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March 19th, 2012, 09:17 PM   #1
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diophantine x + 8x - 6x + 8 = y

Solve in the set of the integers.
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March 20th, 2012, 05:23 PM   #2
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Re: diophantine x + 8x - 6x + 8 = y

  • Wolfram says that the unique solution is
    [/*:m:2tfzl1qp]
  • This problem is supposed to be solved using factorization. [/*:m:2tfzl1qp]
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March 20th, 2012, 05:37 PM   #3
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Re: diophantine x + 8x - 6x + 8 = y

I can't see a good factorization (into linear polinomials for example), since x + 8x - 6x + 8, 8x - 6x + 8 and x + 8x - 6x have no rationals roots, except for 0 on the last one.
We can try observing that and , but I don't think that this is a good choice.
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March 20th, 2012, 05:52 PM   #4
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Re: diophantine x + 8x - 6x + 8 = y

Start from here and you will see that Wolfram's solution is right

y^3-8=(y-2)(y^2+2y+4)

x(x^2+8x-6)=(y-2)(y^2+2y+4)
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March 20th, 2012, 06:25 PM   #5
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Re: diophantine x + 8x - 6x + 8 = y

Did you managed to solve the problem?
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March 20th, 2012, 06:28 PM   #6
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Re: diophantine x + 8x - 6x + 8 = y

It is easy to use bounding arguments for x,y positive : for example. Try to get a lower bound and you should be settled.
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March 20th, 2012, 08:10 PM   #7
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Re: diophantine x + 8x - 6x + 8 = y

Thanks Proglote, it worked xP
I didn't like the solution, thought... but I don't think that there is another solution without inequalities.

Also, voc conseguiu algo na OBM ano passado? ^^
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March 21st, 2012, 02:20 AM   #8
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Re: diophantine x + 8x - 6x + 8 = y

(0,2)and(9,11)
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March 21st, 2012, 07:43 AM   #9
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Re: diophantine x + 8x - 6x + 8 = y

Quote:
Originally Posted by Pell's fish
I didn't like the solution, thought... but I don't think that there is another solution without inequalities.

Also, voc conseguiu algo na OBM ano passado? ^^
There might be other solutions, but this way it is safer.

I got an honourable mention on OBM - nvel 3, I hope this year I can get a medal (it's my last chance). How did you do?
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March 23rd, 2012, 12:07 AM   #10
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Re: diophantine x + 8x - 6x + 8 = y

I didn't took the test last year. But now I have my last chance too, I just want at least a honorable mention... :P
studying for the test is already something cool, so it will be nice anyway...
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