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Let $n$ be a positive integer and $f(x)$ be a polynomial with nonnegative integer coefficients. We prove that ${rm lcm}_{lceil n/2rceil le ile n} {f(i)}ge 2^n$ except that $f(x)=x$ and $n=1, 2, 3, 4, 6$ and that $f(x)=x^s$ with $sge 2$ being an integer and $n=1$, where $lceil n/2rceil$ denotes the smallest integer which is not less than $n/2$. This improves and extends the lower bounds obtained by Nair in 1982, Farhi in 2007 and Oon in 2013.
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For relatively prime positive integers $u_0$ and $r$, we consider the least common multiple $L_n:=mathrm{lcm}(u_0,u_1,ldots, u_n)$ of the finite arithmetic progression ${u_k:=u_0+kr}_{k=0}^n$. We derive new lower bounds on $L_n$ which improve upon those obtained previously when either $u_0$ or $n$ is large. When $r$ is prime, our best bound is sharp up to a factor of $n+1$ for $u_0$ properly chosen, and is also nearly sharp as $ntoinfty$.
For relatively prime positive integers u_0 and r, we consider the arithmetic progression {u_k := u_0+k*r} (0 <= k <= n). Define L_n := lcm{u_0,u_1,...,u_n} and let a >= 2 be any integer. In this paper, we show that, for integers alpha,r >= a and n >= 2*alpha*r, we have L_n >= u_0*r^{alpha+a-2}*(r+1)^n. In particular, letting a = 2 yields an improvement to the best previous lower bound on L_n (obtained by Hong and Yang) for all but three choices of alpha,r >= 2.
Erd{o}s and Niven proved in 1946 that for any positive integers $m$ and $d$, there are at most finitely many integers $n$ for which at least one of the elementary symmetric functions of $1/m, 1/(m+d), ..., 1/(m+(n-1)d)$ are integers. Recently, Wang and Hong refined this result by showing that if $ngeq 4$, then none of the elementary symmetric functions of $1/m, 1/(m+d), ..., 1/(m+(n-1)d)$ is an integer for any positive integers $m$ and $d$. Let $f$ be a polynomial of degree at least $2$ and of nonnegative integer coefficients. In this paper, we show that none of the elementary symmetric functions of $1/f(1), 1/f(2), ..., 1/f(n)$ is an integer except for $f(x)=x^{m}$ with $mgeq2$ being an integer and $n=1$.
We approximate intersection numbers $biglangle psi_1^{d_1}cdots psi_n^{d_n}bigrangle_{g,n}$ on Deligne-Mumfords moduli space $overline{mathcal M}_{g,n}$ of genus $g$ stable complex curves with $n$ marked points by certain closed-form expressions in $d_1,dots,d_n$. Conjecturally, these approximations become asymptotically exact uniformly in $d_i$ when $gtoinfty$ and $n$ remains bounded or grows slowly. In this note we prove a lower bound for the intersection numbers in terms of the above-mentioned approximatingexpressions multiplied by an explicit factor $lambda(g,n)$, which tends to $1$ when $gtoinfty$ and $d_1+dots+d_{n-2}=o(g)$.
Let $K$ be a 1-dimensional function field over an algebraically closed field of characteristic $0$, and let $A/K$ be an abelian surface. Under mild assumptions, we prove a Lehmer-type lower bound for points in $A(bar{K})$. More precisely, we prove that there are constants $C_1,C_2>0$ such that the normalized Bernoulli-part of the canonical height is bounded below by $$ hat{h}_A^{mathbb{B}}(P) ge C_1bigl[K(P):Kbigr]^{-2} $$ for all points $Pin{A(bar{K})}$ whose height satisfies $0<hat{h}_A(P)le{C_2}$.
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