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Mar 8, 2023 · The primitive root of a prime number n is an integer r between [1, n-1] such that the values of r^x (mod n) where x is in the range [0, n-2] are different. Return -1 if n is a non-prime number. Examples: Input : 7. Output : Smallest primitive root = 3. Explanation: n = 7. 3^0(mod 7) = 1. 3^1(mod 7) = 3. 3^2(mod 7) = 2.
When primitive roots exist, it is often very convenient to use them in proofs and explicit constructions; for instance, if \( p \) is an odd prime and \( g \) is a primitive root mod \( p \), the quadratic residues mod \( p \) are precisely the even powers of the primitive root.
5 days ago · A primitive root of a prime is an integer such that (mod ) has multiplicative order (Ribenboim 1996, p. 22). More generally, if ( and are relatively prime ) and is of multiplicative order modulo where is the totient function , then is a primitive root of (Burton 1989, p. 187).
In modular arithmetic, a number g is a primitive root modulo n if every number a coprime to n is congruent to a power of g modulo n. That is, g is a primitive root modulo n if for every integer a coprime to n, there is some integer k for which g k ≡ a (mod n). Such a value k is called the index or discrete logarithm of a to the base g modulo n.
Moreover, if \(r\) is a primitive root modulo \(p^2\), then \(r\) is a primitive root modulo \(p^m\) for all positive integers \(m\). By Theorem 62, we know that any prime \(p\) has a primitive root \(r\) which is also a primitive root modulo \(p^2\), thus \[\label{1} p^2\nmid (r^{p-1}-1).\]
We say that an integer a a is a root of f(x) f ( x) modulo m m if f(a) ≡ 0(mod m) f ( a) ≡ 0 ( m o d m). Notice that x ≡ 3(mod 11) x ≡ 3 ( m o d 11) is a root for f(x) = 2x2 + x + 1 f ( x) = 2 x 2 + x + 1 since f(3) = 22 ≡ 0(mod 11) f ( 3) = 22 ≡ 0 ( m o d 11). We now introduce Lagrange’s theorem for primes.
If the positive integer \(m\) has a primitive root, then it has a total of \(\phi(\phi(m))\) incongruent primitive roots. Let \(r\) be a primitive root modulo \(m\). By Theorem 56, we see that \(\{r^1,r^2,...,r^{\phi(m)}\}\) form a reduced residue system modulo \(n\).
De nition 9.1. A generator of (Z=p) is called a primitive root mod p. Example: Take p= 7. Then 23 1 mod 7; so 2 has order 3 mod 7, and is not a primitive root. However, 32 2 mod 7;33 6 1 mod 7: Since the order of an element divides the order of the group, which is 6 in this case, it follows that 3 has order 6 mod 7, and so is a primitive root.
Primitive Roots. Theorem 1: If g is a primitive root of m, then the least residues of g, g2, …, gΦ (m) (mod m) are a permutation of the Φ (m) positive integers relatively prime to m. Example 2. Example 3. Lemma 1: If t is the order of a (mod m), then t is the order of ak (mod m) if and only if (k, t) = 1.
When there is such a nice residue as $2$ is here, it’s called a primitive root, and it’s a serious Theorem that when $n$ is a prime, there always is a primitive root. For instance, $n=5$ has $2$ for a p.r., $n=7$ can’t use $2$, but $3$ is a good p.r.
