1 < α = β < −1/2.     

Exponential polynomials as solutions of certain nonlinear difference equations

Recently, C.-C. Yang and I. Laine have investigated finite order entire solutions f of nonlinear differential-difference equations of the form fn + L(z, f ) = h, where n ≥ 2 is an integer. In particular, it is known that the equation f(z)2 + q(z)f (z + 1) = p(z), where p(z), q(z) are polynomials, has no transcendental entire solutions of finite order. Assuming that Q(z) is also a polynomial and c ∈ C, equations of the form f(z)n + q(z)e Q(z) f(z + c) = p(z) do posses finite order entire solutions. A classification of these solutions in terms of growth and zero distribution will be given. In particular, it is shown that any exponential polynomial solution must reduce to a rather specific form. This reasoning relies on an earlier paper due to N. Steinmetz.     

On a conjecture of Kippenhahn about the characteristic polynomial of a pencil generated by two Hermitian matrices. I

Let A be an n × n complex matrix, and write A = H + iK, where i² = ?1 and H and K are Hermitian matrices. The characteristic polynomial of the pencil xH + yK is f(x, y, z) = det(zI ? xH ? yK). Suppose f(x, y, z) is factored into a product of irreducible polynomials. Kippenhahn [5, p. 212] conjectured that if there is a repeated factor, then there is a unitary matrix U such that U^?1AU is block diagonal. We prove that if f(x, y, z) has a linear factor of multiplicity greater than n?3, then H and K have a common eigenvector. This may be viewed as a special case of Kippenhahn’s conjecture.     

On sizes of complete caps in projective spaces PG(n, q) and arcs in planes PG(2, q)

More than thirty new upper bounds on the smallest size t ₂(2, q) of a complete arc in the plane PG(2, q) are obtained for (169 ≤ q ≤ 839. New upper bounds on the smallest size t ₂(n, q) of the complete cap in the space PG(n, q) are given for n = 3 and 25 ≤ q ≤ 97, q odd; n = 4 and q = 7, 8, 11, 13, 17; n = 5 and q = 5, 7, 8, 9; n = 6 and q = 4, 8. The bounds are obtained by computer search for new small complete arcs and caps. New upper bounds on the largest size m ₂(n, q) of a complete cap in PG(n, q) are given for q = 4, n = 5, 6, and q = 3, n = 7, 8, 9. The new lower bound 534 ≤ m ₂(8, 3) is obtained by finding a complete 534-cap in PG(8, 3). Many new sizes of complete arcs and caps are obtained. The updated tables of upper bounds for t ₂(n, q), n ≥ 2, and of the spectrum of known sizes for complete caps are given. Interesting complete caps in PG(3, q) of large size are described. A proof of the construction of complete caps in PG(3, 2 ^h ) announced in previous papers is given; this is modified from a construction of Segre. In PG(2, q), for q = 17, δ = 4, and q = 19, 27, δ = 3, we give complete ${(\frac{1}{2}(q + 3) + \delta)}$ -arcs other than conics that share ${\frac{1}{2}(q + 3)}$ points with an irreducible conic. It is shown that they are unique up to collineation. In PG(2, q), ${{q \equiv 2}}$ (mod 3) odd, we propose new constructions of ${\frac{1}{2} (q + 7)}$ -arcs and show that they are complete for q ≤ 3701.     

The exponential Lebesgue-Nagell equation X 2 + P 2m = Y n

Let p be an odd prime. In this paper, a complete classification of all positive integer solutions (x, y, m, n) of the equation x ²+p ^2m = y ⁿ , gcd(x, y) = 1, n > 2, is given. As a consequence, we solve the equation for certain interesting cases.     

The estimate for mean values on prime numbers relative to \frac{4} {p} = \frac{1} {{n_1 }} + \frac{1} {{n_2 }} + \frac{1} {{n_3 }}

If n is a positive integer,let f （n） denote the number of positive integer solutions （n 1,n 2,n 3） of the Diophantine equation 4/n=1/n₁ + 1/n₂ + 1/n₃.For the prime number p,f （p） can be split into f 1 （p） + f 2 （p）,where f i （p） （i=1,2） counts those solutions with exactly i of denominators n 1,n 2,n 3 divisible by p.In this paper,we shall study the estimate for mean values ∑ p相似文献   

Uniqueness implies existence and uniqueness conditions for nonlocal boundary value problems for nth order differential equations

For the nth order differential equation, y(n)=f(x,y,y^′,…,y(n−1)), we consider uniqueness implies existence results for solutions satisfying certain nonlocal (k+2)-point boundary conditions, 1?k?n−1. Uniqueness of solutions when k=n−1 is intimately related to uniqueness of solutions when 1?k?n−2. These relationships are investigated as well.     

EXISTENCE OF FAST-DECAY GROUND STATE OF P-LAPLACIAN EQUATION

Using the shooting argument and an approximating method, this paper is concerned with the existence of fast-decay ground state of p-Laplacian equation: Δ_pu + f(u) = 0, in Rⁿ, where f(u) behaves just like f(u) = u^q – u^s, as s > q >np/(n – p) – 1.     

Conditional equations for quadratic functions

We consider quadratic functions f that satisfy the additional equation y² f(x) =  x² f(y) for the pairs \({ (x,y) \in \mathbb{R}^2}\) that fulfill the condition P(x, y) =  0 for some fixed polynomial P of two variables. If P(x, y) =  ax +  by +  c with \({ a , b , c \in \mathbb{R}}\) and \({(a^2 + b^2)c \neq 0}\) or P(x,y) =  x ⁿ ? y with a natural number \({n \geq 2}\), we prove that f(x) =  f(1) x² for all \({x \in \mathbb{R}}\). Some related problems, admitting quadratic functions generated by derivations, are considered as well.