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Find all possible positive integers $m$ and $m$ primes ${{p}_{1}}<{{p}_{2}}<\cdots <{{p}_{m}}$ such that $\frac{1}{{{p}_{1}}}+\frac{1}{{{p}_{2}}}+\cdots +\frac{1}{{{p}_{m}}}+\frac{1}{{{p}_{1}}{{p}_{2}}\cdots {{p}_{m}}}$ is an integer. I think $m=2,{{p}_{1}}=2,{{p}_{2}}=3$ is the only solution, but I don’t know how to give a proof.

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No, it is not the only solution. There are at least two more.

$$ 1/2+1/3+1/7 + 1/(2\cdot3\cdot7) = 1 $$

$$ 1/2+1/3+1/11+1/23+1/31 + 1/(2*3*11*23*31) = 1 $$

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  • $\begingroup$ Two more examples are given by the sets of primes $\{2,3,7,43\}$ and $\{2, 3, 11, 23, 31, 47059\}$. $\endgroup$ Apr 10, 2014 at 15:18
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These numbers are tabulated at the Online Encyclopedia of Integer Sequences. Some discussion and various links are given. Apparently, it is unknown whether there are infinitely many such integers, and unknown whether there are any odd numbers with this property.

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