Number TheoryUnit sum composed of unit fractions

conscipost

Member
Is it necessary for a unit sum composed of unit fractions to include 1/2? Doing maple runs this seems to be the case, but it is not evident to me how this could be

Edit: In fact it seems it could not be, given the Erdos Graham problem Erd?s?Graham problem - Wikipedia, the free encyclopedia

But considering an arbitrary fraction and one minus it, it seems the unit fraction representation of one of these two's parts is bound to include 1/2.

I feel a bit mixed up here.

EditEdit:1/3+1/4+1/5+1/6+1/20 does it. I think I thought that distinct unit fraction representations were unique. But this is not the case clearly.

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mathbalarka

Well-known member
MHB Math Helper
Is it necessary for a unit sum composed of unit fractions to include 1/2?
No. 1/3 + 1/3 + 1/3 = 1.

EDIT : Nontrivial 1/3 + 1/4 + 1/5 + 1/6 + 1/20 = 1.

conscipost

Member
No. 1/3 + 1/3 + 1/3 = 1.
I'm afraid I'm being awfully careless in the statement. thank you,

mathbalarka

Well-known member
MHB Math Helper
I also gave a non-trivial example there, you might want to look at that.

conscipost

Member
I also gave a non-trivial example there, you might want to look at that.
Thanks. I figured that out and edited the first post just before you posted.

I did not think that there are infinitely many representations of a non unit fraction in terms of distinct unit fractions and so thought that given one I had the only one that would do so.

mathbalarka

Well-known member
MHB Math Helper
conscipost said:
Given one I had the only one that would do so.
$$\frac{1}{3} + \frac{1}{5} + \frac{1}{7} + \frac{1}{9} + \frac{1}{11} + \frac{1}{15} + \frac{1}{33} + \frac{1}{45} + \frac{1}{385} = 1$$

There exists trivially infinitely many unit fractions with not just without 2 but with odd denominator which sum up to unity.