The zero-multiplicity of third-order linear recurrences associated to the Tribonacci sequence

In this paper, we consider third-order linear recurrences _{un}n≥0${\left\{u_n\right\}}_{n\ge 0}$ satisfying the recurrence relation _un+3=_un+2+_un+1+_un$u_{n+3}=u_{n+2}+u_{n+1}+u_n$ for all n≥0$n\ge 0$ and investigate the multiplicity of its zeros. We prove that _{un}n≥0${\left\{u_n\right\}}_{n\ge 0}$ has zero-multiplicity at most 2, except for nonzero multiples of shifts of the Tribonacci sequence which has zero-multiplicity 4 when the indices are extended to all the integers.     

Quartic Thue equations

We will give upper bounds upon the number of integral solutions to binary quartic Thue equations. We will also study the geometric properties of a specific family of binary quartic Thue equations to establish sharper upper bounds.     

Solving superelliptic Diophantine equations by Baker's method

We describe a method for complete solution of the superelliptic Diophantine equation ay^p=f(x). The method is based on Baker's theory of linear forms in the logarithms. The characteristic feature of our approach (as compared with the classical method) is that we reduce the equation directly to the linear forms in logarithms, without intermediate use of Thue and linear unit equations. We show that the reduction method of Baker and Davenport [3] is applicable for superelliptic equations, and develop a very efficient method for enumerating the solutions below the reduced bound. The method requires computing the algebraic data in number fields of degree pn(n-1)/2 at most; in many cases this number can be reduced. Two examples with p=3 and n=4 are given.     

关于指数丢番图方程a~x+(3a~2-1)~y=(4a~2-1)~z的正整数解(英文)

本文通过计算Jacobi符号,运用代数数的对数线性型的下界估计,证明了:当整数a>1时,指数丢番图方程a~x+(3a~2-1)~y=(4a~2-1)~z仅有正整数解(x,y,z)=(2,1,1).     

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     

On the number of solutions to systems of Pell equations

We prove that, for positive integers a, b, c and d with c≠d, a>1, b>1, the number of simultaneous solutions in positive integers to ax²−cz²=1, by²−dz²=1 is at most two. This result is the best possible one. We prove a similar result for the system of equations x²−ay²=1, z²−bx²=1.     

Linear Forms in two m-adic Logarithms and Applications to Diophantine Problems

We give sharp, explicit estimates for linear forms in two logarithms, simultaneously for several non-Archimedean valuations. We present applications to explicit lower bounds for the fractional part of powers of rational numbers, and to the Diophantine equation (x ⁿ – 1)/(x – 1) = y ^q .     

On the ratio of two blocks of consecutive integers

Under certain assumptions, it is shown that eq. (2) has only finitely many solutions in integersx≥0,y≥0,k≥2,l≥0. In particular, it is proved that (2) witha=b=1, l=k implies thatx=7,y=0,k=3.     

A hypergeometric approach,via linear forms involving logarithms,to criteria for irrationality of Euler’s constant

Using an integral of a hypergeometric function, we give necessary and sufficient conditions for irrationality of Euler’s constant γ. The proof is by reduction to known irrationality criteria for γ involving a Beukers-type double integral. We show that the hypergeometric and double integrals are equal by evaluating them. To do this, we introduce a construction of linear forms in 1, γ, and logarithms from Nesterenko-type series of rational functions. In the Appendix, S. Zlobin gives a change-of-variables proof that the series and the double integral are equal.      

关于丢番图方程$x^y+y^x=z^z$

利用$p$-adic对数线性型估计, 证明了方程x^y+y^x=z^z满足x,y,z均大于1的整数解(x,y,z)必然两两互素且有z<2.8*10^9.     

On sets of integers whose shifted products are powers

Let N be a positive integer and let A be a subset of {1,…,N} with the property that aa^′+1 is a pure power whenever a and a^′ are distinct elements of A. We prove that |A|, the cardinality of A, is not large. In particular, we show that |A|?(logN)^2/3(loglogN)^1/3.     

Higher discriminants of binary forms

We propose a method for constructing systems of polynomial equations that define submanifolds of degenerate binary forms of an arbitrary degeneracy degree. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 153, No. 2, pp. 147–157, November, 2007.     

Computing partial information out of intractable: Powers of algebraic numbers as an example

We consider an algorithmic problem of computing the first, i.e., the most significant digits of n² (in base 3) and of the nth Fibonacci number. While the decidability is trivial, efficient algorithms for those problems are not immediate. We show, based on Baker's inapproximability results of transcendental numbers that both of the above problems can be solved in polynomial time with respect to the length of n. We point out that our approach works also for much more general expressions of algebraic numbers.