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Michael Dagg
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 Sum of integers whose reciprocals sum to 1   « on: Nov 16th, 2008, 11:47am » Quote Modify

Prove/disprove: Every integer greater than 23 can be written as
the sum of integers whose reciprocals sum to 1.
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Michael Dagg
John_Thomas
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 Re: Sum of integers whose reciprocals sum to 1   « Reply #1 on: Dec 7th, 2008, 12:32pm » Quote Modify

All integers can be written as the sum of integers whose reciprocals sum to 1.

Given a set of integers that sums to x and whose reciprocals sum to 1, a set that sums to x+3 (whose reciprocals still sum to 1) can be formed by adding 2, 2, and -1 to the set.  A set that sums to x-3 (whose reciprocals still sum to 1) can be formed by adding -2, -2, and 1 to the set.

Since there are solutions for 9 (3 + 3 + 3), 10 (2 + 4 + 4),  and 11 (2 + 3 + 6), there are solutions for all integers.

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towr
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 Re: Sum of integers whose reciprocals sum to 1   « Reply #2 on: Dec 7th, 2008, 1:22pm » Quote Modify

Heh. I wish I'd spotted that.

But how about if the sum needs to consist solely of positive integers?
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River Phoenix
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 Re: Sum of integers whose reciprocals sum to 1   « Reply #3 on: Dec 9th, 2008, 4:40pm » Quote Modify

on Dec 7th, 2008, 1:22pm, towr wrote:
 Heh. I wish I'd spotted that.   But how about if the sum needs to consist solely of positive integers?

What about distinct integers? Just curious.
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towr
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 Re: Sum of integers whose reciprocals sum to 1   « Reply #4 on: Dec 10th, 2008, 12:50am » Quote Modify

on Dec 9th, 2008, 4:40pm, River Phoenix wrote:
 What about distinct integers? Just curious.
http://mathworld.wolfram.com/EgyptianNumber.html
Every number over (and including) 78 can be written as the sum of distinct integers whose reciprocals sum to 1

I wouldn't know how to prove it though.
 « Last Edit: Dec 10th, 2008, 12:50am by towr » IP Logged

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