Through Wikipedia, I discovered the following Arxiv.org article on Least Common Multiple.
I have not seen the notation referenced and I am not clear how $\gamma$ is defined since it stated to be a non-zero integer.
I don't see how $\gamma$ can be defined such that it is both a sum of the inverse product and also a non-zero integer.
Lemma 1 says:
Let $(\alpha_i)_{i\in I}$ and $(\beta_i)_{i\in I}$ be two finite sequences of non-zero integers such that: $$\sum_{i \in I}\frac{1}{\alpha_i\beta_i}=\frac{1}{\gamma}$$ for some non-zero integer $\gamma$. Then, the integer lcm$\{\alpha_i,i \in I\}.$lcm$\{\beta_i,i \in I\}$ is a multiple of $\gamma$.
Here is my understanding. Please let me know if got anything incorrect:
- $I$ is a set of consecutive integers such as $\{ 1,2,3,\dots, n\}$
- $(\alpha_i), (\beta_i)$ are sequences of $n$ elements where each element is a non-zero integer.
- $\frac{1}{\gamma}$ is the sum of the inverse product of correlated elements in each sequence.
- $\gamma$ is a non-zero integer.
- lcm$\{\alpha_i,i \in I\}$ is the lcm of all elements in $(\alpha_i)_{i\in I}$
- lcm$\{\beta_i,i \in I\}$ is the lcm of all elements in $(\beta_i)_{i\in I}$
- lcm$\{\alpha_i,i\in I\}.$lcm$\{\beta_i,i\in I\}$ is the product of the lcm of $(\alpha_i)_{i\in I}$ and the lcm of $(\beta_i)_{i\in I}$
Here is an example to show what I mean for term:
Let:
- $I = \{1, 2, 3\}$
- $(\alpha_i)_{i\in I} = \{4,5,6\}$
- $(\beta_i)_{i\in I} = \{7,8,9\}$
Then:
- $\frac{1}{\gamma} = \frac{1}{4\times7} + \frac{1}{5\times8} + \frac{1}{6\times9}$ but then $\gamma$ is not an integer.
- lcm$\{\alpha_i,i \in I\} = 60$
- lcm$\{\beta_i,i \in I\} = 504$
- lcm$\{\alpha_i,i\in I\}.$lcm$\{\beta_i,i\in I\} = 30,240$
How is $\gamma$ defined? What did I get right in my understanding? What did I get wrong?
You're right. Not all sequences of $\alpha_i$ and $\beta_i$ will be valid inputs to this lemma; you've demonstrated two sequences to which the lemma does not apply.
A few examples of sequences which are valid inputs: