What is a primitive root of a number?
Every integer relatively prime to n is congruent to g mod n, where g and n are primitive roots. In other words, the integer g is a primitive root (mod n) if for every value a relatively prime to n, there is an integer z such that it is a primitive root.
How do you find the primitive root of a number?
In Euler's Totient Function, phi = n-1. We assume n is prime. 1. Then, use phi/prime-factors one at a time to calculate all powers. Calculate the n - 1 power for all i=2 to n-1 powers by modulating (i* powers).
What is the primitive root of 11?
6573, 592, 5103, 7112, 6, 7, 8
What are the primitive roots of 31?
primitive roots moduloexponent (OEIS: A002322)286292, 3, 8, 10, 11, 14, 15, 18, 19, 21, 26, 2728304313, 11, 12, 13, 17, 21, 22, 2430
How do you find the primitive root of a number?
An integer r modulo n has the same multiplicative order as Euler Totient Function /(n), and if n is a prime number, it has the same Euler Totient Function as n-1. In the case of Euler's totient function, phi = n-1, find all its prime factors, assuming n is a prime number.
What is primitive root give example?
A few examples. Among the order of one, the order of three and five is six, the order of nine and eleven is three, and the order of thirteen is two. Three and five are the primitive roots of 14 modulo three. [1, 2, 4, 7, 8, 11, 13, 14] are congruence classes; they all make up *(15) = 8 classes.
How do you find the primitive root of 13?
In addition to the primitive roots mod p, there are also (p*1). Taking the case p = 13 from the table, as an example. The reciprocal of (p*1) = (12) = (192) = (p*1/2)(1*1/3) = 4; and the reciprocal of (202) = (p*1). Based on [b1, b5, b7, b11] as a primitive root mod 13, the set of primitive roots is complete.
How do you find the primitive root of 11?
In modified 11 form, the primitive roots are 2, 6, 7, 8. In this case, we would simply compute the first *(11) = 10 powers of each unit modulo 11, and then check whether or not all these powers are present.
How do you find the primitive root of 29?
A primitive root is the power of 2n mod 29, such that gcd(n, 28) = 1. Therefore, the primitive roots are 2, 8, 3, 19, 18, 14, 27, 21, 26, 10, 11, 15 (1, 3, 5, 9, 11, 13, 15, 17, 19, 23, 25, 27), i.e., [2n : n = 1, 3, 5, 9, 11, 13, 15, 17, 19, 23, 25, 27].
What are the primitive roots of 12?
primitive roots moduloorder (OEIS: A000010)112, 6, 7, 810124132, 6, 7, 1112143, 56
How do you show that 2 is a primitive root of 11?
In the case of 5, 257 is a primitive root. Display that root modulo 11. The order of the modulus of 11 will divide (11), so (mod11) = 2. To measure this we check 22*4, mod11, and 25*10.
What is the primitive root of 13?
73, 592, 5103, 7112, 6, 7, 8132, 6, 7, 11
What are the primitive roots of 23?
*(23) = 22, so in order to determine if a is a primitive root, we will also need to check a2 * 1 (mod 23) and a11 * 1. 52 * 2 (mod 23) gives 5 as a primitive root.
What are the primitive roots of 17?
Based on problem 7, since (3)=16, the other primitive roots are power three odd. particular, one has 3, 33 = 10, 35 = 5, 37 = 11, 39 = 14, 311 = 7, 313 = 12, and 315 = 6 have all been modified to 7.