MHB Unit sum composed of unit fractions

AI Thread Summary
The discussion centers on whether a unit sum composed of unit fractions must include 1/2. Initial thoughts suggest that it might be necessary, but examples demonstrate that it is not. The Erdos-Graham problem is referenced to support the idea that distinct unit fraction representations can exist without including 1/2. Additionally, participants clarify that there are infinitely many representations of non-unit fractions as sums of distinct unit fractions. Ultimately, it is established that a unit sum can be achieved without the inclusion of 1/2.
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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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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.
 
mathbalarka said:
No. 1/3 + 1/3 + 1/3 = 1.

I'm afraid I'm being awfully careless in the statement. thank you,
 
I also gave a non-trivial example there, you might want to look at that.
 
mathbalarka said:
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.
 
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.
 
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