Consider a prime \(p\neq 2\) and let \(s\) is a positive integer, then \(2p^s\) has a primitive root. 2. Information about your device and internet connection, including your IP address, Browsing and search activity while using Verizon Media websites and apps. We now list the set of integers that do not have primitive roots. Example 1. By Theorem 54, we know that \(n\mid \phi(p^m)\). Given a prime .The task is to count all the primitive roots of .. A primitive root is an integer x (1 <= x < p) such that none of the integers x – 1, x 2 – 1, …., x p – 2 – 1 are divisible by but x p – 1 – 1 is divisible by .. We prove the result by induction. Have questions or comments? Hence, \(ord_{p^2}s\) equals either \(p-1\) or \(p(p-1)\). In the following theorem, we prove that no power of 2, other than 2 or 4, has a primitive root and that is because when \(m\) is an odd integer, \(ord_2^km\neq \phi(2^k)\) and this is because \(2^k\mid (a^{\phi(2^k)/2}-1)\). 3. So it has order if . Given a prime .The task is to count all the primitive roots of .. A primitive root is an integer x (1 <= x < p) such that none of the integers x – 1, x 2 – 1, …., x p – 2 – 1 are divisible by but x p – 1 – 1 is divisible by .. Moreover, if \(r\) is a primitive root modulo \(p^2\), then \(r\) is a primitive root modulo \(p^m\) for all positive integers \(m\). Let \(n=ord_{p^m}r\). Missed the LibreFest? This means that, if α is such a root, then α p m −1 = 1 and α i ≠ 1 for 0 < i < p m − 1. As a result, by Theorem 63, Theorem 65 and Theorem 66, we see that, The positive integer \(m\) has a primitive root if and only if \(n=2,4, p^s\) or \(2p^s\), for prime \(p\neq 2\) and \(s\) is a positive integer. Dr. Wissam Raji, Ph.D., of the American University in Beirut. The proof of that tells you how to find all the others, given one. Assume now that \[2^k\mid (m^{2^{k-2}}-1).\] Then there is an integer \(q\) such that \[m^{2^{k-2}}=1+q.2^{k}.\] Thus squaring both sides, we get \[m^{2^{k-1}}=1+q.2^{k+1}+q^22^{2k}.\] Thus \[2^{k+1}\mid (m^{2^{k-1}}-1).\]. Then . If \(r\) is odd, then \[2\mid (r^{\phi(2p^s)}-1).\] Thus by Theorem 56, we get \[2p^s\mid (r^{\phi(2p^s)}-1).\] It is important to note that no smaller power of \(r\) is congruent to 1 modulo \(2p^s\). Suppose you are searching for a 1024-bit safe prime. It will calculate the primitive roots of your number. Which of the following integers 4, 12, 28, 36, 125 have a primitive root. If \(m\) has a primitive root \(r\) then \(r\) and \(m\) are relatively prime and \(ord_mr=\phi(m)\). Although there can be multiple primitive root for a prime number but we are only concerned for smallest one.If you want to find all roots then continue the process till p-1 instead of breaking up on finding first primitive root. From wiki... psi (25) = 20. We hence have everything we need to calculate the number of primitive roots that a prime has. How many primitive roots are there mod 263? Problem 15: For any positive integer , Solution: This is not easy unless you get the idea that . If \(n=p^s(p-1)\) with \(s\leq m-2\), then \[p^k\mid r^{p^{m-2}(p-1)}-1,\] which is a contradiction. Hence \(n\mid p^m(p-1)\). Thus using the Chinese Remainder Theorem, we get \[m\mid (r^L-1),\] which leads to \(ord_mr=\phi(m)\leq L\). Note now that 2 and 4 have primitive roots 1 and 3 respectively. 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 will prove by induction that \[\label{2} p^m\nmid (r^{p^{m-2}(p-1)}-1)\] for all integers \(m\geq 2\). In this section, we demonstrate which integers have primitive roots. Now because \[\phi(m)=\phi(p_1^{s_1})\phi(p_2^{s_2})...\phi(p_n^{s_n})\leq [\phi(p_1^{s_1}),\phi(p_2^{s_2}),...,\phi(p_n^{s_n})].\] Now the inequality above holds only if \[\phi(p_1^{s_1}),\phi(p_2^{s_2}),...,\phi(p_n^{s_n})\] are relatively prime. 4- If it is 1 then 'i' is not a primitive root of n. 5- If it is never 1 then return i;. Then \(s\) is also a primitive root modulo \(p\). Let \(p\) be an odd prime. Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. Notice that 7 has 3 as a primitive root. Finding Other Primitive Roots (mod p) Suppose that we have a primitive root, g. For example, 2 is a primitive root of 59. Primitive n th n^\text{th} n th roots of unity are roots of unity whose multiplicative order is n. n. n. They are the roots of the n th n^\text{th} n th cyclotomic polynomial, and are central in many branches of number theory, especially algebraic number theory. Notice now that by Theorem 41, \[\phi(p_1^{s_1}),\phi(p_2^{s_2}),...,\phi(p_n^{s_n})\] are not relatively prime unless \(m=p^s\) or \(m=2p^s\) where \(p\) is an odd prime and \(t\) is any positive integer. We also have, we have \((r,p^s)=1\) where \(p^s\) is of the primes in the factorization of \(m\). Due to this, Find all primitive roots modulo 22. Show that there are the same number of primitive roots modulo \(2p ^s\), 5.4: Introduction to Quadratic Residues and Nonresidues. Find all primitive roots modulo 25. For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org. Primitive roots do not necessarily exist mod n n n for any n n n. Here is a complete classification: There are primitive roots mod n n n if and only if n = 1, 2, 4, p k, n = 1,2,4,p^k, n = 1, 2, 4, p k, or 2 p k, 2p^k, 2 p k, where p p p is an odd prime. Then any power of \(p\) is a primitive root. 25 = 5^5. If \(m\) is not \(p^a\) or \(2p^a\), then \(m\) does not have a primitive root. Hence \[ord_{p^m}r=\phi(p^m).\], We prove now ([2]) by induction. Also, we know that \(\phi(p^m)=p^m(p-1)\). You can change your choices at any time by visiting Your Privacy Controls. Enter a prime number into the box, then click "submit." Hence \(ord_{49}3=42\). Note that \[\begin{aligned} s^{p-1}=(r+p)^{p-1}&=&r^{p-1}+(p-1)r^{p-2}p+...+p^{p-1}\\&=& r^{p-1}+(p-1)p.r^{p-2}(mod \ p^2).\end{aligned}\] Hence \[p^2\mid s^{p-1}-(1-pr^{p-2}.\] Note also that if \[p^2 \mid (s^{p-1}-1),\] then \[p^2\mid pr^{p-2}.\] Thus we have \[p\mid r^{p-2}\] which is impossible because \(p\nmid r\).
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