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In modular arithmetic, a branch of number theory, a number g is a primitive root modulo n if every number acoprime to n is congruent to a power of g modulo n. That is, g is a primitive root mod n if for every integer a coprime to n, there is an integer k such that g^k ≡ a (mod n). Such a value k is called the index or discrete logarithm of a to the base g modulo n. Note that g is a primitive root mod n if and only if g is a generator of the multiplicative group of integers modulo n.

Gauss defined primitive roots in Article 57 of the Disquisitiones Arithmeticae (1801), where he credited Euler with coining the term. In Article 56 he stated that Lambert and Euler knew of them, but he was the first to rigorously demonstrate that primitive roots exist for a primen. In fact, the Disquisitiones contains two proofs: the one in Article 54 is a nonconstructive existence proof, while the other in Article 55 is constructive.

3Examples
5Order of magnitude of primitive roots

Elementary example[edit]

The number 3 is a primitive root modulo 7^[1] because

\begin{array}{rcrcrcrcrcr} 3^{1} & = & 3 & = & 3^{0} \times 3 & \equiv & 1 \times 3 & = & 3 & \equiv & 3 (\mod 7) \\ 3^{2} & = & 9 & = & 3^{1} \times 3 & \equiv & 3 \times 3 & = & 9 & \equiv & 2 (\mod 7) \\ 3^{3} & = & 27 & = & 3^{2} \times 3 & \equiv & 2 \times 3 & = & 6 & \equiv & 6 (\mod 7) \\ 3^{4} & = & 81 & = & 3^{3} \times 3 & \equiv & 6 \times 3 & = & 18 & \equiv & 4 (\mod 7) \\ 3^{5} & = & 243 & = & 3^{4} \times 3 & \equiv & 4 \times 3 & = & 12 & \equiv & 5 (\mod 7) \\ 3^{6} & = & 729 & = & 3^{5} \times 3 & \equiv & 5 \times 3 & = & 15 & \equiv & 1 (\mod 7) \\ 3^{7} & = & 2187 & = & 3^{6} \times 3 & \equiv & 1 \times 3 & = & 3 & \equiv & 3 (\mod 7) \end{array} {displaystyle {begin{array}{rcrcrcrcrcr}3^{1}&=&3&=&3^{0}times 3&equiv &1times 3&=&3&equiv &3{pmod {7}}3^{2}&=&9&=&3^{1}times 3&equiv &3times 3&=&9&equiv &2{pmod {7}}3^{3}&=&27&=&3^{2}times 3&equiv &2times 3&=&6&equiv &6{pmod {7}}3^{4}&=&81&=&3^{3}times 3&equiv &6times 3&=&18&equiv &4{pmod {7}}3^{5}&=&243&=&3^{4}times 3&equiv &4times 3&=&12&equiv &5{pmod {7}}3^{6}&=&729&=&3^{5}times 3&equiv &5times 3&=&15&equiv &1{pmod {7}}3^{7}&=&2187&=&3^{6}times 3&equiv &1times 3&=&3&equiv &3{pmod {7}}end{array}}}

Here we see that the period of 3^k modulo 7 is 6. The remainders in the period, which are 3, 2, 6, 4, 5, 1, form a rearrangement of all nonzero remainders modulo 7, implying that 3 is indeed a primitive root modulo 7. This derives from the fact that a sequence (g^k modulo n) always repeats after some value of k, since modulo n produces a finite number of values. If g is a primitive root modulo n and n is prime, then the period of repetition is n-1. Curiously, permutations created in this way (and their circular shifts) have been shown to be Costas arrays.

Definition[edit]

If n is a positive integer, the integers between 0 and n − 1 that are coprime to n (or equivalently, the congruence classes coprime to n) form a group with multiplication modulon as the operation; it is denoted by Z_n^× and is called the group of units modulo n or the group of primitive classes modulo n. As explained in the article multiplicative group of integers modulo n, this group is cyclic if and only if n is equal to 2, 4, p^k, or 2p^k where p^k is a power of an odd prime number.^[2]^[3]^[4] A generator of this cyclic group is called a primitive root modulo n,^[5] or a primitive element of Z_n^×.

The order of (i.e., the number of elements in) Z_n^× is given by Euler's totient functionφ(n) (sequence A000010 in the OEIS). Euler's theorem says that a^φ(n) ≡ 1 (mod n) for every a coprime to n; the lowest power of a which is congruent to 1 modulo n is called the multiplicative order of a modulo n. In particular, for a to be a primitive root modulo n, φ(n) has to be the smallest power of a which is congruent to 1 modulo n.

Examples[edit]

For example, if n = 14 then the elements of Z_n^× are the congruence classes {1, 3, 5, 9, 11, 13}; there are φ(14) = 6 of them. Here is a table of their powers modulo 14:

The order of 1 is 1, the orders of 3 and 5 are 6, the orders of 9 and 11 are 3, and the order of 13 is 2. Thus, 3 and 5 are the primitive roots modulo 14.

For a second example let n = 15. The elements of Z₁₅^× are the congruence classes {1, 2, 4, 7, 8, 11, 13, 14}; there are φ(15) = 8 of them.

Since there is no number whose order is 8, there are no primitive roots modulo 15. Indeed, λ(15) = 4, where λ is the Carmichael function. (sequence A002322 in the OEIS)

Table of primitive roots[edit]

Numbers which have a primitive root are

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 17, 18, 19, 22, 23, 25, 26, 27, 29, 31, 34, 37, 38, 41, 43, 46, 47, 49, 50, 53, 54, 58, 59, 61, 62, 67, 71, 73, 74, 79, 81, 82, 83, 86, 89, 94, 97, 98, 101, 103, 106, 107, 109, 113, 118, 121, 122, 125, 127, 131, 134, 137, 139, 142, 146, 149, .. (sequence A033948 in the OEIS)

This is Gauss's table of the primitive roots from the Disquisitiones. Unlike most modern authors he did not always choose the smallest primitive root. Instead, he chose 10 if it is a primitive root; if it isn't, he chose whichever root gives 10 the smallest index, and, if there is more than one, chose the smallest of them. This is not only to make hand calculation easier, but is used in § VI where the periodic decimal expansions of rational numbers are investigated.

The rows of the table are labelled with the prime powers (excepting 2, 4, and 8) less than 100; the second column is a primitive root modulo that number. The columns are labelled with the primes less than 100. The entry in row p, column q is the index of q modulo p for the given root.

For example, in row 11, 2 is given as the primitive root, and in column 5 the entry is 4. This means that 2⁴ = 16 ≡ 5 (mod 11).

For the index of a composite number, add the indices of its prime factors.

For example, in row 11, the index of 6 is the sum of the indices for 2 and 3: 2^{1 + 8} = 512 ≡ 6 (mod 11). The index of 25 is twice the index 5: 2⁸ = 256 ≡ 25 (mod 11). (Of course, since 25 ≡ 3 (mod 11), the entry for 3 is 8).

The table is straight forward for the odd prime powers. But the powers of 2 (16, 32, and 64) do not have primitive roots; instead, the powers of 5 account for one-half of the odd numbers less than the power of 2, and their negatives modulo the power of 2 account for the other half. All powers of 5 are ≡ 5 or 1 (mod 8); the columns headed by numbers ≡ 3 or 7 (mod 8) contain the index of its negative. (Sequence A185189 and A185268 in OEIS)

For example, modulo 32 the index for 7 is 2, and 5² = 25 ≡ −7 (mod 32), but the entry for 17 is 4, and 5⁴ = 625 ≡ 17 (mod 32).

Primitive roots and indices
(other columns are the indices of integers under respective column headings)
n	root	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59	61	67	71	73	79	83	89	97
3	2	1
5	2	1	3
7	3	2	1	5
9	2	1	*	5	4
11	2	1	8	4	7
13	6	5	8	9	7	11
16	5	*	3	1	2	1	3
17	10	10	11	7	9	13	12
19	10	17	5	2	12	6	13	8
23	10	8	20	15	21	3	12	17	5
25	2	1	7	*	5	16	19	13	18	11
27	2	1	*	5	16	13	8	15	12	11
29	10	11	27	18	20	23	2	7	15	24
31	17	12	13	20	4	29	23	1	22	21	27
32	5	*	3	1	2	5	7	4	7	6	3	0
37	5	11	34	1	28	6	13	5	25	21	15	27
41	6	26	15	22	39	3	31	33	9	36	7	28	32
43	28	39	17	5	7	6	40	16	29	20	25	32	35	18
47	10	30	18	17	38	27	3	42	29	39	43	5	24	25	37
49	10	2	13	41	*	16	9	31	35	32	24	7	38	27	36	23
53	26	25	9	31	38	46	28	42	41	39	6	45	22	33	30	8
59	10	25	32	34	44	45	28	14	22	27	4	7	41	2	13	53	28
61	10	47	42	14	23	45	20	49	22	39	25	13	33	18	41	40	51	17
64	5	*	3	1	10	5	15	12	7	14	11	8	9	14	13	12	5	1	3
67	12	29	9	39	7	61	23	8	26	20	22	43	44	19	63	64	3	54	5
71	62	58	18	14	33	43	27	7	38	5	4	13	30	55	44	17	59	29	37	11
73	5	8	6	1	33	55	59	21	62	46	35	11	64	4	51	31	53	5	58	50	44
79	29	50	71	34	19	70	74	9	10	52	1	76	23	21	47	55	7	17	75	54	33	4
81	11	25	*	35	22	1	38	15	12	5	7	14	24	29	10	13	45	53	4	20	33	48	52
83	50	3	52	81	24	72	67	4	59	16	36	32	60	38	49	69	13	20	34	53	17	43	47
89	30	72	87	18	7	4	65	82	53	31	29	57	77	67	59	34	10	45	19	32	26	68	46	27
97	10	86	2	11	53	82	83	19	27	79	47	26	41	71	44	60	14	65	32	51	25	20	42	91	18
n	root	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59	61	67	71	73	79	83	89	97

The following table lists the primitive roots modulo n for n ≤ 72:

${displaystyle n}$	primitive roots modulo ${displaystyle n}$	order (OEIS: A000010)	${displaystyle n}$	primitive roots modulo ${displaystyle n}$	order (OEIS: A000010)
1	0	1	37	2, 5, 13, 15, 17, 18, 19, 20, 22, 24, 32, 35	36
2	1	1	38	3, 13, 15, 21, 29, 33	18
3	2	2	39	24
4	3	2	40	16
5	2, 3	4	41	6, 7, 11, 12, 13, 15, 17, 19, 22, 24, 26, 28, 29, 30, 34, 35	40
6	5	2	42	12
7	3, 5	6	43	3, 5, 12, 18, 19, 20, 26, 28, 29, 30, 33, 34	42
8	4	44	20
9	2, 5	6	45	24
10	3, 7	4	46	5, 7, 11, 15, 17, 19, 21, 33, 37, 43	22
11	2, 6, 7, 8	10	47	5, 10, 11, 13, 15, 19, 20, 22, 23, 26, 29, 30, 31, 33, 35, 38, 39, 40, 41, 43, 44, 45	46
12	4	48	16
13	2, 6, 7, 11	12	49	3, 5, 10, 12, 17, 24, 26, 33, 38, 40, 45, 47	42
14	3, 5	6	50	3, 13, 17, 23, 27, 33, 37, 47	20
15	8	51	32
16	8	52	24
17	3, 5, 6, 7, 10, 11, 12, 14	16	53	2, 3, 5, 8, 12, 14, 18, 19, 20, 21, 22, 26, 27, 31, 32, 33, 34, 35, 39, 41, 45, 48, 50, 51	52
18	5, 11	6	54	5, 11, 23, 29, 41, 47	18
19	2, 3, 10, 13, 14, 15	18	55	40
20	8	56	24
21	12	57	36
22	7, 13, 17, 19	10	58	3, 11, 15, 19, 21, 27, 31, 37, 39, 43, 47, 55	28
23	5, 7, 10, 11, 14, 15, 17, 19, 20, 21	22	59	2, 6, 8, 10, 11, 13, 14, 18, 23, 24, 30, 31, 32, 33, 34, 37, 38, 39, 40, 42, 43, 44, 47, 50, 52, 54, 55, 56	58
24	8	60	16
25	2, 3, 8, 12, 13, 17, 22, 23	20	61	2, 6, 7, 10, 17, 18, 26, 30, 31, 35, 43, 44, 51, 54, 55, 59	60
26	7, 11, 15, 19	12	62	3, 11, 13, 17, 21, 43, 53, 55	30
27	2, 5, 11, 14, 20, 23	18	63	36
28	12	64	32
29	2, 3, 8, 10, 11, 14, 15, 18, 19, 21, 26, 27	28	65	48
30	8	66	20
31	3, 11, 12, 13, 17, 21, 22, 24	30	67	2, 7, 11, 12, 13, 18, 20, 28, 31, 32, 34, 41, 44, 46, 48, 50, 51, 57, 61, 63	66
32	16	68	32
33	20	69	44
34	3, 5, 7, 11, 23, 27, 29, 31	16	70	24
35	24	71	7, 11, 13, 21, 22, 28, 31, 33, 35, 42, 44, 47, 52, 53, 55, 56, 59, 61, 62, 63, 65, 67, 68, 69	70
36	12	72	24

Artin's conjecture on primitive roots states that a given integer a which is neither a perfect square nor −1 is a primitive root modulo infinitely many primes.

The sequence of smallest primitive roots mod n (which is not the same as the sequence of primitive roots in Gauss's table) are

0, 1, 2, 3, 2, 5, 3, 0, 2, 3, 2, 0, 2, 3, 0, 0, 3, 5, 2, 0, 0, 7, 5, 0, 2, 7, 2, 0, 2, 0, 3, 0, 0, 3, 0, 0, 2, 3, 0, 0, 6, 0, 3, 0, 0, 5, 5, 0, 3, 3, 0, 0, 2, 5, 0, 0, 0, 3, 2, 0, 2, 3, 0, 0, 0, 0, 2, 0, 0, 0, 7, 0, 5, 5, 0, .. (sequence A046145 in the OEIS)

For prime n, they are

1, 2, 2, 3, 2, 2, 3, 2, 5, 2, 3, 2, 6, 3, 5, 2, 2, 2, 2, 7, 5, 3, 2, 3, 5, 2, 5, 2, 6, 3, 3, 2, 3, 2, 2, 6, 5, 2, 5, 2, 2, 2, 19, 5, 2, 3, 2, 3, 2, 6, 3, 7, 7, 6, 3, 5, 2, 6, 5, 3, 3, 2, 5, 17, 10, 2, 3, 10, 2, 2, 3, 7, 6, 2, 2, .. (sequence A001918 in the OEIS)

The largest primitive roots modulo n are

0, 1, 2, 3, 3, 5, 5, 0, 5, 7, 8, 0, 11, 5, 0, 0, 14, 11, 15, 0, 0, 19, 21, 0, 23, 19, 23, 0, 27, 0, 24, 0, 0, 31, 0, 0, 35, 33, 0, 0, 35, 0, 34, 0, 0, 43, 45, 0, 47, 47, 0, 0, 51, 47, 0, 0, 0, 55, 56, 0, 59, 55, 0, 0, 0, 0, 63, 0, 0, 0, 69, 0, 68, 69, 0, .. (sequence A046146 in the OEIS)

For prime n, they are

1, 2, 3, 5, 8, 11, 14, 15, 21, 27, 24, 35, 35, 34, 45, 51, 56, 59, 63, 69, 68, 77, 80, 86, 92, 99, 101, 104, 103, 110, 118, 128, 134, 135, 147, 146, 152, 159, 165, 171, 176, 179, 189, 188, 195, 197, 207, 214, 224, 223, .. (sequence A071894 in the OEIS)

Number of primitive roots mod n are

1, 1, 1, 1, 2, 1, 2, 0, 2, 2, 4, 0, 4, 2, 0, 0, 8, 2, 6, 0, 0, 4, 10, 0, 8, 4, 6, 0, 12, 0, 8, 0, 0, 8, 0, 0, 12, 6, 0, 0, 16, 0, 12, 0, 0, 10, 22, 0, 12, 8, 0, 0, 24, 6, 0, 0, 0, 12, 28, 0, 16, 8, 0, 0, 0, 0, 20, 0, 0, 0, 24, 0, 24, 12, 0, .. (sequence A046144 in the OEIS)

For prime n, they are

1, 1, 2, 2, 4, 4, 8, 6, 10, 12, 8, 12, 16, 12, 22, 24, 28, 16, 20, 24, 24, 24, 40, 40, 32, 40, 32, 52, 36, 48, 36, 48, 64, 44, 72, 40, 48, 54, 82, 84, 88, 48, 72, 64, 84, 60, 48, 72, 112, 72, 112, 96, 64, 100, 128, 130, 132, 72, 88, 96, .. (sequence A008330 in the OEIS)

Smallest prime > n with primitive root n are

2, 3, 5, 0, 7, 11, 11, 11, 0, 17, 13, 17, 19, 17, 19, 0, 23, 29, 23, 23, 23, 31, 47, 31, 0, 29, 29, 41, 41, 41, 47, 37, 43, 41, 37, 0, 59, 47, 47, 47, 47, 59, 47, 47, 47, 67, 59, 53, 0, 53, .. (sequence A023049 in the OEIS)

Smallest prime (not necessarily exceeding n) with primitive root n are

2, 3, 2, 0, 2, 11, 2, 3, 2, 7, 2, 5, 2, 3, 2, 0, 2, 5, 2, 3, 2, 5, 2, 7, 2, 3, 2, 5, 2, 11, 2, 3, 2, 19, 2, 0, 2, 3, 2, 7, 2, 5, 2, 3, 2, 11, 2, 5, 2, 3, 2, 5, 2, 7, 2, 3, 2, 5, 2, 19, 2, 3, 2, 0, 2, 7, 2, 3, 2, 19, 2, 5, 2, 3, 2, .. (sequence A056619 in the OEIS)

Arithmetic facts[edit]

Gauss proved^[6] that for any prime number p (with the sole exception of p = 3), the product of its primitive roots is congruent to 1 modulo p.

He also proved^[7] that for any prime number p, the sum of its primitive roots is congruent to μ(p − 1) modulo p, where μ is the Möbius function.

For example,

p = 3, μ(2) = −1. The primitive root is 2.

p = 5, μ(4) = 0. The primitive roots are 2 and 3.

p = 7, μ(6) = 1. The primitive roots are 3 and 5.

p = 31, μ(30) = −1. The primitive roots are 3, 11, 12, 13, 17 ≡ −14, 21 ≡ −10, 22 ≡ −9, and 24 ≡ −7.

Their product 970377408 ≡ 1 (mod 31) and their sum 123 ≡ −1 (mod 31).

3 × 11 = 33 ≡ 2

12 × 13 = 156 ≡ 1

(−14) × (−10) = 140 ≡ 16

(−9) × (−7) = 63 ≡ 1, and 2 × 1 × 16 × 1 = 32 ≡ 1 (mod 31).

What about adding up elements of this multiplicative group? As it happens, sums (or differences) of two primitive roots add up to all elements of the index 2 subgroup of Z/nZ for even n, and to the whole group Z/nZ when n is odd:

Finding primitive roots[edit]

No simple general formula to compute primitive roots modulo n is known.^[9]^[10] There are however methods to locate a primitive root that are faster than simply trying out all candidates. If the multiplicative order of a number m modulo n is equal to ${displaystyle varphi left(nright)}$ (the order of Z_n^×), then it is a primitive root. In fact the converse is true: If m is a primitive root modulo n, then the multiplicative order of m is ${displaystyle varphi left(nright)}$ . We can use this to test for primitive roots.

First, compute ${displaystyle varphi left(nright)}$ . Then determine the different prime factors of ${displaystyle varphi left(nright)}$ , say p₁, .., p_k. Now, for every element m of Z_n^×, compute

{displaystyle m^{varphi (n)/p_{i}}mod nqquad {mbox{ for }}i=1,ldots ,k}

using a fast algorithm for modular exponentiation such as exponentiation by squaring. A number m for which these k results are all different from 1 is a primitive root.

The number of primitive roots modulo n, if there are any, is equal to^[11]

{displaystyle varphi left(varphi left(nright)right)}

since, in general, a cyclic group with r elements has ${displaystyle varphi left(rright)}$ generators. For prime n, this equals ${displaystyle varphi (n-1)}$ , and since ${displaystyle n/varphi (n-1)=O(log log n)}$ the generators are very common among {2,…,n-1} and thus it is relatively easy to find one.^[12]

If g is a primitive root modulo p, then g is a primitive root modulo all powers p^k unless g^{p – 1} ≡ 1 (mod p²); in that case, g + p is.^[13]

If g is a primitive root modulo p^k, then g or g + p^k (whichever one is odd) is a primitive root modulo 2p^k.^[13]

Finding primitive roots modulo p is also equivalent to finding the roots of the (p-1)^thcyclotomic polynomial modulo p.

Order of magnitude of primitive roots[edit]

The least primitive root g_p modulo p (in the range 1, 2, .., p − 1) is generally small.

Upper bounds[edit]

Burgess (1962) proved^[14] that for every ε > 0 there is a C such that ${displaystyle g_{p}leq Cp^{{frac {1}{4}}+epsilon }.}$

Grosswald (1981) proved^[14] that if ${displaystyle p>e^{e^{24}}}$ , then ${displaystyle g_{p}<semantics><mrow><mstyle displaystyle=$ .

Shoup (1990, 1992) proved,^[15] assuming the generalized Riemann hypothesis, that g_p = O(log⁶p).

Lower bounds[edit]

Fridlander (1949) and Salié (1950) proved^[14] that there is a positive constant C such that for infinitely many primes g_p > C log p.

It can be proved^[14] in an elementary manner that for any positive integer M there are infinitely many primes such that M < g_p < p − M.

Applications[edit]

A primitive root modulo n is often used in cryptography, including the Diffie–Hellman key exchange scheme. Sound diffusers have been based on number-theoretic concepts such as primitive roots and quadratic residues.^[16]^[17]

Notes[edit]

^https://web.archive.org/web/20170703173446/http://www.brynmawr.edu:80/math/people/stromquist/numbers/primitive.html
^Weisstein, Eric W.'Modulo Multiplication Group'. MathWorld.
^Primitive root, Encyclopedia of Mathematics.
^Vinogradov 2003, pp. 105–121, § VI PRIMITIVE ROOTS AND INDICES.
^Vinogradov 2003, p. 106.
^Gauss & Clarke 1986, arts. 80.
^Gauss & Clarke 1986, arts. 81.
^Emmanuel Amiot, Music Through Fourier Space, p. 38 (Springer, CMS Series, 2016).
^von zur Gathen & Shparlinski 1998, pp. 15–24: 'One of the most important unsolved problems in the theory of finite fields is designing a fast algorithm to construct primitive roots.'
^Robbins 2006, p. 159: 'There is no convenient formula for computing [the least primitive root].'
^(sequence A010554 in the OEIS)
^Donald E. Knuth, The Art of Computer Programming, vol. 2: Seminumerical Algorithms, 3rd edition, section 4.5.4, p. 391 (Addison–Wesley, 1998).
^ ^a^bCohen 1993, p. 26.
^ ^a^b^c^dRibenboim 1996, p. 24.
^Bach & Shallit 1996, p. 254.
^Walker, R. 'The design and application of modular acoustic diffusing elements'(PDF). BBC Research Department. Retrieved 25 March 2019.
^Eliot Feldman (July 1995). 'A reflection grating that nullifies the specular reflection: A cone of silence'. J. Acoust. Soc. Am. 98 (1): 623–634. doi:10.1121/1.413656.

References[edit]

The Disquisitiones Arithmeticae has been translated from Gauss's Ciceronian Latin into English and German. The German edition includes all of his papers on number theory: all the proofs of quadratic reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished notes.

Amiot, Emmanuel (2016), Music Through Fourier Space, Zürich: Springer, ISBN978-3-319-45581-5.
Bach, Eric; Shallit, Jeffrey (1996), Algorithmic Number Theory (Vol I: Efficient Algorithms), Cambridge: The MIT Press, ISBN978-0-262-02405-1.
Cohen, Henri (1993), A Course in Computational Algebraic Number Theory, Berlin: Springer, ISBN978-3-540-55640-4.
Gauss, Carl Friedrich; Clarke, Arthur A. (translator) (1986), Disquisitiones Arithmeticae (2nd, corrected ed.), New York: Springer, ISBN978-0-387-96254-2 [in English].
Gauss, Carl Friedrich; Maser, H. (translator) (1965), Untersuchungen über höhere Arithmetik [Studies on higher arithmetic] (2nd ed.), New York: Chelsea, ISBN978-0-8284-0191-3 [in German].
Ribenboim, Paulo (1996), The New Book of Prime Number Records, New York: Springer, ISBN978-0-387-94457-9.
Robbins, Neville (2006), Beginning Number Theory, Jones & Bartlett Learning, ISBN978-0-7637-3768-9.
Vinogradov, I. M. (2003), '§ VI PRIMITIVE ROOTS AND INDICES', Elements of Number Theory, Mineola, NY: Dover Publications, pp. 105–121, ISBN978-0-486-49530-9.
von zur Gathen, Joachim; Shparlinski, Igor (1998), 'Orders of Gauss periods in finite fields', Applicable Algebra in Engineering, Communication and Computing, 9 (1): 15–24, CiteSeerX10.1.1.46.5504, doi:10.1007/s002000050093, MR1624824.