“Primitive Root”的概念、定义、习题解题方法及翻译-数学辞典

单词 Primitive Root

释义

Primitive Root

A primitive root of a Prime is an Integer satisfying such that the residue classes of ,, , ..., are all distinct, i.e., (mod ) has Order (Ribenboim1996, p. 22). If is a Prime Number, then there are exactly incongruent primitive roots of (Burton 1989,p. 194).

More generally, if ( and are Relatively Prime) and is of Order modulo , where is the Totient Function, then is a primitive root of (Burton 1989, p. 187). In otherwords, has as a primitive root if , but (mod ) for all positive integers. A primitive root of a number (but not necessarilythe smallest primitive root for composite ) can be computed using the Mathematica routine NumberTheory`NumberTheoryFunctions`PrimitiveRoot[n].

If has a primitive root, then it has exactly of them (Burton 1989, p. 188). For , 2, ..., thefirst few values of are 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 4, 2, 4, 2, 4, 4, 8, ... (Sloane's A010554). has aprimitive root if it is of the form 2, 4, a power , or twice a power , where is an Odd Prime and (Burton 1989, p. 204). The first few for which primitive roots exist are 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 17, 18, 19,22, ... (Sloane's A033948), so the number of primitive root of order for , 2, ... are 0, 1, 1, 1, 2, 1, 2, 0, 2, 2,4, 0, 4, (Sloane's A046144).

The smallest primitive roots for the first few Integers are given in the following table (Sloane's A046145),which omits when does not exist.

2	1	38	3	94	5	158	3
3	2	41	6	97	5	162	5
4	3	43	3	98	3	163	2
5	2	46	5	101	2	166	5
6	5	47	5	103	5	167	5
7	3	49	3	106	3	169	2
9	2	50	3	107	2	173	2
10	3	53	2	109	6	178	3
11	2	54	5	113	3	179	2
13	2	58	3	118	11	181	2
14	3	59	2	121	2	191	19
17	3	61	2	122	7	193	5
18	5	62	3	125	2	194	5
19	2	67	2	127	3	197	2
22	7	71	7	131	2	199	3
23	5	73	5	134	7	202	3
25	2	74	5	137	3	206	5
26	7	79	3	139	2	211	2
27	2	81	2	142	7	214	5
29	2	82	7	146	5	218	11
31	3	83	2	149	2	223	3
34	3	86	3	151	6	226	3
37	2	89	3	157	5	227	2

Let be any Odd Prime , and let

(1)

Then

(2)

For numbers

with primitive roots, all

satisfying

are representable as

(3)

where

, 1, ...,

is known as the index, and

is an Integer. Kearnes showed that for anyPositive Integer

, there exist infinitely many Primes

such that

(4)

Call the least primitive root

. Burgess (1962) proved that

(5)

for

and

Positive constants and

sufficiently large.
References

Abramowitz, M. and Stegun, C. A. (Eds.). ``Primitive Roots.'' §24.3.4 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 827, 1972.

Burgess, D. A. ``On Character Sums and -Series.'' Proc. London Math. Soc. 12, 193-206, 1962.Guy, R. K. ``Primitive Roots.'' §F9 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 248-249, 1994.

Sloane, N. J. A. SequencesA046145 andA001918/M0242in ``An On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.

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