quadripple equation
- Tahmid Hasan
- Posts:665
- Joined:Thu Dec 09, 2010 5:34 pm
- Location:Khulna,Bangladesh.
find all natural number solutions of this equation
$a^4+b^4=2007^{2007}$
$a^4+b^4=2007^{2007}$
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Re: quadripple equation
"Inspiration is needed in geometry, just as much as in poetry." -- Aleksandr Pushkin
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Please install LaTeX fonts in your PC for better looking equations,
learn how to write equations, and don't forget to read Forum Guide and Rules.
- Tahmid Hasan
- Posts:665
- Joined:Thu Dec 09, 2010 5:34 pm
- Location:Khulna,Bangladesh.
Re: quadripple equation
$a^4+b^4 \equiv 0,1,2 \pmod{4}$, but $2007^{2007} \equiv (-1)^{odd} \equiv 3 \pmod{4}$.
"Inspiration is needed in geometry, just as much as in poetry." -- Aleksandr Pushkin
Please install LaTeX fonts in your PC for better looking equations,
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Please install LaTeX fonts in your PC for better looking equations,
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- Tahmid Hasan
- Posts:665
- Joined:Thu Dec 09, 2010 5:34 pm
- Location:Khulna,Bangladesh.
Re: quadripple equation
yes there is no natural number sol,i did it with the digit of the powers
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Re: quadripple equation
could you tell me how that is?Tahmid Hasan wrote:yes there is no natural number sol,i did it with the digit of the powers
One one thing is neutral in the universe, that is $0$.
Re: quadripple equation
Moon via,why have you taken $mod 4$???
"Questions we can't answer are far better than answers we can't question"
Re: quadripple equation
He took $a^4+b^4$ which is congruent to $0,1$ or $2$ (mod $4$) which kills the problem.tanmoy wrote:Moon via,why have you taken $mod 4$???
Last edited by Labib on Mon Mar 10, 2014 2:11 am, edited 1 time in total.
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- Phlembac Adib Hasan
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Re: quadripple equation
Did you want to know how Moon vai realized mod 4 would kill this problem?tanmoy wrote:Moon via,why have you taken $mod 4$???
That's intuition. Taking $\bmod\; 3,4,6,7,9$ is always very helpful. Sometimes the problem itself provides some valuable information. For example, here we had to deal with some fourth powers. So $\bmod \;3,4,8$ are the smartest choices, since $x^2 \equiv 0,1,4 \pmod {3,4,8}$. If there are some cubes, we may consider $\bmod\; 7,9$.
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