ﻻ يوجد ملخص باللغة العربية
In 1956, Je$acute{s}$manowicz conjectured that, for positive integers $m$ and $n$ with $m>n, , gcd(m,, n)=1$ and $m otequiv npmod{2}$, the exponential Diophantine equation $(m^2-n^2)^x+(2mn)^y=(m^2+n^2)^z$ has only the positive integer solution $(x,,y,, z)=(2,,2,,2)$. Recently, Ma and Chen cite{MC17} proved the conjecture if $4 ot|mn$ and $yge2$. In this paper, we present an elementary proof of the result of Ma and Chen cite{MC17}.
We present an elementary proof for Ljunggren equation
Let $Omega(n)$ denote the number of prime factors of $n$. We show that for any bounded $fcolonmathbb{N}tomathbb{C}$ one has [ frac{1}{N}sum_{n=1}^N, f(Omega(n)+1)=frac{1}{N}sum_{n=1}^N, f(Omega(n))+mathrm{o}_{Ntoinfty}(1). ] This yields a new elementary proof of the Prime Number Theorem.
A renowned information-theoretic formula by Shannon expresses the mutual information rate of a white Gaussian channel with a stationary Gaussian input as an integral of a simple function of the power spectral density of the channel input. We give in
A sequence of random variables is called exchangeable if the joint distribution of the sequence is unchanged by any permutation of the indices. De Finettis theorem characterizes all ${0,1}$-valued exchangeable sequences as a mixture of sequences of i
We give an elementary proof of Grothendiecks non-vanishing Theorem: For a finitely generated non-zero module $M$ over a Noetherian local ring $A$ with maximal ideal $m$, the local cohomology module $H^{dim M}_{m}(M)$ is non-zero.