$x^{13}$ = 2 mod 23. I know how to use the Fermat's theorem, but in this case, we don't know the value of "a". So I am unsure of how to even start?
Here all I know is that p = 23 and thus $a^{22}$ = 1 mod 23
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$\begingroup$ What do you mean???????????? $\endgroup$– user717038Nov 7, 2019 at 23:15
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$\begingroup$ What is the problem to be solved here? Find $x$? $\endgroup$– bjorn93Nov 7, 2019 at 23:18
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$\begingroup$ Yeah it just says to solve the equation $\endgroup$– user717038Nov 7, 2019 at 23:34
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1 Answer
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Since $13\times17=221\equiv1\bmod22, $
$x^{13}\equiv2\bmod23\implies x^{13\times17}\equiv x\equiv 2^{17}\bmod 23.$
answered Nov 7, 2019 at 23:21
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$\begingroup$ Note: $x^{13\times17}\equiv x$ follows from Fermat's Little Theorem, which says $x^{22}\equiv1\bmod23$; $2^{11}\equiv1\bmod23$ so $x\equiv2^6=64\bmod 23$ $\endgroup$ Nov 7, 2019 at 23:31
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$\begingroup$ So this means $64^{13}$ divided by 23 will give me a remainder of 2? $\endgroup$– user717038Nov 7, 2019 at 23:58
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$\begingroup$ Ok so is there a formal way you came up with that 17? $\endgroup$– user717038Nov 8, 2019 at 0:10
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