Interested Article - Первообразный корень (теория чисел)

????????????? ?????? ?? ?????? m ? ????? ????? g ?????, ???

g^{\varphi (m)}\equiv 1{\pmod {m}}

?

g^{\ell }\not \equiv 1{\pmod {m}}

???

1\leq \ell <\varphi (m),

??? $\varphi (m)$ ? ??????? ?????? . ??????? ???????, ????????????? ?????? — ??? ?????????? ??????? ????????????????? ?????? ?????? ??????? ?? ?????? m .

????? ?? ????????? ??? $\ell$ ?? $1$ ?? $\varphi (m)$ , ?????????? ????????? ??? ???????:

???????? ?? $g$ ?????? ??????? ??????? ? $m$ , ? ???? ??? - ?? ??? ?? ????????????? ??????.
??? ??? $\varphi (m)$ - ?????? ?????? ????? ??? ???? $m>2$ , ?? $\varphi (m)$ ????? ??? ??????? ???? ??????? ???????? - ??????? ????? $2$ , ?????????????, ??? ????, ????? ??????? ???????????? ?????????? ??-??????, ?????????? ????????? $g^{\varphi (m)/2}\not \equiv 1{\pmod {m}}$ ??? ?????, ??????????? ?? ????????????? ?????? ?? ?????? $m$ . ???? ????????? +1 $\equiv$ m , ?? g - ?? ??????, ? ???? ?????? ????????? -1 $\equiv$ m, ????? g - ??? ???????? ????????????? ??????.
????? ??????? ?????????, ??? $g^{\ell }\not \equiv 1{\pmod {m}}$ ??? ???? $\ell ={\frac {\varphi (m)}{p}}$ , ??? $p$ - ??????? ???????? ????? $\varphi (m)$ , ?????????? ? ?????????? ??? ????????????.

????????

?????????????

????????????? ????? ?????????? ?????? ?? ??????? $m$ ????

m=2,4,p^{a},2p^{a}

,

??? $p>2$ ? ??????? ????? , $a\geqslant 1$ ? ???????????. ?????? ? ???? ??????? ????????????????? ?????? ?????? ??????? ?? ?????? m ???????? ??????????? ??????? ??????? $\varphi (m)$ .

??????????

???? ?? ?????? $m$ ?????????? ????????????? ?????? $g$ , ?? ????? ?????????? $\varphi (\varphi (m))$ ????????? ????????????? ?????? ?? ?????? m , ?????? ??? ??? ????? ??? $g^{k}$ , ??? $1\leq k<\varphi (m)$ ? $(k,\varphi (m))=1$ .

?????? ????? ?? ??????

??? ?????????????? ????? $g$ ??? ??????? $g^{\varphi (m)}=1,g,\dots g^{\varphi (m)-1}$ ?????????? ????? ????? ?? ?????? m ? ???????? ??????????? ??????? ??????? ?? ?????? m . ??????? ??? ??????? ????? $a$ , ??????? ???????? ? $m$ , ???????? ?????????? $\ell ,0\leq \ell <\varphi (m)$ , ?????, ???

g^{\ell }\equiv a{\pmod {m}}.

????? ????? $\ell$ ?????????? ???????? ????? a ?? ????????? g .

??????????? ??????

???????????? ??????????? ????????, ??? ?????????? ????? ????????? $C$ , ??? ??? ??????? ???????? $p$ ?????????? ????????????? ?????? $g<C{\sqrt {p}}$ . ??????? ???????, ??? ??????? ??????? $p$ ??????????? ????????????? ?????? ????? ??????? $O({\sqrt {p}})$ . ????????? ?? ???????????? ??????? ???????, ??? ???? « ?????????? ???????? ?????? » ?????, ?? ????????????? ?????? ???? ????? ?????? $O(\log ^{6}{p})$ ????? ???????????? ???? .

???????

????????????? ????? ??? ??????? ??????? $p$ ???? ??????? ??????? , ?? ????????????? ????????????? ?????? ??? ????? ??????? ??????? $p$ ???? ???????? ???? ??????? ? « ?????????????? ????????????? » (1801 ???).

???????

????? 3 ???????? ????????????? ?????? ?? ?????? 7. ????? ? ???? ?????????, ?????????? ?????? ????? ?? 1 ?? 6 ??????????? ??? ????????? ??????? ?????? ?? ?????? 7:

3^{1}\equiv 3\ {\pmod {7}}

3^{2}\equiv 2\ {\pmod {7}}

3^{3}\equiv 6\ {\pmod {7}}

3^{4}\equiv 4\ {\pmod {7}}

3^{5}\equiv 5\ {\pmod {7}}

3^{6}\equiv 1\ {\pmod {7}}

??????? ?????????? ????????????? ?????? ?? ?????? m (?????????????????? ? OEIS ):

?????? m	2	3	4	5	6	7	8	9	10	11	12	13	14
????????????? ??????	1	2	3	2	5	3	—	2	3	2	—	2	3

??. ?????

??????????

. cp-algorithms.com . ???? ?????????: 27 ??????? 2020. 24 ??????? 2020 ????.
Bach, Eric; Shallit, Jeffrey. Algorithmic Number Theory (Vol I: Efficient Algorithms). — Cambridge: The MIT Press, 1996. — P. 254. — ISBN 978-0-262-02405-1 .

??????????

?????????? ?. ?. ????? 6. ????????????? ????? ? ??????? // ?????? ?????? ?????. — 1952. — 182 ?.
?????????? ?. ?. ????? 7. ????????????? ????? ? ??????? // ?????? ?????. — ?. : «????????», 2008. — 464 ?.