On a conjecture on exponential Diophantine equations

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

Elementary Trigonometric Sums related to Quadratic Residues

Let p be a prime = 3 (mod 4). A number of elegant number-theoretical properties of the sums T(p) = \sqrt{p}sum_{n=1}^{(p-1)/2} tan(n^2\pi/p) and C(p) = \sqrt{p}sum_{n=1}^{(p-1)/2} cot(n^2\pi/p) are proved. For example, T(p) equals p times the excess of the odd quadratic residues over the even ones in the set {1,2,...,p-1}; this number is positive if p = 3 (mod 8) and negative if p = 7 (mod 8). In this revised version the connection of these sums with the class-number h(-p) is also discussed. For example, a very simple formula expressing h(-p) by means of the aforementioned excess is proved. The bibliography has been considerably enriched. This article is of an expository nature.Comment: A number of misprints have been corrected and one or two improvements have been done to the previous version of the paper with same title. The paper will appear to Elem. der Mat

An inequality about irreducible factors of integer polynomials

AbstractWe give a new upper bound for the height of an irreducible factor of an integer polynomial. This paper also contains several bounds for the case of polynomials with complex coefficients

Applications of the representation of finite fields by matrices

AbstractWe consider the matrix well-known representation of K[X]/(P), when P is monic irreducible polynomial, with coefficients in K. This representation enables us to give a fast algorithm to solve the equation xd=a in a finite field

On the diophantine equation xp−x=yq−yx^p-x=y^q-y

We consider the diophantine equation x^p-x=y^q-y \tag"$(*)$" in integers $(x,p,y,q)$. We prove that for given $p$ and $q$ with 2\le p < q $(*)$ has only finitely many solutions. Assuming the abc-conjecture we can prove that $p$ and $q$ are bounded. In the special case $p=2$ and $y$ a prime power we are able to solve $(*)$ completely