1,308 research outputs found
On a conjecture on exponential Diophantine equations
We study the solutions of a Diophantine equation of the form ,
where , and . The main
result is that if there exists a solution with odd then
this is the only solution in integers greater than 1, with the possible
exception of finitely many values . We also prove the uniqueness of such
a solution if any of , , 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
We consider the diophantine equation
x^p-x=y^q-y
\tag"$(*)$"
in integers . We prove that for given and with 2\le p < q has only finitely many solutions. Assuming the abc-conjecture we can prove that and are bounded. In the special case and a prime power we are able to solve completely
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