10 research outputs found
A unique pair of triangles
A rational triangle is a triangle with sides of rational lengths. In this
short note, we prove that there exists a unique pair of a rational right
triangle and a rational isosceles triangle which have the same perimeter and
the same area. In the proof, we determine the set of rational points on a
certain hyperelliptic curve by a standard but sophisticated argument which is
based on the 2-descent on its Jacobian variety and Coleman's theory of -adic
abelian integrals.Comment: 5 pages, to appear in Journal of Number Theory, Some modifications
are added to the article published onlin
A note on the Diophantine equation and application to odd perfect numbers
Let be an odd perfect number. Then, Euler proved that there exist some
integers and a prime such that , , and . In this note, we prove that the
ratio is neither a square nor a square times
a single prime unless . It is a direct consequence of a certain
property of the Diophantine equation , where
denotes one or a prime, whose proof is based on the prime ideal
factorization in the quadratic orders and the
primitive solutions of generalized Fermat equations . We give also a slight generalization to odd multiply perfect numbers.Comment: 6 pages. Comments welcome
Galois trace forms of type for odd
Let be an odd prime number and . Then, it is
well-known that the -root lattice can be realized as the (Hermitian)
trace form of the -th cyclotomic extension
restricted to the fractional ideal generated
by . In this paper, in contrast with the case of the
-root lattice, we prove the following theorem: Let be an odd
positive integer and be a Galois extension of degree . Then,
there exist no fractional ideals of such that the restricted
trace form is of type
. The proof is done by the prime ideal factorization of
fractional ideals of with care of certain 2-adic obstruction. Additionally,
we prove that every cyclic cubic field contains infinitely many distinct sub
-lattices of type (i.e., normalized face centered cubic
lattices) with normal -bases. The latter fact is in contrast with
another fact that among quadratic fields only contain
sub -lattices of type
Absolute zeta functions arising from ceiling and floor Puiseux polynomials
For the -lift of a monoid scheme of finite
type, Deitmar-Koyama-Kurokawa calculated its absolute zeta function by
interpolating for all prime powers using the
Fourier expansion. This absolute zeta function coincides with the absolute zeta
function of a certain polynomial. In this article, we characterize the
polynomial as a ceiling polynomial of the sequence
, which we introduce
independently. Extending this idea, we introduce a certain pair of absolute
zeta functions of a separated scheme of finite type over by
means of a pair of Puiseux polynomials which estimate ""
for sufficiently large . We call them the ceiling and floor Puiseux
polynomials of . In particular, if is an elliptic curve, then our
absolute zeta functions of do not depend on its isogeny class