For any positive integers $k$ and $l$, does the equation $$(\sum_{i=1}^k \frac{1}{p_i}) (\sum_{j=1}^l \frac{1}{q_j}) = 1$$ have solutions in distinct primes, that is, $p_1, p_2, \dots, p_k, q_1, q_2, \dots, q_l$ are distinct?

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    $\begingroup$ Hi & welcome to MSE. Does the statement "all $p_i$ and $q_j$ are distinct mean among just each group separately or all together (i.e., the extra condition of there not being any $p_i$ equal to any $q_j$)? $\endgroup$ – John Omielan Jan 7 at 6:02
  • $\begingroup$ Thanks for the question! I have adjusted the original post so the question is more clear. $\endgroup$ – xiaopv Jan 7 at 14:48
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    $\begingroup$ You are welcome for the question, and thanks for making this clear. Please help use to better help you by providing some context such as where this question comes from, what you've tried so far, any particular issues you're having, etc. Thanks. $\endgroup$ – John Omielan Jan 7 at 17:26
  • $\begingroup$ Seems to be a tough problem; maybe, try to post it on MathOverflow (indicating that it has been previously posted on Math StackExchange, but has not got solved). $\endgroup$ – W-t-P Jan 11 at 21:04
  • $\begingroup$ Now posted, without informing either site, to MO, mathoverflow.net/questions/320838/… $\endgroup$ – Gerry Myerson Jan 14 at 14:33

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