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   Sum of integers whose reciprocals sum to 1
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   Author  Topic: Sum of integers whose reciprocals sum to 1  (Read 7028 times)
Michael Dagg
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Sum of integers whose reciprocals sum to 1  
« on: Nov 16th, 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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Re: Sum of integers whose reciprocals sum to 1  
« Reply #1 on: Dec 7th, 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 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 »
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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 9th, 2008, 4:40pm »
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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 »
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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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