Find all pairs

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Phlembac Adib Hasan
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Find all pairs

Unread post by Phlembac Adib Hasan » Wed Feb 01, 2012 8:56 pm

Find all pairs of positive integers $x,y$ satisfying the equation \[x^{y^2}=y^{x}\]
(If it's not been posted yet) A nice problem.I solved it about one year ago.Number theory lovers must solve it,I think.
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sakibtanvir
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Re: Find all pairs

Unread post by sakibtanvir » Sat Feb 04, 2012 7:41 pm

An amount of certain opposition is a great help to a man.Kites rise against,not with,the wind.

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Phlembac Adib Hasan
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Re: Find all pairs

Unread post by Phlembac Adib Hasan » Sat Feb 04, 2012 9:51 pm

Hei, it's an IMO problem and you can't use wolframalpha there.I want a complete mathematical proof.And always remember that pure mathematicians do not like calculators.
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Masum
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Re: Find all pairs

Unread post by Masum » Fri Feb 10, 2012 6:22 pm

Phlembac Adib Hasan wrote:Find all pairs of positive integers $x,y$ satisfying the equation \[x^{y^2}=y^{x}\]
(If it's not been posted yet) A nice problem.I solved it about one year ago.Number theory lovers must solve it,I think.
But the calculation is tedious. Just saying that $x,y$ must have same prime factors. Then some little arguments should come up with a solution.
One one thing is neutral in the universe, that is $0$.

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Re: Find all pairs

Unread post by Phlembac Adib Hasan » Sun Feb 12, 2012 9:07 pm

Masum Vaia wrote: But the calculation is tedious. Just saying that $x,y$ must have same prime factors. Then some little arguments should come up with a solution.
Yes, I agree.But what will you say if anyone uses calculators to prove?Like, if I use computer to prove $2011^{2012}>2012^{2011}$, will you admire it?Though calculators can work for large numbers, but not at soooooo.....muuuuuuch large range.So we must use logic at last for complete proof.
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