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 $p$-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 2ln2=1+q+⋯+qα2ln^{2} = 1+q+ \cdots +q^{\alpha} and application to odd perfect numbers

Let $N$ be an odd perfect number. Then, Euler proved that there exist some integers $n, \alpha$ and a prime $q$ such that $N = n^{2}q^{\alpha}$, $q \nmid n$, and $q \equiv \alpha \equiv 1 \bmod 4$. In this note, we prove that the ratio $\frac{\sigma(n^{2})}{q^{\alpha}}$ is neither a square nor a square times a single prime unless $\alpha = 1$. It is a direct consequence of a certain property of the Diophantine equation $2ln^{2} = 1+q+ \cdots +q^{\alpha}$, where $l$ denotes one or a prime, whose proof is based on the prime ideal factorization in the quadratic orders $\mathbb{Z}[\sqrt{1-q}]$ and the primitive solutions of generalized Fermat equations $x^{\beta}+y^{\beta} = 2z^{2}$. We give also a slight generalization to odd multiply perfect numbers.Comment: 6 pages. Comments welcome

Galois trace forms of type An,Dn,EnA_{n}, D_{n}, E_{n} for odd nn

Let $p$ be an odd prime number and $\zeta_{p} := \exp(2\pi i/p)$. Then, it is well-known that the $A_{p-1}$-root lattice can be realized as the (Hermitian) trace form of the $p$-th cyclotomic extension $\mathbb{Q}(\zeta_{p})/\mathbb{Q}$ restricted to the fractional ideal generated by $(1-\zeta_{p})^{-(p-3)/2}$. In this paper, in contrast with the case of the $A_{p-1}$-root lattice, we prove the following theorem: Let $n$ be an odd positive integer and $F/\mathbb{Q}$ be a Galois extension of degree $n$. Then, there exist no fractional ideals $\Lambda$ of $F$ such that the restricted trace form $(\Lambda, \mathrm{Tr}|_{\Lambda \times \Lambda})$ is of type $A_{n}, D_{n}, E_{n}$. The proof is done by the prime ideal factorization of fractional ideals of $F$ with care of certain 2-adic obstruction. Additionally, we prove that every cyclic cubic field contains infinitely many distinct sub $\mathbb{Z}$-lattices of type $A_{3}$ (i.e., normalized face centered cubic lattices) with normal $\mathbb{Z}$-bases. The latter fact is in contrast with another fact that among quadratic fields only $\mathbb{Q}(\sqrt{\pm3})$ contain sub $\mathbb{Z}$-lattices of type $A_{2}$

Absolute zeta functions arising from ceiling and floor Puiseux polynomials

For the $\mathbb{Z}$-lift $X_\mathbb{Z}$ of a monoid scheme $X$ of finite type, Deitmar-Koyama-Kurokawa calculated its absolute zeta function by interpolating $\#X_\mathbb{Z}(\mathbb{F}_q)$ for all prime powers $q$ 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 $\left(\#X_\mathbb{Z}(\mathbb{F}_q)\right)_q$, which we introduce independently. Extending this idea, we introduce a certain pair of absolute zeta functions of a separated scheme $X$ of finite type over $\mathbb{Q}$ by means of a pair of Puiseux polynomials which estimate "$\#X(\mathbb{F}_{p^m})$" for sufficiently large $p$. We call them the ceiling and floor Puiseux polynomials of $X$. In particular, if $X$ is an elliptic curve, then our absolute zeta functions of $X$ do not depend on its isogeny class