Primitive roots in number theory 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 integer and p p p the odd prime number. 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 primitive root modulo n, 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 group 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, integers 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), order (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.