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April 4th, 2011, 09:57 PM  #1 
Newbie Joined: Mar 2011 Posts: 8 Thanks: 0  Alternating Sums of powers of 2
Hello, Here's an interesting problem that I have solved, but would like someone else's input simply to see if people come up with the same solution or possibly find a more elegant one. I hope anyone who tries this has fun with the problem! Which numbers can be written by selecting a subset of the powers of two and alternating them (positivenegative,..., negativepositive,...) and list them in increasing order to form a sum. *A sum cannot be made by simply one power of two, namely n=n. SO, can ALL positive integers be written this way? how many ways can a number 'n' be written as an alternating sum of powers of two? Thanks in advance, Rutzer Last edited by skipjack; February 18th, 2019 at 05:55 AM. 
April 4th, 2011, 11:48 PM  #2 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Alternating Sums of powers of 2
I think that all numbers can be written in this form, yes. I think there's a nice bijection with the binary digits of the number by taking the consecutive 1 bits and representing them as the difference of two powers of two.

April 5th, 2011, 12:37 PM  #3 
Newbie Joined: Mar 2011 Posts: 8 Thanks: 0  Re: Alternating Sums of powers of 2
Excellent, it seems we think alike! I took the binary approach as well. but I also saw the possibility for an induction... thanks for your response. rutzer Last edited by skipjack; February 18th, 2019 at 05:52 AM. 
February 18th, 2019, 05:35 AM  #4 
Newbie Joined: Feb 2019 From: chengdu china Posts: 1 Thanks: 0 
I agree that all numbers seem to be expressible in this way and this can be shown with binary. I also think there are 2 ways to express all nonpowers of 2, 1 ways to express powers of 2.
Last edited by skipjack; February 18th, 2019 at 06:00 AM. 

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