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June 30th, 2008, 11:39 AM  #1 
Newbie Joined: Jun 2008 Posts: 4 Thanks: 0  Problem solving an equation
How do I show that the equation y^9 + y^8  7y^7  5y^6 + 21y^5 + 11y^4  27y^3  5y^2 + 20y + 4 = 0, where y > 0 has no integers solutions? I think estimating the left term may help to show that the left term is larger than zero for each y > 0. Thanks in advance, frank 
June 30th, 2008, 11:55 AM  #2 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Problem solving an equation
You can check that it fails for y = 1 or 2. Otherwise y >= 3. The expression on the left can be rewritten as y^6(y^3 + y^2  7y  5) + y^2(21y^3 + 11y^2  27y  5) + 20y + 4 From here it suffices to notice that each term is positive: y^3 + y^2 >= 12y, so y^3 + y^2  7y  5 >= 5y  5 >= 10 21y^3 + 11y^2 >= 222y, so 21y^3 + 11y^2  27y  5 >= 195y  5 >= 580 20y >= 60 4 = 4 
June 30th, 2008, 09:59 PM  #3 
Newbie Joined: Jun 2008 Posts: 4 Thanks: 0  Re: Problem solving an equation
Thank you for the fast reply! There is another problem: How can I show that this equation has no solutions which are larger than zero? 
July 1st, 2008, 04:05 AM  #4 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Problem solving an equation
What do you mean by that?

July 1st, 2008, 04:13 AM  #5 
Senior Member Joined: Oct 2007 From: Chicago Posts: 1,701 Thanks: 3  Re: Problem solving an equation
I think he means 
July 1st, 2008, 05:55 AM  #6 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Problem solving an equation
If you mean y, I already showed that. If you mean x, well... I suppose I also showed that.

July 24th, 2008, 12:22 PM  #7 
Newbie Joined: Jun 2008 Posts: 4 Thanks: 0  Re: Problem solving an equation
Thank you! I got the solution of the equation. 

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