Put \(2^6+2^9+2^n=576+2^n=m^2\)

\(\Rightarrow2^n=m^2-576=m^2-24^2=\left(m-24\right)\left(m+24\right)\)

Put \(\left\{{}\begin{matrix}m-24=2^a\\m+24=2^b\end{matrix}\right.\) ( a < b ; a + b = n)

Then \(\Rightarrow2^b-2^a=\left(m+24\right)-\left(m-24\right)\)

\(\Rightarrow2^a\left(2^{b-a}-1\right)=48=2^4.3\)

\(2^{b-a}-1⋮̸2\Rightarrow2^{b-a}-1=3\)

\(\Rightarrow\left\{{}\begin{matrix}2^a=2^4\\2^{b-a}=3+1=4=2^2\end{matrix}\right.\)

\(\Rightarrow\left\{{}\begin{matrix}a=4\\b-a=2\Rightarrow b=6\end{matrix}\right.\)

\(\Rightarrow n=a+b=4+6=10\)

Check the result: \(2^6+2^9+2^{10}=40^2\).

Put 26+29+2n=576+2n=m2

⇒2n=m2−576=m2−242=(m−24)(m+24)

Put {m−24=2am+24=2b

 ( a < b ; a + b = n)

Then ⇒2b−2a=(m+24)−(m−24)

⇒2a(2b−a−1)=48=24.3

2b−a−1⋮̸2⇒2b−a−1=3

⇒{2a=242b−a=3+1=4=22

⇒{a=4b−a=2⇒b=6

⇒n=a+b=4+6=10

Check the result: 26+29+210=402

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