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Is there any integer $n\ge 2$ such that $2^n\equiv 3 \bmod n$? I understand that $n$ must be an odd non-prime. I checked up to a million with no success (but $2^n\equiv 5 \bmod n$ and $2^n\equiv 7 \bmod n$ have solutions).

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The smallest such $n$ is $n=4700063497$. A few others are known. J. Crump found $n=8365386194032363$ in 2000. Max Alekseyev found $n=3468371109448915$. Joe Crump found $n=10991007971508067$. Some information on these is at http://www.cs.ucla.edu/~klinger/newno.html

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    $\begingroup$ And using OEIS, one can find least solutions for $2^n=2k+1\pmod n$ for $k=1,\ldots,33$: oeis.org/A124977 $\endgroup$ – Yoav Kallus Jun 28 '13 at 2:55

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