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Primitive roots in pdf

Continue When using a primitive root g gg mod n n, elements n∗ mathbb (_n) bn∗ can be written as 1.g,g2,...,gφ (n) 1, g, g'2, ldots, g'phi (n)-1 1,g2,...,gφ(n) Multiplying the two elements corresponds to the addition of their fashion exhibitors φ (n). Fi (n). φ (n). That is, the multiplier arithmetic in N∗ mathbb _n n∗ comes down to additive arithmetic in Zφ (n) mathbb phi (n) Zφ (n) . Let k k k be an and p p p the odd . How many non-zero kthk'text'th'kth powers have mod p? p?p? The question certainly depends on the relationship between k kk and p pp. When kz2, k2, k'2, the answer is p-12 frak-r-12 2p-1 (see square residues), but when kp-1, k q p-1, k'p-1, answer 1 1 1 (small theorem). As an illustration, take k'9,p-13 k 9, p 13 to 9, page 13. Then it's easy to check that g'2 g'2 g'2 is a primitive root mod p pp. Ninth powers mod p p are 20,29,218,227,... 2'0, 2'9, 2'{18}, 2'{27}, 20,29,218,227,..., but we can consider mod exhibitors 12 12 12, because 212≡1 2'{12} Equiv 1 212≡1. So we get 20,29,26,23,20,...,2'0, 2'9, 2'6, 2'3, 2'0, 'ldots,20,29,26,23,20,..., so there are four 9th9 'text'th'9th mod powers 13 13 13 13. Exhibitors are multiples of 3 33, which is gcd (9.13-1) (9,13-1) (9,13-1). There are 13-134frac 13-1'3 and 4313'1 No 4 of them. In general, since each non-grain element pmathbb _p p can be written as ga g'a ga mod p p p for some integer x x x, equation xk≡y x≡y x'k q≡y mod p p p can be rewritten (ga≡k≡gb, (g'a) q q equiv g'b, (ga)k≡gb, or gak'b≡1 g'ak-b'equiv 1 gak≡1 mod p. Since g g g g g is a primitive root This happens if and only if (p'1) ∣ (ak'b) (p-1) (ak-b) (p.1)∣ (akb), so that ak≡b a ≡c equv b a≡b mod p-1 p-1 p. This raises the question: How does the range over p1 mathbb p-1 (p-1), how many values can ak ak mod p'1 p'1 p'1 take over? that ak (p:1)c:k∈ c∈ big'ak (p-1)c: k, c (in mathbb ∈ is a set of multiples of gcd (k,p-1) (k,p-1) (k,p-1) (k,p-1). so that's the answer. □□ Here's another problem where she can help write elements of n∗ matebb _p n∗ as the forces of primitive root. , branches of number theory, number g is a primitive root module n, if each number of coprime to n coincides with the force of g modulo n. That is, g is a , if for each integer coprime to n, there is an integer k such that gk ≡ a (mod n). This value k is called an index or discrete logarith of the base g modulo n. Note that g is a primitive root module n if and only if Mr. the multiplied of integrators modulo n. Gauss identified primitive roots in article 57 of the Arithmeticae Discivism (1801), where he credited Euler with coining the term. In article 56, he stated that Lambert and Euler knew about them, but he was the first to strictly demonstrate that primitive roots exist for prime-n. In fact, disquisitions contain two evidence: one in article 54 is unconstructive proof of existence and the other in article 55 is constructive. Elementary example number 3 is a primitive root module 7.1, because 3 1 and 3 0 × 3 ≡ 1 × 3 x 3 ≡ 3 (Mod 7) 3 2 x 9 - 3 1 × 3 ≡ 3 × 3 and 9 ≡ 2 (Mod 7) 3 x 27th 3 3 3 2 × 3 ≡ 2 × 3 and 6 ≡ 6 (Mod 7) 3 4 81 3 3 × 3 ≡ 3 × 3 - 18 ≡ 4 (Mod 7) 3 5 x 243 - 3 3 4 × 3 ≡ 4 × 3 and 12 ≡ 5 (Mod 7) 3 6 x 729 - 3 5 × 3 ≡ 5 × 5 3 and 15 ≡ 1 (Mod 7) 3 7 and 2187 3 6 × 3 ≡ 1 × 3 x 3 ≡ 3 (Mod 7) Display Style Startrcrcrcrcr 3{1} 33'{0}'times 3 Equiv 3'3'equiv (3'pmod {7})3'3'{2} 9'3'3'{1} times 3'equiv (3'time 3) 9equiv 2pmod {7} 3{3} 273'{2} times 3'equiv 2time 3 6'equiv 6'pmod {7}'3'{4}'81'3'{3} times 3'equiv 6 'times 3'18'equi {7}v 3'{5} 243'3'{4} times 3'equiv 4'times 3'12'equi {6} {7} 729-3'{5} times 3'equiv 5 'times 3'15'equiv (1'pmod {7}'3'{7}'21872187s 3{6} times 3'equiv 1 times 3'3'equiv (3'pmod {7}'end) Here we see that the period 3k modulo 7 is 6. Remains in the period that is 3, 2, 6, 4, 5, 1, form a permutation of all the non-grain remnants of modulo 7, implying that 3 is a truly primitive root modulo 7. This stems from the fact that the sequence (gk modulo n) is always repeated after some k, as modulo n produces the final number of values. If g is a primitive root modlo n and n is prime, then the repetition period is n-1. Curiously, the permutations created in this way (and their circular shifts) were shown as Costas arrays. Definition If n is a positive integer, between 0 and n No. 1, who coprime to n (or equivalent, coprime class to n) form a group, multiplying modulo n as surgery; it is denoted by the ×n unit group, or group of primitive modulo n. As explained in the article of the multiplier group integers modulo n, this multiplier group (No×n) is cyclical, if and only if n is 2, 4, pk or 2pk, where PK is the force of a strange number. When (and only when) this group× is cyclical, the generator of this cyclical group is called primitive root modlo n'5 (or in a more complete language by the primitive root of modulo n unity, emphasizing its role as a fundamental solution to the roots of the unity of the polynomial equations Xm No. 1 in the n ring n), or simply primitive. When× × is non-cyclical, there are no such primitive elements of mod n. For any n (regardless of whether× is a z×n cyclical), (i.e. number of elements in) z×n is given to totient euler function φ (n) (A000010 sequence in OEIS). And then, the Euler theorem says that aφ (n) ≡ 1 (mod n) for each coprime to n; the lowest power, which coincides with 1 modulo n, In particular, in order to be a primitive root module n, φ (n) must be the smallest force that coincides with 1 modulo n. Examples For example, if n No. 14, the elements of the z×n are classes of congruence No.1, 3, 5, 9, 11, 13; there are φ (14) and 6 of them. Here is a table of their powers modulo 14: x x, x2, x3, ... (mod 14) 1 : 1 3 : 3, 9, 13, 11, 5, 1 5 : 5, 11, 13, 9, 3, 1 9 : 9, 11, 11: 11, 9, 1 13 : 13, 1 Order 1 is 1, orders 3 and 5 6, orders 9 and 11 3, and order 13 2. Thus, 3 and 5 are the primitive roots of modulo 14. For the second example, let no 15. No×15 are congruence classes No.1, 2, 4, 7, 8, 11, 13, 14; there are φ (15) and 8 of them. x x, x2, x3, ... (mod 15) 1 : 1 2 : 2, 4, 8, 1 4 : 4, 1 7 : 7, 4, 13, 1 8 : 8, 4, 2, 11 : 11, 13 : 13: 13, 4, 7, 14, 1 So there is no room, order of which 8, there are no primitive roots modulo 15. In fact, No (15) No 4, where - carmichael's function. (A002322 in OEIS) Table of primitive roots Numbers that have primitive root 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, 81, 82, 83, 86, 89, 94, 74, 79, 97, 7, 99 , 98 , 101, 103, 106, 107, 109, 113, 118, 121, 122, 125, 127, 131, 134, 137, 139, 142, 146, 149, ... This is the Gauss table of primitive roots from Disquisitiones. Unlike most contemporary authors, it was not always chosen from the smallest primitive roots. Instead, he chose 10 if it was a primitive root; if it is not, he chose the one that the root gives the 10 smallest index, and, if there is more than one, chose the smallest of them. This not only simplifies the calculation of the hand, but is also used in the VI place, where periodic decimal expansion of rational numbers is investigated. Table rows are marked with major credentials (except 2, 4 and 8) less than 100; The second column is a primitive root module of this number. Columns are marked with premieres of less than 100. Write in line p, q column is q modulo p index for a given root. For example, in line 11, 2 is given as a primitive root, and in column 5, the record is 4. This means that 24 and 16 ≡ 5 (mod 11). For the composite index, add indices of its main factors. For example, in line 11, index 6 is the sum of indices for 2 and 3: 21 and 8 and 512 6 (mod 11). The index of 25 doubles the index 5: 28 and 256 ≡ 25 (mod 11). (Of course, with 25 ≡ 3 (mod 11), the record for 3 is 8). The table is simple for the odd prime credentials. But forces 2 (16, 32 and 64) have no primitive roots; instead, the powers of 5 make up half of the odd numbers less than Power 2, and their negatives modulo power 2 falls on the other half. All powers 5 are ≡ 5 or 1 (mod 8); columns led by ≡ 3 or 7 (mod 8) contain an index of its negative. For example, the modulo 32 index for 7 is 2, and the 52th 25 ≡ No. 7 (Mod 32), but the record for 17 is 4, and 54 and 625 ≡ 17 (mod 32). Primitive Roots and Indices (other columns are centner indices under appropriate column titles) n Root 2 3 5 7 11 13 17 19 23 29 31 31 41 43 47 53 59 61 67 71 73 79 83 83 89 97 3 2 1 5 2 2 3 3 3 3 2 1 9 2 2 1st 5 4 11 2 8 8 7 13 6 8 9 7 11 16 5 5 3 1 2 1 3 3 3 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 of modulo n for n ≤ 72: n displaystyle n primitive roots modulo n displaystyle n order (OEIS : A000010) n displaystyle n primitive roots modulo n 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, 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 hypothesis of primitive roots states that this integer, which is neither a perfect square nor a No 1 is a primitive root modelo of infinite prime. The sequence of the smallest primitive roots modulo n (which is not the same as the sequence of primitive roots in the Gauss table) 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, 0, 0, 0, 3, 0, 0, 3, 0, 6, 0, 0, 0. , 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, ... For prime, they are 1, 2, 2, 2, 2, 2, 2, 2, 2, 5, 2, 3, 2, 2, 2, 2, 3, 5, 6, 3, 5, 2, 2, 2, 7, 5, 3, 2, 3, 5, 2, 5, 2, 6, 3, 3, 2, 2, 2, 2, 6, 5, 2, 2, 19, 5, 2, 3, 2, 3, 2, 3, 3, 7, 7, 6, 3, 3, 3, 3, 3, 3 , 5 , 2, 6, 5, 3, 3, 2, 5, 17, 10, 2, 3, 10, 2, 2, 3, 7, 6, 2, 2, ... The largest primitive roots of modulo n are 0, 1, 2, 3, 3, 5, 0, 5, 7, 8, 0, 11, 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, 35, 0, 34, 0, 43, 45, 0, 47, 47, 0, 0, 0, 0, 0, 45, 0, 47, 47, 0, 0, 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, ... (A046146 sequence in OEIS) For prime, they are 1, 2, 3, 5, 8, 11, 14, 15, 21, 27, 24, 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, 159165, 171, 176, 179, 189, 188, 195, 197, 197 207, 214, 224, 223, ... Number of primitive modulo n 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 0, 2, 2, 4, 0, 4, 2, 0, 0, 0, 0, 8, 2, 6, 0, 0, 4, 10, 0, 8, 4, 6, 0, 12, 0, 8, 0, 0, 8, 12, 6, 0, 0, 16, 0, 12, 0, 10, 22, 0, 12, 0, 12, 0, 0, 0, 0, 0, 2 , 6 , 0, 0, 12, 28, 0, 16, 8, 0, 0, 0, 0, 20, 0, 0, 0, 24, 0, 24, 12, 0, ... (A046144 sequence in OEIS) For prime, they're 1, 1, 2, 2, 4, 4, 8, 6, 10, 12, 8, 12, 16, 12, 22, 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, ... (A008330 sequence in OEIS) The smallest prime is qgt; n with primitive root n 2, 3, 5, 0, 7, 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, 49, 47, 47, 67, 59, 53, 0, 53, ... (A023049 sequence in OEIS) The smallest prime (not necessarily exceeding n) with a primitive root n 2, 3, 2, 0, 2, 11, 2, 3, 2, 7, 2, 5, 2, 3, 2, 0, 2, 5, 3, 2, 2, 2, 2, 5, 2, 2, 2, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 19, 2, 0, 2, 3, 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, ... (A056619 sequence in OEIS) Gauss arithmetic facts have proved that for any simple number p (with the sole exception of p No. 3), the product of its primitive roots coincides with 1 modulo p. It also proved that for any simple number p, the sum of its primitive roots coincides with the μ (p 1) modulo p, where μ function of The Meobius. For example, p No. 3, μ (2) No.1. Primitive Root 2. p No 5, μ (4) 0. Primitive Roots 2 and 3. p No 7, μ (6) 1. Primitive roots 3 and 5. p 31, μ (30) No. Primitive roots 3, 11, 12, 13, 17, 21, 22 and 24. Their product 970377408 ≡ 1 (mod 31) and their sum of 123 ≡ No. 1 (mod 31). 3 × 11 and 33 ≡ 2 12 × 13 and 156 ≡ 1 (No14) × (No10) 140 ≡ × 16 (No9 ≡) × (No.7) and 2 × 1 × 16 × 1 and 32 ≡ 1 (Mod 31). What about the amount of elements of this multiplier group? As it happens, the amounts (or differences) of the two primitive roots add up to all elements of the index 2 subgroup for the even n, and for the whole group when n is odd: q/n×× Search for primitive roots No simple common formula for calculating the primitive roots of modulo n is not known. However, there are methods to find a primitive root that are faster than just trying out all the candidates. If the order of the multiplier number m modulo n is equal to φ (n) (display (varphi (n) (order of z×n), then this is a primitive root. In fact the opposite is true: if m is a primitive root modlo n φ φ, then the multiplier order m is φ (n) . We can use this to test the candidate m to see if it is primitive. p1, ... pk. Finally, calculate m φ (n) / p i mod n for i q 1, ... , k 'displaystyle m'varphi (n)/p_'i'bmod n'qquad (mbox) for z'1, zldota, k, using a quick algorithm for экспонентации, такие как экспонентация путем квадратирования. Число м, для которого эти результаты k все отличаются от 1 является примитивным корнем. Количество примитивных корней modulo n, если таковые имеются, равно φ ( φ ( n ) , «дисплей стиль »varphi »слева (Варфи (n) »справа)», так как, в целом, циклическая группа с элементами r имеет φ ( r ) » »дисплей »style varphi (r)» генераторы. Для прайм-н, это равно φ (n q 1 ) »displaystyle »varphi (n-1)», и так как n / φ ( n q 1 ) ∈ O ( журнал ⁡ журнал ⁡ n ) » »displaystyle n/varphi (n-1) ..., n'1 , и, таким образом, это относительно легко найти один. Если g является примитивным корневой модуло p, то g также является примитивным корневим модуло, все силы pk, если gp'1 ≡ 1 (mod p2); в этом случае, г й р есть. Если г является примитивным корневым модуло пк, то либо g, либо g q pk (какой бы ни был странный) является примитивным корневым модуло 2pk. Порядок величины примитивных корней Наименее примитивный корень gp modulo p (в диапазоне 1 , 2, ..., р No 1) обычно мал. Верхние границы Burgess (1962) доказали, что на каждые ε > 0 есть C такой, что g p ≤ C p 1 4 и ε . «Дисплей g_»p'p'leq Cp'frac {1}{4} эпсилон. Гроссвальд (1981) доказал, что если p > e e 24 (дисплей p>e'e{24}), то g p < p= 0.499= {\displaystyle=> <p^{0.499}} .= carella= (2015)= proved[15]= that= there= is= a= c=>0 «displaystyle C>0» такой, что g p ≤ C р 5 / журнал ⁡ журнал ⁡ р «дисплей стиль g_»p'leq Cp'5/«log p» для всех достаточно больших праймов p > 2 «displaystyle p>2». Шуп (1990, 1992) доказал, что, предполагая обобщенную гипотезу Римана, что gp s O(log6 p). Нижние границы Fridlander (1949) и Sali ' (1950) доказали , что есть положительная константа C такова, что для бесконечно многих премьер gp > C журнал р. Это может быть доказано элементарным образом, что для любого положительного integer M < gp=> < p − M. Applications A primitive root modulo n is often used in , including the Diffie–Hellman key exchange scheme. Sound diffusers have been based on number-theoretic concepts such as primitive roots and quadratic residues. [17] [18] See also Artin's conjecture on primitive roots Full reptend prime Gauss's generalization of Wilson's theorem Root of unity modulo n Notes ^ Archived copy. Archived from the original on 2017-07-03. Retrieved 2017-07-03.CS1 maint: archived copy as title (link) ^ 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 p − m.= applications= a= primitive= root= modulo= n= is= often= used= in= cryptography,= including= the= diffie–hellman= key= exchange= scheme.= sound= diffusers= have= been= based= on= concepts= such= as= primitive= roots= and= quadratic= residues. [17] [18]= see= also= artin's= conjecture= on= primitive= roots= dirichlet= character= full= reptend= prime= gauss's= generalization= of= wilson's= theorem= multiplicative= order= quadratic= residue= root= of= unity= modulo= n= notes= ^= archived= copy.= archived= from= the= original= on= 2017-07-03.= retrieved= 2017-07-03.cs1= maint:= archived= copy= as= title= (link)= ^= 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=></ p − M. Applications 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. [17] [18] See also Artin's conjecture on primitive roots Dirichlet character Full reptend prime Gauss's generalization of Wilson's theorem Multiplicative order Quadratic residue Root of unity modulo n Notes ^ Archived copy. Archived from the original on 2017-07-03. Retrieved 2017-07-03.CS1 maint: archived copy as title (link) ^ 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 > Есть бесконечно много премьер, таких как M</p^{0.499}}> M</p^{0.499}}> Art. 81. Emmanuel Amiot, Music through The Fourier Space, page 38 (Springer, CMS Series, 2016). von zur Gaten and Sparlinsky 1998, page 15-24: One of the most important unresolved problems in the theory of the end fields is the development of a fast algorithm for building primitive roots. p. 159: There is no convenient formula for calculating (the least primitive root). (A010554 sequence in OEIS) - Donald E. Knut, The Art of Computer Programming, vol. 2: Seminumer Algorithms, 3rd edition, section 4.5.4, page 391 (Addison-Wesley, 1998). a b Cohen 1993, page 26. a b c d Ribenboim 1996, page 24. Carella, N.A. (2015). Least prime primitive roots. International Journal of Mathematics and Informatics. 10 (2): 185–194. arXiv:1709.01172. Bach and Shalit 1996, page 254. Walker, R. Design and application of modular acoustic diffusive elements (PDF). The BBC's research department. Received on March 25, 2019. Elliot Feldman (July 1995). Reflection of the lattice, which negates the mirror reflections: a cone of silence. J. Akust. Soc. Am. 98 (1): 623-634. Bibkod:1995ASAJ... 98..623F. doi:10.1121/1.413656. Disquisitiones Arithmeticaae references were translated from ciceronic Latin gauss into English and German. The German edition includes all his work on the theory of numbers: all the evidence of square reciprocity, the definition of the Gauss amount mark, the investigation of two-square-metre reciprocity and unpublished notes. Amiot, Emmanuel (2016), Music via The Fourier Space, zurich: Springer, ISBN 978-3-319-45581-5. Bach, Eric; Shalit, Jeffrey (1996), Algorithmic Number Theory (Vol I: Effective Algorithms), Cambridge: MIT Press, ISBN 978-0-262-02405-1. Carella, N.A. (2015), Least Prime, International Journal of Mathematics and Informatics, 10 (2): 185-194, arXiv:1709.01172. Cohen, Henri (1993), Computational Algebraic Room Theory Course, Berlin: Springer, ISBN 978-3-540-55640-4. ; C Clarke, Arthur A. (translator) (1986), Disquisitiones Arithmeticae (2nd place, corrected Ed.), New York: Springer, ISBN 978-0-387-96254-2 (English). Carl Friedrich Gauss; Mather, H. (translator) (1965), Unteuhungen Sber Hehere Arithmetic (Research on Higher Arithmetic) (2nd Ed.), New York: Chelsea, ISBN 978-0-8284-0191-3 (in German). Ribenboim, Paulo (1996), New Prime Room Records Book, New York: Springer, ISBN 978-0-387-94457-9. Robbins, Neville (2006), Origin Number Theory, Jones and Bartlett Learning, ISBN 978-0-7637-3768-9. Vinogradov, I.M. (2003), VI PRIMITIVE ROOTS AND INDICES, Elements of Number Theory, Mineola, NY: Dover Publications, page 105-121, ISBN 978-0-486-49530-9. von zur Gaten, Joachim; Sparlinsky, Igor (1998), Order of Gauss in the final areas, Applied algebra in engineering, communications and computing, 9 (1): CiteSeerX 10.1.1.46.5504, doi:10.1007/s002000050093, MR 1624824, S2CID 19232025. Further ore reading, Oystein (1988), Room Theory and Its History, Dover, 284-302, ISBN 978-0-486-65620-5. External references to Weisstein, Eric V. Primitive Root. Matmir. Square remnants and primitive roots. Michigan Technology. Primitive Roots Calculator. Blutlip. Extracted from the primitive roots in number theory pdf. primitive roots and indices in number theory

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