Narrow Results

Given a genus 2 curve $\mathrm{\text{C}}$ defined over a finite field ${𝔽}_q$ of odd characteristic such that $2|\#\mathrm{\text{Jac}}(\mathrm{\text{C}})({𝔽}_q)$, we study the growth of the 2-adic valuation of the cardinality of the Jacobian over a tower of quadratic extensions of ${𝔽}_q$. In the cases of simpler regularity, we determine the exponents of the 2-Sylow subgroup of $\mathrm{\text{Jac}}(\mathrm{\text{C}})({𝔽}_{q^{2^k}})$.

In this paper, we investigate the 2-adic valuations of the Stirling numbers S(n, k) of the second kind. We show that v₂(S(4i, 5)) = v₂(S(4i + 3, 5)) if and only if i ≢ 7 (mod 32). This confirms a conjecture of Amdeberhan, Manna and Moll raised in 2008. We show also that v₂(S(2ⁿ + 1, k + 1)) = s₂(n) - 1 for any positive integer n, where s₂(n) is the sum of binary digits of n. It proves another conjecture of Amdeberhan, Manna and Moll.

Let $𝜀={({𝜀}_n)}_{n\in \mathbb{N}}$ be an integer sequence and $f(x)={\sum }_{n=0}^{\infty }{𝜀}_n{x}^n$ be its ordinary generating function. In this paper, we study the behavior of 2-adic valuations of the sequence ${(c_m(n))}_{n\in \mathbb{N}}$, where $m\in \mathbb{Z}$ is fixed and

Let $n$ and $k$ be positive integers. We denote by $v_2(n)$ the 2-adic valuation of $n$. The Stirling numbers of the first kind, denoted by $s(n,k)$, count the number of permutations of $n$ elements with $k$ disjoint cycles. In recent years, Lengyel, Komatsu and Young, Leonetti and Sanna, and Adelberg made some progress on the study of the $p$-adic valuations of $s(n,k)$. In this paper, by introducing the concept of $m$th Stirling numbers of the first kind and providing a detailed 2-adic analysis, we show an explicit formula on the 2-adic valuation of $s(2^n,k)$. We also prove that $v_2(s(2^n+1,k+1))=v_2(s(2^n,k))$ holds for all integers $k$ between 1 and $2^n$. As a corollary, we show that $v_2(s(2^n,2^n-k))=2n-2-v_2(k-1)$ if $k$ is odd and $3\le k\le 2^{n-1}+1$. This confirms partially a conjecture of Lengyel raised in 2015. Furthermore, we show that if $k\le 2^n$, then $v_2(s(2^n,k))\le v_2(s(2^n,1))$ and $v_2(H(2^n,k))\le -n$, where $H(n,k)$ stands for the $k$th elementary symmetric functions of $1,1/2,\dots ,1/n$. The latter one supports the conjecture of Leonetti and Sanna suggested in 2017.

Let v₂(n) denote the 2-adic valuation of any positive integer n. Recently, Farhi introduced a curious arithmetic function f defined for any positive integer n by . Farhi showed that the inequality with c = 4.01055487… holds for all positive integer n and conjectured that one can replace the upper bound cⁿ by 4ⁿ in this inequality. In this paper, we show two identities about the product and then use it to prove partially Farhi's conjecture. Finally, we propose a conjecture from which the truth of Farhi's conjecture can be deduced. In particular, we confirm the truth of our conjecture for all positive integers n up to 100000 by using Matlab 7.1.

We verify a conjecture of Farhi concerning the growth of a curious arithmetic function defined by 2-adic valuation. We also prove a stronger version suggested by Zhong and Tan.