$2^{x}=3^{y}+5$
Posted: Sun Sep 01, 2013 12:31 pm
Find all positive integers x,y such that $2^{x}=3^{y}+5$
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It is easy to see that $(x,y)=(3,1),(5,3)$.asif e elahi wrote:Find all positive integers x,y such that $2^{x}=3^{y}+5$
How $y \equiv 11 \pmod{16}$ ?harrypham wrote: Therefore $3^y \equiv 59 \pmod{64}$. It follows that $y \equiv 11 \pmod{16}$. Hence $3^y \equiv 3^{11} \equiv 7 \pmod{17}$.
You note that $3^{16} \equiv 1 \pmod{64}$ and $3^{11} \equiv 59 \pmod{64}$.asif e elahi wrote:How $y \equiv 11 \pmod{16}$ ?harrypham wrote: Therefore $3^y \equiv 59 \pmod{64}$. It follows that $y \equiv 11 \pmod{16}$. Hence $3^y \equiv 3^{11} \equiv 7 \pmod{17}$.