In my textbook its described as finding $k\;$th roots moduluo $m$. How do you solve equations of this form as I can't find any examples anywhere? I am just looking for a step by step example so I can understand how to solve them. An example question could be: solve $x^{11} \equiv 5 \bmod 47$.

Any links to any websites or videos explaining the congruence equations in this form would be greatly appreciated also. Thanks!

  • $\begingroup$ just not got anything to go by, not many examples anywhere of this type, I've tried but what do I look up to find them, any links? $\endgroup$ – Anonymous May 2 at 17:31
  • $\begingroup$ thanks but thats not really what im after, I just want a straightforward step by step solution of a question similar to the one above to help my understanding preferably from different website thats got a few. Appreciate it anyway though! $\endgroup$ – Anonymous May 2 at 17:43

In $\mathbb F_{47}$one has $x^{46}=1$ then $x^{23}=\pm1$

$x^{23}=1\iff x^{11}\cdot x^{12}=1\iff 5\cdot x^{12}=1\Rightarrow x^{12}=\dfrac 15=19$.

Then $\begin{cases}x^{12}=19\\x^{11}=5\end{cases}\Rightarrow x=\dfrac{19}{5}=19\cdot19=34$ and $34^{11}=21$ thus $x=34$ is not solution.

$x^{23}=-1\iff x^{11}\cdot x^{12}=-1\iff 5\cdot x^{12}=-1\Rightarrow x^{12}=\dfrac {-1}{5}=28$.

Then $\begin{cases}x^{12}=28\\x^{11}=5\end{cases}\Rightarrow x=\dfrac{28}{5}=28\cdot19=15$ and $\boxed{15^{11}=5}$

Thus $\color{red}{x=15}$ is the only solution.

  • $\begingroup$ How did you compute $1/5$ and $34^{11}$ and $15^{11}?$ Those omitted steps seem to be where most of the work is this way. $\endgroup$ – Bill Dubuque May 3 at 3:54
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    $\begingroup$ This is from straightforward calculation and it is possible in several ways by successive simplifications $34^2=28; \space 34^3=12; \space 34^9=12^3; \space 34^{11}=12^3\cdot15^2\ne5$ $15^2=37; \space 15^3=38; \space 15^4=6; \space 15^{11}=6^2\cdot15^3=36\cdot38=5$. $\endgroup$ – Piquito May 4 at 13:30
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    $\begingroup$ Besides $5x=47y+1$ easily gives $x=19$ $\endgroup$ – Piquito May 4 at 13:31
  • $\begingroup$ I think in the first line you used Fermat little theorem ..... correct? if so why are you sure that 47 does not divide x (as it is the first condition that must be satisfied for Fermat little theorem to be applied) $\endgroup$ – Secretly Jun 1 at 16:09
  • $\begingroup$ Also I do not understand why in the first line of your answer $x^23 \equiv 1$ (I can see you are using equality sign instead of $\equiv$ sign ..... correct?) $\endgroup$ – Secretly Jun 1 at 17:08

Hint $\bmod 47\!:\,\ x^{\large 11}\equiv 5\,\overset{(\ \ )^{\Large 21}}{\Longrightarrow} x\equiv 5^{\large 21}\ $ by $\,11\cdot 21\equiv 1 \pmod{\!46}\,$ and little Fermat.

Thus $\,x\equiv \dfrac{\color{#c00}{5^{\large 23}}}{5^{\large 2}}\equiv \dfrac{\color{#c00}{\bf -1}}{25}\equiv \dfrac{-2}{50}\equiv\dfrac{45}3\equiv 15,\,$ by $\, \color{#c00}{5^{\large 23}} \equiv \left(\dfrac{5}{47}\right) = \left(\dfrac{47}{5}\right)=\left(\dfrac{2}{5}\right)= \color{#c00}{\bf -1}\,$

Or $\ x^{\large 23}\equiv -1\ $ by $\ (x^{\large 23})^{\large 11} = (x^{\large 11})^{\large 23} \equiv \color{#c00}{5^{\large 23} \equiv\bf -1}\, $ so $\ x\equiv \dfrac{x^{\large 23}}{(x^{\large 11})^{\large 2}}\equiv \dfrac{-1}{25}\equiv 15\,$ as above.

The first method raised $\,x^{\large 11}\equiv 5\,$ to power $\, \dfrac{1}{11}\equiv 21\pmod{\!46}\,$ to get the $11$'th root, using

$\!\bmod 46\!:\,\ \dfrac{1}{11}\equiv \dfrac{5}{55}\equiv \dfrac{5}9\equiv \dfrac{25}{45}\equiv \dfrac{25}{-1}\equiv 21,\, $ computed by Gauss's algorithm, as above.

In the second method instead of using $\,x^{\large 46}\equiv 1\,$ we use $\,x^{\large 23}\equiv -1.\,$ Here $\,1/11\,$ is simpler

$\!\bmod 23\!:\,\ \dfrac{1}{11}\equiv \dfrac{2}{22}\equiv \dfrac{2}{-1}\equiv -2\ $ so $\ 5\equiv x^{\large 11}\,\overset{\large(\ \ )^{\Large -2}\!}\Longrightarrow\, 5^{\large -2}\equiv x^{\large-22}\equiv \dfrac{x}{x^{\large 23}}\equiv -x$

See this theorem for the general result when the power is coprime to the order(s).


Hint at another related way to think about it:

  • $(x^{11})^5\equiv x^9\equiv 5^5\equiv 23 \bmod 47$ Because of $(a^b)^c=a^{bc}$, $a^{p-1}\equiv 1\bmod p$ for prime p, a not a multiple, and $55\equiv 9\bmod 46$ which is valid by extension of the above rules plus $1^n=1$ and the fact that the first multiple of a number greater than a non multiple is of lower remainder if 0 is considered highest. oh and 1 times anything is itself.
  • $(x^9)^6\equiv x^8\equiv 23^6\equiv 6^2\bmod 47$ Because $54\equiv 8\bmod 46$
  • $(x^8)^6\equiv x^2\equiv 6^{12} \bmod 47$ Because $48\equiv 2\bmod 46$
  • $x\equiv -(6^6) \bmod 47$ Because $(-1)^{2n+1}=-1$, $6^6=992(47)+32$,$32^{11}=766570149339658(47)+42$ which is $-5\bmod 47$, but that means the other is $5\equiv 47$

three round of Fermat. two possible cases, but only one stated works because 11 is odd. EDIT : and here's a case that's not true:

$$4^3\equiv 5 \bmod 48$$

It's not true because:

$$y\equiv b\bmod m\implies y=mx+b$$

if y and m share a factor, then: $$y-mx=b$$ shows us, b will have that factor as well, by the inverse of the distributive property ( factoring out).

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    $\begingroup$ It's not clear what you did. You should explain why you choose those particular powers so we can tell whether it is a general method or lucky coincidence that doesn't generalize. $\endgroup$ – Bill Dubuque May 2 at 20:06
  • $\begingroup$ 3 rounds of Fermat chose the powers via the exponent rules and being the least power over 46. and yes I know $-(6^6)\bmod 47$ also can work. $\endgroup$ – Roddy MacPhee May 2 at 20:09
  • $\begingroup$ okay one gives -5 the other 5 because 11 is odd. $\endgroup$ – Roddy MacPhee May 2 at 20:25
  • $\begingroup$ It seems like pure luck. Essentially you are using that $\,11k \equiv 2\pmod{\!46}$ for $\,k = 5\cdot 6\cdot 6\,$ so raising $\,x^{11}\equiv 5\pmod{\!47}$ to power $k$ yields $\, x^2\equiv 5^k.\,$ You got lucky that your computation of $5^k$ yielded on obvious perfect square. That won't work in general. In fact $\,k\equiv 42\pmod{\!46}$ so you compute $\,5^{42},\,$ which is the square of what I did. Nor do you explain how you chose the correct square root (which you originally had wrong). Without any explanation it is magic - not math. $\endgroup$ – Bill Dubuque May 2 at 20:49
  • $\begingroup$ that last part was checked later. all I did was $11\cdot5=55\equiv 9 \bmod 46$ as long as I didn't hit an exponent that was a factor of 46, I knew the next exponent would drop. this tends to bring the exponent down to 2 then you can sqrt it. and figure out which sqrt is required. $\endgroup$ – Roddy MacPhee May 2 at 20:54

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