I want find out the Smallest positive integer r such that $8^{17} \equiv r \pmod{97}$.
Fermat's theorem only tells us $8^{96} \equiv 1 \pmod{97}.$
How can I proceed. Any hint will be appreciated.
Thank you in advance.
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1You wrote mod 97 three times and formatted it three different ways. Are you familiar with successive squaring? – May 05 '16 at 03:00
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1Hint: $8^{16}=2^{48}\equiv \pm 1\pmod{97}$. When is $2^{(p-1)/2}\equiv 1\pmod p$? – Thomas Andrews May 05 '16 at 03:02
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Ohh. r =8 is the answer. – Surojit May 05 '16 at 03:05
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https://en.wikipedia.org/wiki/Euler%27s_criterion – user236182 May 05 '16 at 03:10
1 Answers
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We have that $2$ is a quadratic residue $\pmod p$ when $p \equiv \pm 1 \pmod 8$. In this case, $97 \equiv 1 \pmod 8$, ensuring that $2$ is a quadratic residue.
By Euler's Criterion, we have that $8^{16}=2^{48} \equiv 1 \pmod {97}$. Thus, $8^{17} \equiv 8 \pmod {97}$.
S.C.B.
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