Prob!
- Abdul Muntakim Rafi
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Re: Prob!
\[(a^2c^2+a^2d^2+b^2c^2+b^2d^2) (e^2+f^2)\]
\[e^2(a^2c^2+a^2d^2+b^2c^2+b^2d^2)+f^2(a^2c^2+a^2d^2+b^2c^2+b^2d^2) \]
\[e^2((ac+bd)^2+(ad-bc)^2)+f^2((ac+bd)^2+(ad-bc)^2) \]
Now let $ac+bd=x,ad-bc=y$
\[e^2(x^2+y^2)+f^2(x^2+y^2) \]
\[e^2x^2+e^2y^2+f^2x^2+f^2y^2 \]
\[(ex+fy)^2+(ey-fx)^2\]
\[(e(ac+bd)+f(ad-bc))^2+(e(ad-bc)-f(ac+bd))^2\]
\[e^2(a^2c^2+a^2d^2+b^2c^2+b^2d^2)+f^2(a^2c^2+a^2d^2+b^2c^2+b^2d^2) \]
\[e^2((ac+bd)^2+(ad-bc)^2)+f^2((ac+bd)^2+(ad-bc)^2) \]
Now let $ac+bd=x,ad-bc=y$
\[e^2(x^2+y^2)+f^2(x^2+y^2) \]
\[e^2x^2+e^2y^2+f^2x^2+f^2y^2 \]
\[(ex+fy)^2+(ey-fx)^2\]
\[(e(ac+bd)+f(ad-bc))^2+(e(ad-bc)-f(ac+bd))^2\]
Man himself is the master of his fate...
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Re: Prob!
Proof the general version now,$\prod (a^{2}_{i}+a^{2}_{j})$ can be shown as sum of two squares. (Very easy with a simple trick)
You spin my head right round right round,
When you go down, when you go down down......(-$from$ "$THE$ $UGLY$ $TRUTH$" )
When you go down, when you go down down......(-$from$ "$THE$ $UGLY$ $TRUTH$" )
Re: Prob!
Factorisation!
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Nur Muhammad Shafiullah | Mahi
Use $L^AT_EX$, It makes our work a lot easier!
Nur Muhammad Shafiullah | Mahi
- Abdul Muntakim Rafi
- Posts:173
- Joined:Tue Mar 29, 2011 10:07 pm
- Location:bangladesh,the earth,milkyway,local group.
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- Posts:461
- Joined:Wed Dec 15, 2010 10:05 am
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Re: Prob!
$\prod_{i=1}^{n-1} (a_{i}^2 + a_{i+1}^2)=(a_{1}^2 + a_{2}^2)(a_{3}^2 + a_{4}^2).....(a_{n-1}^2 + a_{n}^2)$
You spin my head right round right round,
When you go down, when you go down down......(-$from$ "$THE$ $UGLY$ $TRUTH$" )
When you go down, when you go down down......(-$from$ "$THE$ $UGLY$ $TRUTH$" )
- Tahmid Hasan
- Posts:665
- Joined:Thu Dec 09, 2010 5:34 pm
- Location:Khulna,Bangladesh.
Re: Prob!
here's the crux move-if two integers are sum of two squares,then their product is also sum of two squares.
বড় ভালবাসি তোমায়,মা
Re: Prob!
Or use complex numbers to write $a^2+b^2=(a+bi)(a-bi)$ and use the fact that multiplications of any two numbers in the form $a+bi$ or $a-bi$ also has that form.So it follows directly.
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Use $L^AT_EX$, It makes our work a lot easier!
Nur Muhammad Shafiullah | Mahi
Use $L^AT_EX$, It makes our work a lot easier!
Nur Muhammad Shafiullah | Mahi
- Abdul Muntakim Rafi
- Posts:173
- Joined:Tue Mar 29, 2011 10:07 pm
- Location:bangladesh,the earth,milkyway,local group.
Re: Prob!
Thanks for telling the meaning... Sourav
and mine is like tahmid's...
\[(a^2+b^2)(c^2+d^2)\]
can be written as
\[(ad+bc)^2+(ac-bd)^2\]
We can continue in this way...
And isn't the two lines I wrote enough to prove
and mine is like tahmid's...
\[(a^2+b^2)(c^2+d^2)\]
can be written as
\[(ad+bc)^2+(ac-bd)^2\]
We can continue in this way...
And isn't the two lines I wrote enough to prove
if two integers are sum of two squares,then their product is also sum of two squares
Man himself is the master of his fate...