Representation of powers of $2$
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- Posts:107
- Joined:Sun Dec 12, 2010 10:46 am
Prove that any positive integral power of $2$ is representable in the form $7x^2+y^2$. where $x,y$ are integers.
- Thanic Nur Samin
- Posts:176
- Joined:Sun Dec 01, 2013 11:02 am
Re: Representation of powers of $2$
Base Case:
$2^3=8=7+1=7(1)^2+1^2$
$2^4=16=7+9=7(1)^2+3^2$
Induction:
For odd powers,
Let $2^p=7a^2+b^2$[$p$ is odd]
$\rightarrow 2^{p+2}=7(2a)^2+(2b)^2$
For even powers,
Let $2^p=7a^2+b^2$[$p$ is even]
$\rightarrow 2^{p+2}=7(2a)^2+(2b)^2$
Proved by induction.
$2^3=8=7+1=7(1)^2+1^2$
$2^4=16=7+9=7(1)^2+3^2$
Induction:
For odd powers,
Let $2^p=7a^2+b^2$[$p$ is odd]
$\rightarrow 2^{p+2}=7(2a)^2+(2b)^2$
For even powers,
Let $2^p=7a^2+b^2$[$p$ is even]
$\rightarrow 2^{p+2}=7(2a)^2+(2b)^2$
Proved by induction.
Hammer with tact.
Because destroying everything mindlessly isn't cool enough.
Because destroying everything mindlessly isn't cool enough.
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- Posts:107
- Joined:Sun Dec 12, 2010 10:46 am
Re: Representation of powers of $2$
For even powers, take $x=0$ and for odd powers take $x=y$. We're done!
Note that through this solution we can also find a representation directly.
Note that through this solution we can also find a representation directly.