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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 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

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 observing that and , 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^38=(y2)(y^2+2y+4) x(x^2+8x6)=(y2)(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 : 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  
Senior Member Joined: Apr 2011 From: Recife, BR Posts: 352 Thanks: 0  Re: diophantine x³ + 8x²  6x + 8 = y³ Quote:
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... 

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8x², diophantine 
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