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wu :: forums    riddles    putnam exam (pure math) (Moderators: Eigenray, towr, william wu, Grimbal, Icarus, SMQ)    Partition naturals into APs « Previous topic | Next topic » Pages: 1  Reply Notify of replies Send Topic Print    Author  Topic: Partition naturals into APs  (Read 1473 times) Aryabhatta Uberpuzzler     Gender: Posts: 1321 Partition naturals into APs   « on: Aug 5th, 2005, 8:52am » Quote Modify Can we partition the set of natural numbers into a finite number (at least 2) of arithmetic progressions, such that no two progressions have the same common difference?   This is probably a well known result, but pretty interesting nevertheless. IP Logged Icarus wu::riddles Moderator Uberpuzzler Boldly going where even angels fear to tread.     Gender: Posts: 4863 Re: Partition naturals into APs   « Reply #1 on: Aug 5th, 2005, 3:05pm » Quote Modify If you demand that your arithmetic progressions all have nth term of form nd for some fixed d, then yes, it is a very well known result! If you allow the terms to have form nd + c for some fixed d, c, then the situation is harder. « Last Edit: Aug 5th, 2005, 3:06pm by Icarus » IP Logged "Pi goes on and on and on ... And e is just as cursed. I wonder: Which is larger When their digits are reversed? " - Anonymous Aryabhatta Uberpuzzler     Gender: Posts: 1321 Re: Partition naturals into APs   « Reply #2 on: Aug 5th, 2005, 5:42pm » Quote Modify It could be nd+c. For instance one partition is 2n, 4n+1, 4n+3. IP Logged Aryabhatta Uberpuzzler     Gender: Posts: 1321 Re: Partition naturals into APs   « Reply #3 on: Aug 12th, 2005, 7:46pm » Quote Modify Hint: sometimes it makes things simple to make things complex : IP Logged Eigenray wu::riddles Moderator Uberpuzzler     Gender: Posts: 1948 Re: Partition naturals into APs   « Reply #4 on: Aug 15th, 2005, 1:08pm » Quote Modify Perhaps people are remaining silent because the problem is already on this site, and they do not wish to spoil it.   Anyway, both the result and its proof are far too lovely to only be listed once! IP Logged Aryabhatta Uberpuzzler     Gender: Posts: 1321 Re: Partition naturals into APs   « Reply #5 on: Aug 15th, 2005, 4:17pm » Quote Modify on Aug 15th, 2005, 1:08pm, Eigenray wrote: Perhaps people are remaining silent because the problem is already on this site, and they do not wish to spoil it.   Aw shoot! Here is the link: http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_put nam;action=display;num=1047927969;start=0#0   Quote: Anyway, both the result and its proof are far too lovely to only be listed once!   Completely agree. The approach is beautiful and is basically the building block for Analytic number theory! IP Logged 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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