I'm primarily a programmer, so forgive me if I don't know the proper nomenclature or notation.
Last night, an old teacher of mine told me about a question that had caused some noodle-scratching for him:
For any two sequences of consecutive integers, can the sums of their reciprocals be equal?
Now, I gather that these sums are called "harmonic numbers" if the consecutive sequence begins with 1. But what if it doesn't begin with 1?
How might we go about proving that for any two such sequences, their sums are unequal?
I have written a quick Python script that returns, in the form of a reduced fraction, the sum of any $\sigma(m, k)$ where $m$ is the first number to consider and $k$ is the length of the sequence.
So, I guess I have two questions:
1) Where can I find out about the current state of research on this question?
2) What are the most likely approaches, or "hooks" that I might grasp onto to arrive at a proof?
(Via the comments, I'll add the following for clarity:)
$\sigma(m,k)$ is $\frac{1}{m} + \frac{1}{m+1}+\dots +\frac{1}{m+k−1}.$ My question is: Can there exist distinct pairs $(m_1,k_1)$ and $(m_2,k_2)$ such that the corresponding sums are equal?