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 March 19th, 2012, 09:17 PM #1 Member   Joined: Aug 2010 From: Vitória ES, Brazil Posts: 71 Thanks: 0 diophantine x³ + 8x² - 6x + 8 = y³ Solve $x^3 + 8x^2 - 6x + 8= y^3$ in the set of the integers.
 March 20th, 2012, 05:23 PM #2 Member   Joined: Aug 2010 From: Vitória ES, Brazil Posts: 71 Thanks: 0 Re: diophantine x³ + 8x² - 6x + 8 = y³ Wolfram says that the unique solution is $(0, 2).$ [/*:m:2tfzl1qp] This problem is supposed to be solved using factorization. [/*:m:2tfzl1qp]
 March 20th, 2012, 05:37 PM #3 Member   Joined: Aug 2010 From: Vitória ES, Brazil Posts: 71 Thanks: 0 Re: diophantine x³ + 8x² - 6x + 8 = y³ I can't see a good factorization (into linear polinomials for example), since x³ + 8x² - 6x + 8, 8x² - 6x + 8 and x³ + 8x² - 6x have no rationals roots, except for 0 on the last one. We can try $x(x^2 + 8x - 6)= (y - 2)(y^2 + 2y + 4)$ observing that $\gcd(x, x^2 + 8x - 6)= \gcd(x, 6)$ and $\gcd(y - 2, y^2 + 2y + 4)= \gcd(y, 2)$, but I don't think that this is a good choice.
 March 20th, 2012, 05:52 PM #4 Senior Member   Joined: Jan 2011 Posts: 560 Thanks: 1 Re: diophantine x³ + 8x² - 6x + 8 = y³ 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)
 March 20th, 2012, 06:25 PM #5 Member   Joined: Aug 2010 From: Vitória ES, Brazil Posts: 71 Thanks: 0 Re: diophantine x³ + 8x² - 6x + 8 = y³ Did you managed to solve the problem?
 March 20th, 2012, 06:28 PM #6 Senior Member   Joined: Apr 2011 From: Recife, BR Posts: 352 Thanks: 0 Re: diophantine x³ + 8x² - 6x + 8 = y³ It is easy to use bounding arguments for x,y positive : $(x+3)^3 > x^3+8x^2-6x+8$ for example. Try to get a lower bound and you should be settled.
 March 20th, 2012, 08:10 PM #7 Member   Joined: Aug 2010 From: Vitória ES, Brazil Posts: 71 Thanks: 0 Re: diophantine x³ + 8x² - 6x + 8 = y³ 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, você conseguiu algo na OBM ano passado? ^^
 March 21st, 2012, 02:20 AM #8 Senior Member   Joined: Oct 2008 Posts: 215 Thanks: 0 Re: diophantine x³ + 8x² - 6x + 8 = y³ (0,2)and(9,11)
March 21st, 2012, 07:43 AM   #9
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Re: diophantine x³ + 8x² - 6x + 8 = y³

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

I got an honourable mention on OBM - nível 3, I hope this year I can get a medal (it's my last chance). How did you do?

 March 23rd, 2012, 12:07 AM #10 Member   Joined: Aug 2010 From: Vitória ES, Brazil Posts: 71 Thanks: 0 Re: diophantine x³ + 8x² - 6x + 8 = y³ 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...

 Tags 8x², diophantine

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