关于方程xy+yz+zx=n的正整数解

本文在广义Ｒｉｅｍａｎｎ猜想成立的条件下证明了：当且仅当正整数ｎ＝１，２，４，６，１０，１８，２２，３０，４２，５８，７０，７８，１０２，１３０，１９０，２１０，３３０，４６２时，方程ｘｙ＋ｙｚ＋ｚｘ＝ｎ无正整数解（ｘ，ｙ，ｚ）．     

方程xy+yz+zx=n的正整数解

本文用 Ｓｉｅｇｅｌ－Ｔａｔｕｚａｗａ定理证明了：当ｎ＞１．２×１０~１１时,至多有两个正 整数ｎ。使方程ｘｕ＋ｙｚ＋ｚｘ＝ｎ无适合（ｘ,ｙ,ｚ）＝１且０＜ｘ＜ｙ＜ｚ的解（ｘ,ｙ,ｚ）, 并给出类数为２的二次域与多项式表素数的一个结果．     

Semigroups satisfying (xy)m = xmym

In this paper we characterize Archimedean semigroups with idempotents satisfying (xy)^m = x^my^m as exactly those semigroups which are a retract extension of a completely simple semigroup satisfying (xy)^m = x^my^m by a nil semigroup satisfying (xy)^m = x^my^m. Regular semigroups satisfying (xy)² = x²y² are exactly those semigroups which are a spined product of a band and a semigroup which is a semilattice of Abelian groups. A semigroup which is a nil extension of a regular semigroup satisfies (xy)² = x²y² if and only if it is a retract extension of a regular semigroup satisfying (xy)² = x²y² by a nil semigroup satisfying (xy)² = x²y²     

关于算子方程AX=XAX与AX=XA=XAX

用A的不变子空间作参数，给出了算子方程AX＝XAX的全部解。当A是单射或稠值域时，或者当A是正规算子时，给出了算子方程AX＝XA＝XAX的全部解。我们还给出正规算子X是算子方程AX＝XZ＝XAX的解的充分必要条件。     

Preserving Zeros of xy and xy*

We describe surjective additive maps θ: A → B which preserve zero products, where A is a ring with a nontrivial idempotent and B is a prime ring. We also characterize surjective additive maps θ: A → B such that for all x, y ∈ A we have θ(x)θ(y)* = 0 if and only if xy* = 0. Here A is a unital prime ring with involution that contains a nontrivial idempotent and B is a prime ring with involution.     

On the Equations zm = F(x, y) and Axp + Byq = Czr

We investigate integer solutions of the superelliptic equation where F is a homogeneous polynomialwith integer coefficients, and of the generalized Fermat equation where A, B and C are non-zero integers.Call an integer solution (x, y, z) to such an equation properif gcd(x, y, z) = 1. Using Faltings' Theorem, we shall givecriteria for these equations to have only finitely many propersolutions. We examine (1) using a descent technique of Kummer, which allowsus to obtain, from any infinite set of proper solutions to (1),infinitely many rational points on a curve of (usually) highgenus, thus contradicting Faltings' Theorem (for example, thisworks if F(t, 1) = 0 has three simple roots and m 4). We study (2) via a descent method which uses unramified coveringsof P₁ \ {0, 1, } of signature (p, q, r), and show that (2) hasonly finitely many proper solutions if l/p + l/q + 1/r <1. In cases where these coverings arise from modular curves,our descent leads naturally to the approach of Hellegouarchand Frey to Fermat's Last Theorem. We explain how their ideamay be exploited for other examples of (2). We then collect together a variety of results for (2) when 1/p+ 1/q + 1/r 1. In particular, we consider ‘local-global’principles for proper solutions, and consider solutions in functionfields.     

Growth series for the group 〈x,y¦x −1 yx=y l 〉

Supported by SERC Postgraduate Award.     

关于Pell方程组X~2-3y~2=1与y~2-Dz~2=1的解

设D=2p_1…P_s(1≤s≤4),P_1…,P_s是互异的奇素数.证明了:Pell方程组x~2-3y~2=1,y~2-Dz~2=1除开D=2×7,2×3×5×7×13外,仅有平凡解(x,y,z)=(±2,±1,0).     

Das goursat-problem füru xy =f (x,y, u)

Ohne Zusammenfassung     

The exponential diophantine equation x y + y x = y 2 with xy odd

For any fixed positive integer D which is not a square, let (u, υ) = (u ₁, υ ₁) be the fundamental solution of the Pell equation u ² ? Dυ ² = 1. Further let $\mathbb{D}$ be the set of all positive integers D such that D is odd, D is not a square and gcd(D, υ ₁) > max(1, √D/8). In this paper we prove that if (x, y, z) is a positive integer solution of the equation x ^y + y ^x = z ² satisfying gcd(x, y) = 1 and xy is odd, then either $x \in \mathbb{D}$ or $y \in \mathbb{D}$ .     

Das Goursat-Problem füru xy =f(x,y, u)

Ohne Zusammenfassung     

关于不定方程组x-1=3py^2，x^2＋x＋1=3z^2

设P为素数，利用同余及高次丢番图方程的一些结果证明了不定方程组x-1=3py^2，x^2＋x＋1=3z^2仅有正整数解（p，x，y，z）=（7，22，1，13）。     

Ringe mitx(yz)=(yx)z

Ohne ZusammenfassungVorgetragen auf der DMV-Tagung in Düsseldorf am 18.9. 1966.