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Zaara

  • 2 years ago

if a and b are two integers with gcd(a,b)=1, then show that gcd(a-b, a^2+ab+b^2)=1 or 3

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  1. Zaara
    • 2 years ago
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    @experimentX

  2. amistre64
    • 2 years ago
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    i wonder a-b = c a^2+ab+b^2 = d a-b+ab = c +ab a^2+ab+ab+b^2 = d +ab a-b+ab = c +ab (a+b)^2 = d +ab

  3. Zaara
    • 2 years ago
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    then how to show gcd equals 1 or 3??????

  4. amistre64
    • 2 years ago
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    .... still looking at it :)

  5. amistre64
    • 2 years ago
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    wondering if a euclidean algorithm would pan out a^2+ab+b^2 = (a-b)(q) + r ...

  6. satellite73
    • 2 years ago
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    i am going to make a guess that it has something to do with \[a^3-b^2=(a-b)(a^2+ab+b^2)\] just a guess though

  7. satellite73
    • 2 years ago
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    not a solution by any means, i just thought it might be the hook in to the problem i have no idea how to solve it

  8. ParthKohli
    • 2 years ago
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    STOP RIGHT THERE. http://openstudy.com/study#/updates/512f5747e4b098bb5fbd07fa

  9. Zaara
    • 2 years ago
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    @ParthKohli thats it!!!

  10. Zaara
    • 2 years ago
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    better if the whole thing is explained....

  11. amistre64
    • 2 years ago
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    a^2+ab+b^2 = (a-b)(q) + r a^2+ab+b^2 - r= (a-b)(q) (a^2+ab+b^2 - r)(a-b) = q i was wondering if there was a method we could employ there; give that r=1 or 3, and gcd(a,b) = 1

  12. Callisto
    • 2 years ago
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    *bookmark*

  13. ParthKohli
    • 2 years ago
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    :-\ \[\dfrac{(a^2 + ab + b^2 - r)}{a - b} = q\]

  14. ParthKohli
    • 2 years ago
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    Can anyone explain?

  15. amistre64
    • 2 years ago
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    \[c|d~\iff~d=cn~;~n \in Z\] \[p = kq+r~\to \frac{p-r}{k}=q~,~q\in Z\]

  16. amistre64
    • 2 years ago
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    the euclidean algorithm has alot of recurrsive:\[a=bq+r\] if \[a=bq+r\\b = r(b)+0\]then gcd(a,b)=r

  17. amistre64
    • 2 years ago
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    hmmm, so if we can show that: a = r(b^2+1) or a/(b^2+1) = r it looks good to me :) is there a flaw in my thought perhaps? if not then: \[\frac{a^2+ab+b^2}{(a-b)^2+1}=1~or~3\]should do it :)

  18. amistre64
    • 2 years ago
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    pffft, misread my own thoughts lol if b=rb then a = rbq+r a = r(bq+1) for q in Z

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