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In this thesis three different topics in number theory are studied. The first part deals with exponential Diophantine equations in positive characteristic. In the second part various statistical properties of class groups are proven. In the final part we apply the circle method to count the number of representations of an odd integer n as the sum of three primes all with a fixed primitive root.

In this paper, we develop a new method for finding all perfect powers which can be expressed as the sum of two rational S-units, where S is a finite set of primes. Our approach is based upon the modularity of Galois representations and, for the most part, does not require lower bounds for linear forms in logarithms. Its main virtue is that it enables to carry out such a program explicitly, at least for certain small sets of primes S; we do so for S = {2, 3} and S = {3, 5, 7}., Comment: Missing solution in Prop. 5.4 added. To appear in Mathematics of Computation

The dynamic mechanical relaxation behavior of Ti36.2Zr30.3Cu8.3Fe4Be21.2 bulk metallic glass with good glass-forming ability was investigated by mechanical spectroscopy. The mechanical relaxation behavior was analyzed in the framework of quasi-point defects model. The experimental results demonstrate that the atomic mobility of the metallic glass is closely associated with the correlation factor χ. The physical aging below the glass transition temperature Tg shows a non-Debye relaxation behavior, which could be well described by stretched Kohlrausch exponential equation. The Kohlrausch exponent $$\beta_{\text{aging}}$$ reflects the dynamic heterogeneities of the metallic glass. Both concentration of “defects” and atomic mobility decrease caused by the in situ successive heating during the mechanical spectroscopy experiments.

In this work, we derive approximate semi-analytical solutions of the steady, creeping, weakly compressible plane and axisymmetric Poiseuille flows of a Herschel-Bulkley fluid. Since the flow is weakly compressible, the radial velocity component is assumed to be zero and the derivatives of the axial velocity with respect to the axial direction are assumed to be much smaller than those with respect to the radial direction. The axial velocity is then given by an expression similar to that holding for the incompressible flow, the only difference being that the pressure-gradient is a function of the axial coordinate and satisfies a non-linear equation involving the density of the fluid. In the present work, a linear as well as an exponential equation of state, relating the density of the fluid to the pressure, are considered. The pressure distribution along the flow direction is calculated by means of numerical integration and the two-dimensional axial velocity can then be constructed. The effects of compressibility, the equation of state, the Bingham number and the power-law exponent on the solutions are investigated. © 2008 Elsevier B.V. All rights reserved. 158 1-3 162 169 Cited By :13

We study the solutions of a Diophantine equation of the form $a^x+b^y=c^z$, where $a\equiv 2 \pmod 4$, $b\equiv 3 \pmod 4$ and $\gcd (a,b,c)=1$. The main result is that if there exists a solution $(x,y,z)=(2,2,r)$ with $r>1$ odd then this is the only solution in integers greater than 1, with the possible exception of finitely many values $(c,r)$. We also prove the uniqueness of such a solution if any of $a$, $b$, $c$ is a prime power. In a different vein, we obtain various inequalities that must be satisfied by the components of a putative second solution.

Let K be a field of characteristic 0. We consider linear equations a1*x1+...+an*xn=1 in unknowns x1,...,xn from G, where a1,...,an are non-zero elements of K, and where G is a subgroup of the multiplicative group of non-zero elements of K. Two tuples (a1,...,an) and (b1,...,bn) of non-zero elements of K are called G-equivalent if there are u1,...,un in G such that b1=a1*u1,..., bn=an*un. Denote by m(a1,...,an,G) the smallest number m such that the set of solutions of a1*x1+...+an*xn=1 in x1,...,xn from G is contained in the union of m proper linear subspaces of K^n. It is known that m(a1,...,an,G) is finite; clearly, this quantity does not change if (a1,...,an) is replaced by a G-equivalent tuple. Gyory and the author proved in 1988 that there is a constant c(n) depending only on the number of variables n, such that for all but finitely many G-equivalence classes (a1,...,an), one has m(a1,...,an,G)< c(n). It is as yet not clear what is the best possible value of c(n). Gyory and the author showed that c(n)=2^{(n+1)!} can be taken. This was improved by the author in 1993 to c(n)=(n!)^{2n+2}. In the present paper we improve this further to c(n)=2^{n+1}, and give an example showing that c(n) can not be smaller than n., 12 pages, latex file

In this paper we consider the Diophantine equation \begin{align*}b^k +\left(a+b\right)^k &+ \cdots + \left(a\left(x-1\right) + b\right)^k=\\ &=d^l + \left(c+d\right)^l + \cdots + \left(c\left(y-1\right) + d\right)^l, \end{align*} where $a,b,c,d,k,l$ are given integers. We prove that, under some reasonable assumptions, this equation has only finitely many integer solutions., This version differs slightly from the published version in its exposition

The aim of this diploma thesis is to develop a good teaching tool for study- ing exponential and logarithmic equations at the secondary school. It is not a textbook but an interactive website which contains dynamic elements and tests that can help stu- dents understand this subject matter. The website should mainly serve secondary school students as a supplementary study material. In the first chapter of the work the existing websites devoted to this theme are discussed and evaluated. On the basis of these and other educational sites the autor created a new website.

Opisane su metode za rješavanje specijalnih eksponencijalnih jednadžbi kod kojih su baza i eksponenti algebarske funkcije. Spomenute metode su ilustrirane na nizu zanimljivih zadataka koji su prilagođeni učenicima srednjih škola., Methods for solving special exponential equations whose bases and exponents are algebraic functions are described. Mentioned methods are illustrated on a number of interesting tasks that have been adapted for high school students.