kill diophantus!!!!!!(BOMC-2)

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Niloy Da Fermat
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kill diophantus!!!!!!(BOMC-2)

Unread post by Niloy Da Fermat » Tue Mar 27, 2012 7:27 pm

পূর্ণ সংখ্যায় সমাধান বের করার জন্য কি কি অস্ত্র আছে সবাই মিলে তার একটা লিস্ট করা যাক ।
kame......hame.......haa!!!!
আব্দুল্লাহ আল মুনইম (নিলয়)
সরকারি কেসি কলেজ, ঝিনাইদহ

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Niloy Da Fermat
Posts:33
Joined:Wed Mar 21, 2012 11:48 am

Re: kill diophantus!!!!!!(BOMC-2)

Unread post by Niloy Da Fermat » Tue Mar 27, 2012 7:31 pm

I am starting-
1.considering the equation in different mods
2.Bezout's identity
3.General solution of Pythagorian triple
4.General solution of Pell-Fermat equation
kame......hame.......haa!!!!
আব্দুল্লাহ আল মুনইম (নিলয়)
সরকারি কেসি কলেজ, ঝিনাইদহ

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Phlembac Adib Hasan
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Re: kill diophantus!!!!!!(BOMC-2)

Unread post by Phlembac Adib Hasan » Tue Mar 27, 2012 8:37 pm

$1.$I use a trick to find solutions in such equations: $p^n+a^r=b^k$ where $p$ is a prime.If somehow I can prove both $r$ and $k$ are even, then I write the equation in this way: $p^n=(b^{\frac {k}{2}}+a^{\frac {r}{2}})(b^{\frac {k}{2}}-a^{\frac {r}{2}})$.Here both of the numbers of RHS are power of $p$.But they differ by $2a^{\frac {r}{2}}$.Which brings the desired solution.
Sometimes it's very helpful in solving difficult problems or lemmas.

$2.$ Another way to find integer solution is to use logarithms.Perhaps most of you know about it.So I am giving just an example.
viewtopic.php?f=20&t=1681
Now two exercises:
$(1)$ Find all the solutions of the equation $a^b=b^a$.
$(2)$ viewtopic.php?f=26&t=1894
These tricks turn the problems to simple jokes.
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User avatar
Phlembac Adib Hasan
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Re: kill diophantus!!!!!!(BOMC-2)

Unread post by Phlembac Adib Hasan » Thu Apr 12, 2012 12:32 pm

Here is another exercise:
viewtopic.php?f=20&t=2027
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