Prob!
 Abdul Muntakim Rafi
 Posts: 173
 Joined: Tue Mar 29, 2011 10:07 pm
 Location: bangladesh,the earth,milkyway,local group.
Re: Prob!
\[(a^2c^2+a^2d^2+b^2c^2+b^2d^2) (e^2+f^2)\]
\[e^2(a^2c^2+a^2d^2+b^2c^2+b^2d^2)+f^2(a^2c^2+a^2d^2+b^2c^2+b^2d^2) \]
\[e^2((ac+bd)^2+(adbc)^2)+f^2((ac+bd)^2+(adbc)^2) \]
Now let $ac+bd=x,adbc=y$
\[e^2(x^2+y^2)+f^2(x^2+y^2) \]
\[e^2x^2+e^2y^2+f^2x^2+f^2y^2 \]
\[(ex+fy)^2+(eyfx)^2\]
\[(e(ac+bd)+f(adbc))^2+(e(adbc)f(ac+bd))^2\]
\[e^2(a^2c^2+a^2d^2+b^2c^2+b^2d^2)+f^2(a^2c^2+a^2d^2+b^2c^2+b^2d^2) \]
\[e^2((ac+bd)^2+(adbc)^2)+f^2((ac+bd)^2+(adbc)^2) \]
Now let $ac+bd=x,adbc=y$
\[e^2(x^2+y^2)+f^2(x^2+y^2) \]
\[e^2x^2+e^2y^2+f^2x^2+f^2y^2 \]
\[(ex+fy)^2+(eyfx)^2\]
\[(e(ac+bd)+f(adbc))^2+(e(adbc)f(ac+bd))^2\]
Man himself is the master of his fate...

 Posts: 461
 Joined: Wed Dec 15, 2010 10:05 am
 Location: Dhaka
 Contact:
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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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.

 Posts: 461
 Joined: Wed Dec 15, 2010 10:05 am
 Location: Dhaka
 Contact:
Re: Prob!
$\prod_{i=1}^{n1} (a_{i}^2 + a_{i+1}^2)=(a_{1}^2 + a_{2}^2)(a_{3}^2 + a_{4}^2).....(a_{n1}^2 + a_{n}^2)$
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 moveif 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 $a^2+b^2=(a+bi)(abi)$ and use the fact that multiplications of any two numbers in the form $a+bi$ or $abi$ 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
\[(ad+bc)^2+(acbd)^2\]
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
\[(ad+bc)^2+(acbd)^2\]
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...