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        riddles >> putnam exam (pure math) >> Sum of integers whose reciprocals sum to 1
        
(Message started by: Michael Dagg on Nov 16^th, 2008, 11:47am)

Title: Sum of integers whose reciprocals sum to 1
Post by Michael Dagg on Nov 16^th, 2008, 11:47am

Prove/disprove: Every integer greater than 23 can be written as
the sum of integers whose reciprocals sum to 1.

Title: Re: Sum of integers whose reciprocals sum to 1
Post by John_Thomas on Dec 7^th, 2008, 12:32pm

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.

Title: Re: Sum of integers whose reciprocals sum to 1
Post by towr on Dec 7^th, 2008, 1:22pm

Heh. I wish I'd spotted that.

But how about if the sum needs to consist solely of positive integers?

Title: Re: Sum of integers whose reciprocals sum to 1
Post by River Phoenix on Dec 9^th, 2008, 4:40pm

on 12/07/08 at 13:22:50, 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.

Title: Re: Sum of integers whose reciprocals sum to 1
Post by towr on Dec 10^th, 2008, 12:50am

on 12/09/08 at 16:40:49, River Phoenix wrote:

What about distinct integers? Just curious.

http://mathworld.wolfram.com/EgyptianNumber.html
[hide]Every number over (and including) 78 can be written as the sum of distinct integers whose reciprocals sum to 1[/hide]

I wouldn't know how to prove it though.