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wu :: forums    riddles    medium (Moderators: Eigenray, ThudnBlunder, towr, SMQ, Grimbal, Icarus, william wu)    Squares: sums and differences « Previous topic | Next topic » Pages: 1  Reply Notify of replies Send Topic Print    Author  Topic: Squares: sums and differences  (Read 1129 times) Christine Full Member     Posts: 159 Squares: sums and differences   « on: Mar 11th, 2013, 1:23pm » Quote Modify My question is: how do you find sums and differences that are both squares?   x + y is a square x - y is a square IP Logged towr wu::riddles Moderator Uberpuzzler Some people are average, some are just mean.     Gender: Posts: 13730 Re: Squares: sums and differences   « Reply #1 on: Mar 11th, 2013, 1:44pm » Quote Modify (u+v)^2-u^2 = 2uv+v^2 = 2y   x = u^2+y So basically we can take any u paired with an even v. IP Logged Wikipedia, Google, Mathworld, Integer sequence DB pex Uberpuzzler     Gender: Posts: 880 Re: Squares: sums and differences   « Reply #2 on: Mar 11th, 2013, 1:46pm » Quote Modify Or, equivalent but perhaps simpler: take any two squares that are either both even or both odd, say a2 and b2, and let x = (a2 + b2)/2 and y = (a2 - b2)/2. IP Logged Immanuel_Bonfils Junior Member     Posts: 114 Re: Squares: sums and differences   « Reply #3 on: Mar 11th, 2013, 5:01pm » Quote Modify No restriction to x and y (unless being real, y guess), so   parity doesn't matter, even b>a   -> (y <0) would be OK IP Logged Grimbal wu::riddles Moderator Uberpuzzler     Gender: Posts: 7527 Re: Squares: sums and differences   « Reply #4 on: Mar 12th, 2013, 4:00pm » Quote Modify It wasn't stated, but usually these problems ask for integer solutions.  If you are working with reals, every number >=0 is a square.   A simple set of solutions is x=a2 for some a and y=0.  It doesn't feel like a proper solution though.   On the other side, pex's method gives all integer solutions. « Last Edit: Mar 12th, 2013, 4:02pm by Grimbal » IP Logged Christine Full Member     Posts: 159 Re: Squares: sums and differences   « Reply #5 on: Mar 19th, 2013, 10:34am » Quote Modify Thank you all for the feedback.   Sorry I did not state that   > I was looking for integer solutions, > for producing interesting results (e.g. squares of prime numbers)   I'll make sure that next time I'll state clearly my questions. 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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