Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userNew Gelfond-Type Transcendental Numbers
62
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
It is well known that value at a non-zero algebraic number of each of the functions $e^{x}, ln x, sin x, cos x, tan x, csc x, sec x, cot x, sinh x,$ $ cosh x,$ $ tanh x,$ and $coth x$ is transcendental number (see Theorem 9.11 of cite{N}). In the work, we show that for any one of the above mentioned functions, $f(x)$, and for a polynomial $g(x)$ with rational coefficients the zero, if any, of the equation $f(x)=g(x)$ is a transcendental number. We also show that if $f(x)$ and $g(x)$ are polynomials with rational coefficients, then a zero of the equation $e^{f(x)}=g(x)$ is a transcendental number. Finally we show that the existence of an abelian group whose non-zero elements are transcendental numbers.
rate research
Read More
It is widely believed that the continued fraction expansion of every irrational algebraic number $alpha$ either is eventually periodic (and we know that this is the case if and only if $alpha$ is a quadratic irrational), or it contains arbitrarily large partial quotients. Apparently, this question was first considered by Khintchine. A preliminary step towards its resolution consists in providing explicit examples of transcendental continued fractions. The main purpose of the present work is to present new families of transcendental continued fractions with bounded partial quotients. Our results are derived thanks to new combinatorial transcendence criteria recently obtained by Adamczewski and Bugeaud.
In contrast to all other known Ramanujan-type congruences, we discover that Ramanujan-type congruences for Hurwitz class numbers can be supported on non-holomorphic generating series. We establish a divisibility result for such non-holomorphic congruences of Hurwitz class numbers. The two keys tools in our proof are the holomorphic projection of products of theta series with a Hurwitz class number generating series and a theorem by Serre, which allows us to rule out certain congruences.
With this paper we introduce a new series representation of $zeta(3)$, which is based on the Clausen representation of odd integer zeta values. Although, relatively fast converging series based on the Clausen representation exist for $zeta(3)$, their convergence behavior is very slow compared to BBP-type formulas, and as a consequence they are not used for explicit numerical computations. The reason is found in the fact that the corresponding Clausen function can be calculated analytically for a few rational arguments only, where $x=frac{1}{6}$ is the smallest one. Using polylogarithmic identities in combination with a polynomial description of the even Bernoulli numbers, the convergence behavior of the Clausen-type representation has been improved to a level that allows us to challenge ultimately all BBP-type formulas available for $zeta(3)$. We present an explicit numerical comparison between one of the best available BBP formulas and our formalism. Furthermore, we demonstrate by an explicit computation using the first four terms in our series representation only that $zeta(3)$ results with an accuracy of $2*10^{-26}$, where our computation guarantees on each approximation level for an analytical expression for $zeta(3)$.
In this paper we determine the number of endomorphism rings of superspecial abelian surfaces over a field $mathbb{F}_q$ of odd degree over $mathbb{F}_p$ in the isogeny class corresponding to the Weil $q$-number $pmsqrt{q}$. This extends earlier works of T.-C. Yang and the present authors on the isomorphism classes of these abelian surfaces, and also generalizes the classical formula of Deuring for the number of endomorphism rings of supersingular elliptic curves. Our method is to explore the relationship between the type and class numbers of the quaternion orders concerned. We study the Picard group action of the center of an arbitrary $mathbb{Z}$-order in a totally definite quaternion algebra on the ideal class set of said order, and derive an orbit number formula for this action. This allows us to prove an integrality assertion of Vigneras [Enseign. Math. (2), 1975] as follows. Let $F$ be a totally real field of even degree over $mathbb{Q}$, and $D$ be the (unique up to isomorphism) totally definite quaternion $F$-algebra unramified at all finite places of $F$. Then the quotient $h(D)/h(F)$ of the class numbers is an integer.
The Apery numbers $A_n$ and the Franel numbers $f_n$ are defined by $$A_n=sum_{k=0}^{n}{binom{n+k}{2k}}^2{binom{2k}{k}}^2 {rm and } f_n=sum_{k=0}^{n}{binom{n}{k}}^3(n=0, 1, cdots,).$$ In this paper, we prove three supercongruences for Apery numbers or Franel numbers conjectured by Z.-W. Sun. Let $pgeq 5$ be a prime and let $nin mathbb{Z}^{+}$. We show that begin{align} otag frac{1}{n}bigg(sum_{k=0}^{pn-1}(2k+1)A_k-psum_{k=0}^{n-1}(2k+1)A_kbigg)equiv0pmod{p^{4+3 u_p(n)}} end{align} and begin{align} otag frac{1}{n^3}bigg(sum_{k=0}^{pn-1}(2k+1)^3A_k-p^3sum_{k=0}^{n-1}(2k+1)^3A_kbigg)equiv0pmod{p^{6+3 u_p(n)}}, end{align} where $ u_p(n)$ denotes the $p$-adic order of $n$. Also, for any prime $p$ we have begin{align} otag frac{1}{n^3}bigg(sum_{k=0}^{pn-1}(3k+2)(-1)^kf_k-p^2sum_{k=0}^{n-1}(3k+2)(-1)^kf_kbigg)equiv0pmod{p^{3}}. end{align}
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
