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From experiments it seems $(1+\sqrt{2})^n+(1-\sqrt{2})^n=2^a-3^b$ has finite solutions $(a,b,n)$, where $a,b,n$ are non-negative integers. From $S$ unit equation we know $(1+\sqrt{2})^n+(1-\sqrt{2})^n=2^a+3^b$ has finite non-negative integer solutions. Is there any such result for $(1+\sqrt{2})^n+(1-\sqrt{2})^n=2^a-3^b$?

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$f(n)=(1+\sqrt2)^n+(1-\sqrt2)^n$ satisfies $f(n+2)=f(n)+2f(n+1)$ so $f(n)$ is even, but $2^a-3^b$ is odd for $a>0$. – barto Jan 25 '13 at 18:53
Thanks. However it seems result would be true for $5^a-3^a$. – user46185 Jan 25 '13 at 21:15

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