The diophantine equation x 2 + 2 a · 17 b = y n

Let ?, ? be the sets of all integers and positive integers, respectively. Let p be a fixed odd prime. Recently, there have been many papers concerned with solutions (x, y, n, a, b) of the equation x ² + 2 ^a p ^b = y ⁿ , x, y, n ε ?, gcd(x, y) = 1, n ? 3, a, b ε ?, a ? 0, b ? 0. And all solutions of it have been determined for the cases p = 3, p = 5, p = 11 and p = 13. In this paper, we mainly concentrate on the case p = 3, and using certain recent results on exponential diophantine equations including the famous Catalan equation, all solutions (x, y, n, a, b) of the equation x ²+2 ^a · 17 ^b = y ⁿ , x, y, n ε ?, gcd(x, y) = 1, n ? 3, a, b ∈ ?, a ? 0, b ? 0, are determined.     

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).     

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).     

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 diophantine equation y2 = 4qn + 4q + 1

It is known that a certain class of [n, k] codes over GF(q) is related to the diophantine equation y² = 4qⁿ + 4q + 1 (1). In Parts I and II of this paper, two different, and in a certain sense complementary, methods of approach to (1) are discussed and some results concerning (1) are given as applications. A typical result is that the only solutions to (1) are (y, n) = (5, 1), (7, 2), (11, 3) when q = 3 and (y, n) = (2q + 1, 2) when q = 3^f, f >- 2.     

Bounds for singular integrals associated with a class of hypersurfaces

For the hypersurface Γ=(y,γ(y)), the singular integral operator along Γ is defined by. $$Tf(x,x_n ) = P.V.\int_{\mathbb{R}^n } {, f(x - y,x_n ) - } \gamma (y))_{\left| y \right|^{n - 1} }^{\Omega (v)} dy$$ where Σ is homogeneous of order 0, $ \int_{\Sigma _{n \lambda } } {\Omega (y')dy'} = 0 $ . For a certain class of hypersurfaces, T is shown to be bounded on L^p(Rⁿ) provided Ω∈L _α ¹ (Σ _n?2),P>1.     

Existence of triple solutions of discrete (n,p) boundary value problems

We consider the following boundary value problem: −Δⁿy = F(k,y, Δy,…,Δⁿ⁻¹y), k ϵ Z[n − 1, N], Δⁱy(0) = 0, 0 ≤ i ≤ n − 2, Δ^py(N + n - p) = 0, where n ≥ 2 and p is a fixed integer satisfying 0 ≤ p ≤ n − 1. Using a fixed-point theorem for operators on a cone, we shall yield the existence of at least three positive solutions.     

On almost-quasi-commutative rings

A ring R is called almost-quasi-commutative if for each x, y ∈ R there exist nonzero relatively prime integers j = j(x, y) and k = k(x, y) and a non-negative integer n = n(x, y) such that jxy = k(yx) ⁿ . We establish some general properties of such rings, study commutativity of almost-quasi-commutative R, and consider several examples.     

Positive integer solutions of the diophantine equation x 2 − L n xy + (−1) n y 2 = ±5 r

In this paper, we consider the equation x ²?L _n x y+(?1) ⁿ y ² = ±5 ^r and determine the values of n for which the equation has positive integer solutions x and y. Moreover, we give all positive integer solutions of the equation x ²?L _n x y+(?1) ⁿ y ² = ±5 ^r when the equation has positive integer solutions.     

Realization of Boolean functions by formulas in continuous bases containing a continuum of constants

For an arbitrary Boolean function of n variables, we show how to construct formulas of complexity O(2 ^n/2) in the bases $$\left\{ {x - y,xy,\left| x \right|} \right\}\bigcup {\left[ {0,1} \right], } \left\{ {x - y,x*y,2x,\left| x \right|} \right\}\bigcup {\left[ {0,1} \right],}$$ , where x * y = max(?1, min(1, x))max(?1, min(1, y)). The obtained estimates are, in general, order-sharp.     

Sharp estimates for the convergence rate of “hyperbolic” partial sums of double fourier series in orthogonal polynomials

Two-variable functions f(x, y) from the class L ₂ = L ₂((a, b) × (c, d); p(x)q(y)) with the weight p(x)q(y) and the norm $$\left\| f \right\| = \sqrt {\int\limits_a^b {\int\limits_c^d {p(x)q(x)f^2 (x,y)dxdy} } }$$ are approximated by an orthonormal system of orthogonal P _n (x)Q _n (y), n, m = 0, 1, ..., with weights p(x) and q(y). Let $$E_N (f) = \mathop {\inf }\limits_{P_N } \left\| {f - P_N } \right\|$$ denote the best approximation of f ?? L ₂ by algebraic polynomials of the form $$\begin{array}{*{20}c} {P_N (x,y) = \sum\limits_{0 < n,m < N} {a_{m,n} x^n y^m ,} } \\ {P_1 (x,y) = const.} \\ \end{array}$$ . Consider a double Fourier series of f ?? L ₂ in the polynomials P _n (x)Q _m (y), n, m = 0, 1, ..., and its ??hyperbolic?? partial sums $$\begin{array}{*{20}c} {S_1 (f;x,y) = c_{0,0} (f)P_o (x)Q_o (y),} \\ {S_N (f;x,y) = \sum\limits_{0 < n,m < N} {c_{n,m} (f)P_n (x)Q_m (y), N = 2,3, \ldots .} } \\ \end{array}$$ A generalized shift operator Fh and a kth-order generalized modulus of continuity ?? _k (A, h) of a function f ?? L ₂ are used to prove the following sharp estimate for the convergence rate of the approximation: $\begin{gathered} E_N (f) \leqslant (1 - (1 - h)^{2\sqrt N } )^{ - k} \Omega _k (f;h),h \in (0,1), \hfill \\ N = 4,5,...;k = 1,2,... \hfill \\ \end{gathered} $ . Moreover, for every fixed N = 4, 9, 16, ..., the constant on the right-hand side of this inequality is cannot be reduced.     

Commutativity of rings with derivations

I. N. Herstein [10] proved that a prime ring of characteristic not two with a nonzero derivation d satisfying d(x)d(y) = d(y)d(x) for all x, y must be commutative, and H. E. Bell and M. N. Daif [8] showed that a prime ring of arbitrary characteristic with nonzero derivation d satisfying d(xy) = d(yx) for all x, y in some nonzero ideal must also be commutative. For semiprime rings, we show that an inner derivation satisfying the condition of Bell and Daif on a nonzero ideal must be zero on that ideal, and for rings with identity, we generalize all three results to conditions on derivations of powers and powers of derivations. For example, let R be a prime ring with identity and nonzero derivation d, and let m and n be positive integers such that, when charR is finite, m ∨ n < charR. If d(x ^m y ⁿ ) = d(y ⁿ x ^m ) for all x, y ∈ R, then R is commutative. If, in addition, charR≠ 2 and the identity is in the image of an ideal I under d, then d(x) ^m d(y) ⁿ = d(y) ⁿ d(x) ^m for all x, y ∈ I also implies that R is commutative.     

On Heuvers’ Logarithmic Functional Equation

?(x + y) - ?(x) - ?(y) = ?(x ^?1 + y ^?l) are identical to those of the Cauchy equation ?(xy) = ?(x) + ?(y) when ? is a function from the positive real numbers into the reals. In the present article, we prove this equivalence for functions mapping the set of nonzero elements of a field (excluding ?₂) .     

Second order linear O.D.E. and Riccati equations

We prove that approximate solutions of the Riccati equation ?′ + ?² = a(x) yield asymptotic solutions y = e^∝x?(s)ds of the second order linear equation y″ = a(x)y. We show that the iterative scheme ?₀ = a, ?_{n + 1}² = a ? ?_n′ leads to asymptotic solutions of the cited linear equation in many interesting cases.     

Solution of the Szili type conjugate problem on the roots of the laguerre polynomials

Let ?1<α≤0 and let $$L_n^{(\alpha )} (x) = \frac{1}{{n!}}x^{ - \alpha } e^x \frac{{d^n }}{{dx^n }}(x^{\alpha + n} e^{ - x} )$$ be the generalizednth Laguerre polynomial,n=1,2,… Letx ₁,x ₂,…,x _n andx*₁,x*₂,…,x* _n?1 denote the roots ofL _n ^(α) (x) andL _n ^(α)′ (x) respectively and putx*₀=0. In this paper we prove the following theorem: Ify ₀,y ₁,…,y _n ?1 andy ₁ ^′ ,…,y _n ^′ are two systems of arbitrary real numbers, then there exists a unique polynomialP(x) of degree 2n?1 satisfying the conditions $$\begin{gathered} P\left( {x_k^* } \right) = y_k (k = 0,...,n - 1) \hfill \\ P'\left( {x_k } \right) = y_k^\prime (k = 1,...,n). \hfill \\ \end{gathered} $$ .     

The integral points on elliptic curves y 2 = x 3 + (36n 2 − 9)x − 2(36n 2 − 5)

Let n be a positive odd integer. In this paper, combining some properties of quadratic and quartic diophantine equations with elementary analysis, we prove that if n > 1 and both 6n ² ? 1 and 12n ² + 1 are odd primes, then the general elliptic curve y ² = x ³+(36n ²?9)x?2(36n ²?5) has only the integral point (x, y) = (2, 0). By this result we can get that the above elliptic curve has only the trivial integral point for n = 3, 13, 17 etc. Thus it can be seen that the elliptic curve y ² = x ³ + 27x ? 62 really is an unusual elliptic curve which has large integral points.     

Sur l'équation diophantienne

We announce several new theorems on the Diophantine equation (xⁿ−1)/(x−1)=y^q, which include the fact that any integer greater than 2 and with all digits equal to 1 in base ten cannot be a pure power. We then apply these results to solve this equation for any integer x of the form x=z¹ with 2 ≤ z ≤ 10000 and t ≥ 1.     

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.     

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.