Expand all Collapse all | Results 1 - 22 of 22 |
1. CMB Online first
JeÅmanowicz' Conjecture with Congruence Relations. II Let $a,b$ and $c$ be primitive Pythagorean numbers such that
$a^{2}+b^{2}=c^{2}$ with $b$ even.
In this paper, we show that if $b_0 \equiv \epsilon \pmod{a}$
with $\epsilon \in \{\pm1\}$
for certain positive divisors $b_0$ of $b$,
then the Diophantine equation $a^{x}+b^{y}=c^z$ has only the
positive solution $(x,y,z)=(2,2,2)$.
Keywords:exponential Diophantine equations, Pythagorean triples, Pell equations Categories:11D61, 11D09 |
2. CMB 2011 (vol 56 pp. 258)
The Smallest Pisot Element in the Field of Formal Power Series Over a Finite Field Dufresnoy and Pisot characterized the smallest
Pisot number of degree $n \geq 3$ by giving explicitly its minimal
polynomial. In this paper, we translate Dufresnoy and Pisot's
result to the Laurent series case.
The
aim of this paper is to prove that the minimal polynomial
of the smallest Pisot element (SPE) of degree $n$ in the field of
formal power series over a finite field
is given by $P(Y)=Y^{n}-\alpha XY^{n-1}-\alpha^n,$ where $\alpha$
is the least element of the finite field $\mathbb{F}_{q}\backslash\{0\}$
(as a finite total ordered set). We prove that the sequence of
SPEs of degree $n$ is decreasing and converges to $\alpha X.$
Finally, we show how to obtain explicit continued fraction
expansion of the smallest Pisot element over a finite field.
Keywords:Pisot element, continued fraction, Laurent series, finite fields Categories:11A55, 11D45, 11D72, 11J61, 11J66 |
3. CMB 2011 (vol 56 pp. 251)
Sign Changes of the Liouville Function on Quadratics Let $\lambda (n)$ denote the Liouville function. Complementary to the prime number theorem, Chowla conjectured
that
\begin{equation*}
\label{a.1}
\sum_{n\le x} \lambda (f(n)) =o(x)\tag{$*$}
\end{equation*}
for any polynomial $f(x)$ with integer coefficients which is not of
form $bg(x)^2$.
When $f(x)=x$, $(*)$ is equivalent to the prime number theorem.
Chowla's conjecture has been proved for linear functions,
but for degree
greater than 1, the conjecture seems
to be extremely hard and remains wide open.
One can consider a weaker form
of Chowla's conjecture.
Conjecture 1.
[Cassaigne et al.]
If $f(x) \in \mathbb{Z} [x]$ and is not in the form of $bg^2(x)$
for some $g(x)\in \mathbb{Z}[x]$, then $\lambda (f(n))$
changes sign infinitely often.
Clearly, Chowla's conjecture implies Conjecture 1.
Although weaker,
Conjecture 1 is still wide open for polynomials of degree $\gt 1$.
In this article, we study Conjecture 1 for
quadratic polynomials. One of our main theorems is the following.
Theorem 1
Let $f(x) = ax^2+bx +c $ with $a\gt 0$ and $l$
be a positive integer such that $al$ is
not a perfect square. If the
equation $f(n)=lm^2 $ has one solution
$(n_0,m_0) \in \mathbb{Z}^2$, then it has infinitely
many positive solutions $(n,m) \in \mathbb{N}^2$.
As a direct consequence of Theorem 1, we prove the following.
Theorem 2
Let $f(x)=ax^2+bx+c$ with $a \in \mathbb{N}$ and $b,c \in \mathbb{Z}$. Let
\[
A_0=\Bigl[\frac{|b|+(|D|+1)/2}{2a}\Bigr]+1.
\]
Then either the binary sequence $\{ \lambda (f(n)) \}_{n=A_0}^\infty$ is
a constant sequence or it changes sign infinitely often.
Some partial results of Conjecture 1 for quadratic polynomials are also proved using Theorem 1.
Keywords:Liouville function, Chowla's conjecture, prime number theorem, binary sequences, changes sign infinitely often, quadratic polynomials, Pell equation Categories:11N60, 11B83, 11D09 |
4. CMB 2011 (vol 55 pp. 842)
The Rank of Jacobian Varieties over the Maximal Abelian Extensions of Number Fields: Towards the Frey-Jarden Conjecture |
The Rank of Jacobian Varieties over the Maximal Abelian Extensions of Number Fields: Towards the Frey-Jarden Conjecture Frey and Jarden asked if
any abelian variety over a number field $K$
has the infinite Mordell-Weil rank over
the maximal abelian extension $K^{\operatorname{ab}}$.
In this paper,
we give an affirmative answer to their conjecture
for the Jacobian variety
of any smooth projective curve $C$
over $K$
such that $\sharp C(K^{\operatorname{ab}})=\infty$
and for any abelian variety of $\operatorname{GL}_2$-type with trivial character.
Keywords:Mordell-Weil rank, Jacobian varieties, Frey-Jarden conjecture, abelian points Categories:11G05, 11D25, 14G25, 14K07 |
5. CMB 2011 (vol 55 pp. 774)
Pell Equations: Non-Principal Lagrange Criteria and Central Norms We provide a criterion for the central norm to be
any value in the simple continued fraction expansion of $\sqrt{D}$
for any non-square integer $D>1$. We also provide a simple criterion
for the solvability of the Pell equation $x^2-Dy^2=-1$ in terms of
congruence conditions modulo $D$.
Keywords:Pell's equation, continued fractions, central norms Categories:11D09, 11A55, 11R11, 11R29 |
6. CMB 2011 (vol 55 pp. 435)
A Note on the Diophantine Equation $x^2 + y^6 = z^e$, $e \geq 4$ We consider the diophantine equation $x^2 + y^6 = z^e$, $e \geq 4$.
We show that, when $e$ is a multiple of $4$ or $6$, this equation
has no solutions in positive integers with $x$ and $y$ relatively prime.
As a corollary, we show
that there exists no primitive Pythagorean triangle one of whose
leglengths is a perfect cube, while the hypotenuse length is an
integer square.
Keywords:diophantine equation Category:11D |
7. CMB 2009 (vol 52 pp. 63)
Small Zeros of Quadratic Forms Avoiding a Finite Number of Prescribed Hyperplanes We prove a new upper bound for the smallest zero $\mathbf{x}$
of a quadratic form over a number field with the additional
restriction that $\mathbf{x}$ does not lie in a finite number of $m$ prescribed
hyperplanes. Our bound is polynomial in the height of the quadratic
form, with an exponent depending only on the number of variables but
not on $m$.
Categories:11D09, 11E12, 11H46, 11H55 |
8. CMB 2008 (vol 51 pp. 337)
Differences between Perfect Powers We apply the hypergeometric method of Thue and Siegel to prove
that if $a$ and $b$ are positive integers, then the inequality $
0 <| a^x - b^y | < \frac{1}{4} \, \max \{ a^{x/2}, b^{y/2} \}$
has at most a single solution in positive integers $x$ and $y$.
This essentially sharpens a classic result of LeVeque.
Categories:11D61, 11D45 |
9. CMB 2008 (vol 51 pp. 134)
Numerical Semigroups Having a Toms Decomposition We show that the class of system proportionally modular numerical semigroups
coincides with the class of numerical semigroups having a Toms
decomposition.
Categories:20M14, 11D75 |
10. CMB 2007 (vol 50 pp. 191)
Every Real Algebraic Integer Is a Difference of Two Mahler Measures We prove that every real
algebraic integer $\alpha$ is expressible by a
difference of two Mahler measures of integer polynomials.
Moreover, these polynomials can be chosen in such a way that they
both have the same degree as that of $\alpha$, say
$d$, one of these two polynomials is irreducible and
another has an irreducible factor of degree $d$, so
that $\alpha=M(P)-bM(Q)$ with irreducible polynomials
$P, Q\in \mathbb Z[X]$ of degree $d$ and a
positive integer $b$. Finally, if $d \leqslant 3$, then one can take $b=1$.
Keywords:Mahler measures, Pisot numbers, Pell equation, $abc$-conjecture Categories:11R04, 11R06, 11R09, 11R33, 11D09 |
11. CMB 2006 (vol 49 pp. 560)
A K3 Surface Associated With Certain Integral Matrices Having Integral Eigenvalues In this article we will show that there are infinitely many
symmetric, integral $3 \times 3$ matrices, with zeros on the
diagonal, whose eigenvalues are all integral. We will do this by
proving that the rational points on a certain non-Kummer, singular
K3 surface
are dense. We will also compute the entire N\'eron--Severi group of
this surface and find all low degree curves on it.
Keywords:symmetric matrices, eigenvalues, elliptic surfaces, K3 surfaces, NÃ©ron--Severi group, rational curves, Diophantine equations, arithmetic geometry, algebraic geometry, number theory Categories:14G05, 14J28, 11D41 |
12. CMB 2006 (vol 49 pp. 481)
On Sequences of Squares with Constant Second Differences The aim of this paper is to study sequences of integers
for which the second differences between their squares are
constant. We show that there are infinitely many nontrivial
monotone sextuples having this property and discuss some related
problems.
Keywords:sequence of squares, second difference, elliptic curve Categories:11B83, 11Y85, 11D09 |
13. CMB 2005 (vol 48 pp. 636)
Correction to: On the Diophantine Equation $n(n+d)\cdots(n+(k-1)d)=by^l$ In the article under consideration
(Canad. Math. Bull. \textbf{47} (2004), pp.~373--388),
Lemma 6 is not true in the form presented there.
Lemma 6 is used only in the proof of part (i) of Theorem 9.
We note, however, that part (i) of Theorem 9 in question is a special
case of a theorem by Bennet, Bruin, Gy\H{o}ry and Hajdu.
Category:11D41 |
14. CMB 2005 (vol 48 pp. 121)
Necessary and Sufficient Conditions for the Central Norm to Equal $2^h$ in the Simple Continued Fraction Expansion of $\sqrt{2^hc}$ for Any Odd $c>1$ |
Necessary and Sufficient Conditions for the Central Norm to Equal $2^h$ in the Simple Continued Fraction Expansion of $\sqrt{2^hc}$ for Any Odd $c>1$ We look at the simple continued fraction expansion of $\sqrt{D}$
for any $D=2^hc $ where $c>1$ is odd with a goal of
determining necessary and
sufficient conditions for the central norm (as determined by
the infrastructure of the underlying real quadratic order therein) to be
$2^h$. At the end of the paper, we also address the case where $D=c$
is odd and the central norm of $\sqrt{D}$ is equal to $2$.
Keywords:quadratic Diophantine equations, simple continued fractions,, norms of ideals, infrastructure of real quadratic fields Categories:11A55, 11D09, 11R11 |
15. CMB 2004 (vol 47 pp. 373)
On the Diophantine Equation $n(n+d)\cdots(n+(k-1)d)=by^l$ We show that the product of four or five consecutive positive
terms in arithmetic progression can never be a perfect power whenever the
initial term is coprime to the common difference of the arithmetic
progression. This is a generalization of the results of Euler and Obl\'ath
for the case of squares, and an extension of a theorem of Gy\H ory on three
terms in arithmetic progressions. Several other results concerning the
integral solutions of the equation of the title are also obtained. We extend
results of Sander on the rational solutions of the equation in $n,y$ when
$b=d=1$. We show that there are only finitely many solutions in $n,d,b,y$
when $k\geq 3$, $l\geq 2$ are fixed and $k+l>6$.
Category:11D41 |
16. CMB 2004 (vol 47 pp. 264)
Counting Rational Points on Ruled Varieties In this paper, we prove a general result computing the number of rational points
of bounded height on a projective variety $V$ which is covered by lines. The
main technical result used to achieve this is an upper bound on the number of
rational points of bounded height on a line. This upper bound is such that it
can be easily controlled as the line varies, and hence is used to sum the counting
functions of the lines which cover the original variety $V$.
Categories:11G50, 11D45, 11D04, 14G05 |
17. CMB 2003 (vol 46 pp. 26)
Remarques sur les points rationnels des variÃ©tÃ©s de Fermat Soit $K$ un corps de nombres de degr\'e sur $\mathbb{Q}$ inf\'erieur
ou \'egal \`a $2$. On se propose dans ce travail de faire quelques
remarques sur la question de l'existence de deux \'el\'ements non nuls
$a$ et $b$ de $K$, et d'un entier $n\geq 4$, tels que l'\'equation
$ax^n + by^n = 1$ poss\`ede au moins trois points distincts non
triviaux. Cette \'etude se ram\`ene \`a la recherche de points
rationnels sur $K$ d'une vari\'et\'e projective dans $\mathbb{P}^5$ de
dimension $3$, ou d'une surface de $\mathbb{P}^3$.
Category:11D41 |
18. CMB 2003 (vol 46 pp. 71)
The Number of Fields Generated by the Square Root of Values of a Given Polynomial The $abc$-conjecture is applied to various questions involving the
number of distinct fields $\mathbb{Q} \bigl( \sqrt{f(n)} \bigr)$, as we vary
over integers $n$.
Categories:11N32, 11D41 |
19. CMB 2002 (vol 45 pp. 428)
Criteria for Simultaneous Solutions of $X^2 - DY^2 = c$ and $x^2 - Dy^2 = -c$ The purpose of this article is to provide criteria for the
simultaneous solvability of the Diophantine equations $X^2 - DY^2 =
c$ and $x^2 - Dy^2 = -c$ when $c \in \mathbb{Z}$, and $D \in
\mathbb{N}$ is not a perfect square. This continues work in
\cite{me}--\cite{alfnme}.
Keywords:continued fractions, Diophantine equations, fundamental units, simultaneous solutions Categories:11A55, 11R11, 11D09 |
20. CMB 2002 (vol 45 pp. 247)
On a Few Diophantine Equations Related to Fermat's Last Theorem We combine the deep methods of Frey, Ribet, Serre and Wiles with some
results of Darmon, Merel and Poonen to solve certain explicit
diophantine equations. In particular, we prove that the area of a
primitive Pythagorean triangle is never a perfect power, and that each
of the equations $X^4 - 4Y^4 = Z^p$, $X^4 + 4Y^p = Z^2$ has no
non-trivial solution. Proofs are short and rest heavily on results
whose proofs required Wiles' deep machinery.
Keywords:Diophantine equations Category:11D41 |
21. CMB 2000 (vol 43 pp. 218)
Continued Fractions, Jacobi Symbols, and Quadratic Diophantine Equations The results herein continue observations on norm form equations and
continued fractions begun and continued in the works
\cite{chows}--\cite{mol}, and \cite{mvdpw}--\cite{schinz}.
Categories:11R11, 11D09, 11R29, 11R65 |
22. CMB 1998 (vol 41 pp. 158)
Power integral bases in composits of number fields In the present paper we consider the problem of finding power
integral bases in number fields which are composits of two
subfields with coprime discriminants. Especially, we consider
imaginary quadratic extensions of totally real cyclic number
fields of prime degree. As an example we solve the index form
equation completely in a two parametric family of fields of degree
$10$ of this type.
Categories:11D57, 11R21 |