论文标题
某些二次序列的最小常见倍数的非平凡有效下限
Nontrivial effective lower bounds for the least common multiple of some quadratic sequences
论文作者
论文摘要
本文致力于研究数字$ l_ {c，m，n}：= \ mathrm {lcm} \ {m^2+c，（m+1）^2+c，\ dots，n^2+c \} $，$ c，m，m，n $是$ m \ m \ leq n $。确切地说，我们证明$ l_ {c，m，n} $是有理数\ [\ frac {\ displaystyle \ prod_ {k = m}^{n}^{n} \ left（k^2+c \ right）} {c \ cdot （n-m）！\ displayStyle \ prod_ {k = 1}^{n-m} \ left（k^2+4c \ right）}，\]，我们（作为结果）$ l_ {c，m，n} $的某些非平地下限。例如，我们证明，如果$ n- \ frac {1} {2} n^{2/3} \ leq m \ leq n $，那么我们有$ l_ {c，m，m，n} \geqλ（c）\ cdot n e^e^e^e^{3（n-m）} $，$ c） π^2} {3} c - \ frac {5} {12}}}}} {（2π）^{3/2} c} $。此外，必须指出的是，我们的方法（关注交换代数）是新的，与Farhi，Oon和Hong先前使用的方法不同。
This paper is devoted to studying the numbers $L_{c,m,n} := \mathrm{lcm}\{m^2+c ,(m+1)^2+c , \dots , n^2+c\}$, where $c,m,n$ are positive integers such that $m \leq n$. Precisely, we prove that $L_{c,m,n}$ is a multiple of the rational number \[\frac{\displaystyle\prod_{k=m}^{n}\left(k^2+c\right)}{c \cdot (n-m)!\displaystyle\prod_{k=1}^{n-m}\left(k^2+4c\right)} ,\] and we derive (as consequences) some nontrivial lower bounds for $L_{c,m,n}$. We prove for example that if $n- \frac{1}{2} n^{2/3} \leq m \leq n$, then we have $L_{c,m,n} \geq λ(c) \cdot n e^{3 (n - m)}$, where $λ(c) := \frac{e^{- \frac{2 π^2}{3} c - \frac{5}{12}}}{(2 π)^{3/2} c}$. Further, it must be noted that our approach (focusing on commutative algebra) is new and different from those using previously by Farhi, Oon and Hong.
