Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userAn application of the modular method and the symplectic argument to a Lebesgue-Nagell equation
112
0
0.0
(
0
)
Authors
Angelos Koutsianas
Ask ChatGPT about the research
No Arabic abstract
In this paper, we study the generalized Lebesgue-Nagell equation [ x^2+7^{2k+1}=y^n. ] This is the last case of equations of the form $x^2+q^{2k+1}=y^n$ with $kgeq0$ and $q>0$ where $mathbb{Q}(sqrt{-q})$ has class number one. Our proof is based on the modular method and the symplectic argument.
rate research
Read More
It is well-known that for $p=1, 2, 3, 7, 11, 19, 43, 67, 163$, the class number of $mathbb{Q}(sqrt{-p})$ is one. We use this fact to determine all the solutions of $x^2+p^m=4y^n$ in non-negative integers $x, y, m$ and $n$.
Let $n$ be a positive integer. In 1915, Theisinger proved that if $nge 2$, then the $n$-th harmonic sum $sum_{k=1}^nfrac{1}{k}$ is not an integer. Let $a$ and $b$ be positive integers. In 1923, Nagell extended Theisingers theorem by showing that the reciprocal sum $sum_{k=1}^{n}frac{1}{a+(k-1)b}$ is not an integer if $nge 2$. In 1946, ErdH{o}s and Niven proved a theorem of a similar nature that states that there is only a finite number of integers $n$ for which one or more of the elementary symmetric functions of $1,1/2, ..., 1/n$ is an integer. In this paper, we present a generalization of Nagells theorem. In fact, we show that for arbitrary $n$ positive integers $s_1, ..., s_n$ (not necessarily distinct and not necessarily monotonic), the following reciprocal power sum $$sumlimits_{k=1}^{n}frac{1}{(a+(k-1)b)^{s_{k}}}$$ is never an integer if $nge 2$. The proof of our result is analytic and $p$-adic in character.
We introduce Galois families of modular forms. They are a new kind of family coming from Galois representations of the absolute Galois groups of rational function fields over the rational field. We exhibit some examples and provide an infinite Galois family of non-liftable weight one Katz modular eigenforms over an algebraic closure of F_p for p in {3,5,7,11}.
For every normalized newform f in S_2(Gamma_1(N)) with complex multiplication, we study the modular parametrizations of elliptic curves C from the abelian variety A_f. We apply the results obtained when C is Grosss elliptic curve A(p).
A semiclassical approximation approach based on the Maslov complex germ method is considered in detail for the 1D nonlocal Fisher-Kolmogorov-Petrovskii-Piskunov equation under the supposition of weak diffusion. In terms of the semiclassical formalism developed, the original nonlinear equation is reduced to an associated linear partial differential equation and some algebraic equations for the coefficients of the linear equation with a given accuracy of the asymptotic parameter. The solutions of the nonlinear equation are constructed from the solutions of both the linear equation and the algebraic equations. The solutions of the linear problem are found with the use of symmetry operators. A countable family of the leading terms of the semiclassical asymptotics is constructed in explicit form. The semiclassical asymptotics are valid by construction in a finite time interval. We construct asymptotics which are different from the semiclassical ones and can describe evolution of the solutions of the Fisher-Kolmogorov-Petrovskii-Piskunov equation at large times. In the example considered, an initial unimodal distribution becomes multimodal, which can be treated as an example of a space structure.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
