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wu :: forums    riddles    putnam exam (pure math) (Moderators: william wu, towr, Icarus, Eigenray, Grimbal, SMQ)    Sum of integers whose reciprocals sum to 1 « Previous topic | Next topic » Pages: 1  Reply Notify of replies Send Topic Print    Author  Topic: Sum of integers whose reciprocals sum to 1  (Read 7149 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 Pages: 1  Reply Notify of replies Send Topic Print Forum Jump: ----------------------------- riddles -----------------------------  - easy   - medium   - hard   - what am i   - what happened   - microsoft   - cs => putnam exam (pure math)   - suggestions, help, and FAQ   - general problem-solving / chatting / whatever ----------------------------- general -----------------------------  - guestbook   - truth   - complex analysis   - wanted   - psychology   - chinese « Previous topic | Next topic »

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