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wu :: forums    riddles    easy (Moderators: SMQ, william wu, Grimbal, Icarus, ThudnBlunder, Eigenray, towr)    Difference of Two Squares « Previous topic | Next topic » Pages: 1  Reply Notify of replies Send Topic Print    Author  Topic: Difference of Two Squares  (Read 257 times) ThudnBlunder wu::riddles Moderator Uberpuzzler The dewdrop slides into the shining Sea     Gender: Posts: 4489 Difference of Two Squares   « on: Nov 28th, 2003, 3:45am » Quote Modify Find the condition that a positive integer be expressible as the difference of two squares. « Last Edit: Nov 28th, 2003, 5:20am by ThudnBlunder » IP Logged THE MEEK SHALL INHERIT THE EARTH.....................................................................er, if that's all right with the rest of you. TenaliRaman Uberpuzzler I am no special. I am only passionately curious.     Gender: Posts: 1001 Re: Difference of Two Squares   « Reply #1 on: Nov 28th, 2003, 4:49am » Quote Modify :: n = x*y a+b = x a-b = y 2a = x+y 2b = x-y implies, n should be such that it can be factored into two even factors or two odd factors :: IP Logged Self discovery comes when a man measures himself against an obstacle - Antoine de Saint Exupery Sir Col Uberpuzzler impudens simia et macrologus profundus fabulae     Gender: Posts: 1825 Re: Difference of Two Squares   « Reply #2 on: Nov 28th, 2003, 5:05am » Quote Modify :: x2–y2=(x–y)(x+y)   If x=y+1, x2–y2=(y+1–y)(y+1+y)=2y+1.   Hence all odd integers can be expressed as the difference of two squares; for example, 17=2y+1, y=8, x=9, and 92–82=81–64=17.   If x=y+2, x2–y2=(y+2–y)(y+2+y)=4(y+1).   Hence all double evens (multiples of four), can be expressed as the difference of two squares; for example, 48=4(y+1), y=11, x=13, and 132–112=169–121=48.   All we need do now, is prove that single evens (non-multiples of four), cannot be expressed as the difference of two squares.   Consider x2–y2=(x–y)(x+y). If x+y is even, then so too will be x–y, hence (x–y)(x+y) will be a multiple of four. :: IP Logged mathschallenge.net / projecteuler.net LZJ Junior Member     Gender: Posts: 82 Re: Difference of Two Squares   « Reply #3 on: Nov 28th, 2003, 5:38am » Quote Modify Just for clarity, single evens means the integers that have only a single factor of 2. Those that are divisible by 8, 32...can still be expressed as the difference of 2 squares, I think. 9 - 1 = 8, 36 - 4 = 32... 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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