Author |
Topic: Sum of integers whose reciprocals sum to 1 (Read 7150 times) |
|
Michael Dagg
Senior Riddler
Gender:
Posts: 500
|
|
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.
|
|
IP Logged |
Regards, Michael Dagg
|
|
|
John_Thomas
Newbie
Posts: 2
|
|
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.
|
|
IP Logged |
|
|
|
towr
wu::riddles Moderator Uberpuzzler
Some people are average, some are just mean.
Gender:
Posts: 13730
|
|
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?
|
|
IP Logged |
Wikipedia, Google, Mathworld, Integer sequence DB
|
|
|
River Phoenix
Junior Member
Gender:
Posts: 125
|
|
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.
|
|
IP Logged |
|
|
|
towr
wu::riddles Moderator Uberpuzzler
Some people are average, some are just mean.
Gender:
Posts: 13730
|
|
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 |
Wikipedia, Google, Mathworld, Integer sequence DB
|
|
|
|