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wu :: forums    riddles    putnam exam (pure math) (Moderators: towr, Eigenray, Grimbal, Icarus, william wu, SMQ)    Integral Solutions « Previous topic | Next topic » Pages: 1  Reply Notify of replies Send Topic Print    Author  Topic: Integral Solutions  (Read 1383 times) l4z3r Newbie     Posts: 10 Integral Solutions   « on: Aug 29th, 2008, 4:46am » Quote Modify a function is defined as:                                           f: Z(+)  -->  Z                                         f(m,n) =  (n3 + 1)/ (mn - 1)     where Z(+) denotes the set of positive integers and Z the set of integers.   Find all the solutions for (m,n)   EDIT: f(m,n) not f(x) « Last Edit: Aug 29th, 2008, 5:42am by l4z3r » IP Logged towr wu::riddles Moderator Uberpuzzler Some people are average, some are just mean.     Gender: Posts: 13730 Re: Integral Solutions   « Reply #1 on: Aug 29th, 2008, 5:23am » Quote Modify Shouldn't the x in f(x) come into it somewhere? IP Logged Wikipedia, Google, Mathworld, Integer sequence DB l4z3r Newbie     Posts: 10 Re: Integral Solutions   « Reply #2 on: Aug 29th, 2008, 5:42am » Quote Modify ah. meant f(m,n). sorry. IP Logged SMQ wu::riddles Moderator Uberpuzzler     Gender: Posts: 2084 Re: Integral Solutions   « Reply #3 on: Aug 29th, 2008, 5:58am » Quote Modify So, in other words, "find all , such that ( + 1) / ( - 1) ", right?   --SMQ IP Logged --SMQ l4z3r Newbie     Posts: 10 Re: Integral Solutions   « Reply #4 on: Aug 29th, 2008, 7:00am » Quote Modify yes. hint:use n3+1   1(mod3) and mn-1   -1 (mod n) (number theory) IP Logged Eigenray wu::riddles Moderator Uberpuzzler     Gender: Posts: 1948 Re: Integral Solutions   « Reply #5 on: Aug 29th, 2008, 11:53am » Quote Modify If (m,n) is a solution, then (m, (m2+n)/(mn-1)) is also; then use the ideas that appear here (and which should appear here).   Or, you can proceed more directly by writing n3+1 = (mn-1)((an-m)n-1) and bounding. IP Logged l4z3r Newbie     Posts: 10 Re: Integral Solutions   « Reply #6 on: Aug 30th, 2008, 5:34am » Quote Modify hmm. good one. I agree with the first part.   If (m,n) is a solution, then (m, (m2+n)/(mn-1)) is also   but, instead of (mn-1)((an-m)n-1) i feel a better alternative would be (kn-1)(mn-1). Gives the answer in lesser steps, i think. 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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