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Topic: Sum of integers whose reciprocals sum to 1 (Read 6950 times) 

Michael Dagg
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Sum of integers whose reciprocals sum to 1
« on: Nov 16^{th}, 2008, 11:47am » 
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Prove/disprove: Every integer greater than 23 can be written as the sum of integers whose reciprocals sum to 1.


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John_Thomas
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Re: Sum of integers whose reciprocals sum to 1
« Reply #1 on: Dec 7^{th}, 2008, 12:32pm » 
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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 x3 (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 7^{th}, 2008, 1:22pm » 
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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 9^{th}, 2008, 4:40pm » 
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on Dec 7^{th}, 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 10^{th}, 2008, 12:50am » 
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on Dec 9^{th}, 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 10^{th}, 2008, 12:50am by towr » 
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