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We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Hence $ord_{p^m}r=\phi(p^m).$, We prove now () by induction. We now show that any power of an odd prime has a primitive root. Otherwise, we have $$m=p-1$$ and thus $r^{p-1}\equiv 1(mod \ p^2).$ Let $$s=r+p$$. There are such numbers, hence has primitive roots. This means that the equation 57=2^x(mod83). Notice that since $$r$$ is a primitive root modulo $$p$$, then $ord_pr=\phi(p)=p-1.$ Let $$m=ord_{p^2}r$$, then $r^m\equiv 1(mod \ p^2).$ Thus $r^m\equiv 1(mod \ p).$ By Theorem 54, we have $p-1\mid m.$ By Exercise 7 of section 6.1, we also have that $m\mid \phi(p^2).$ Also, $$\phi(p^2)=p(p-1)$$ and thus $$m$$ either divides $$p$$ or $$p-1$$. In the previous problem, you've shown that 2 is a primitive root （mod83). Hence, $m^2=4n^2+4n+1=4n(n+1)+1.$ It follows that $$8\mid (m^2-1)$$. A primitive polynomial of degree m has m different roots in GF(p m), which all have order p m − 1. We now list the set of integers that do not have primitive roots. Let $$m=p_1^{s_1}p_2^{s_2}...p_i^{s_i}$$. 25 = 5^5. We also have, we have $$(r,p^s)=1$$ where $$p^s$$ is of the primes in the factorization of $$m$$. Show that there are the same number of primitive roots modulo $$2p ^s$$ as there are modulo $$p^s$$, where $$p$$ is an odd prime and $$s$$ is a positive integer. Thus we have now \begin{aligned} r^{p^{m-1}(p-1)}&=&(1+kp^{m-1})^p\\ &\equiv&1+kp^m(mod \ p^{m+1})\end{aligned} Because $$p\nmid k$$, we have $p^{m+1}\nmid (r^{p^{m-1}(p-1)}-1).$. Because $$p^s\mid (r+p^s-r)$$, we see that $p^s\mid ((r+p^s)^{\phi(2p^s)}-1).$ As a result, we see that $$2p^s\mid ((r+p^s)^{\phi(2p^s)}-1)$$ and since for no smaller power of $$r+p^s$$ is congruent to 1 modulo $$2p^s$$, we see that $$r+p^s$$ is a primitive root modulo $$2p^s$$. We now show that all integers of the form $$m=2p^s$$ have primitive roots. Proof: Let be a primitive root of . Yahoo is part of Verizon Media. Once we prove the above congruence, we show that $$r$$ is also a primitive root modulo $$p^m$$. Then $p^m\nmid (r^{p^{m-2}(p-1)}-1).$ Because $$(r,p)=1$$, we see that $$(r,p^{m-1})=1$$. If $$m$$ is an odd integer, and if $$k\geq 3$$ is an integer, then $m^{2^{k-2}}\equiv 1(mod \ 2^k).$. Let $$p$$ be an odd prime. Exercises. The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. $\endgroup$ – lulu Mar 23 '19 at 16:58 It has order . Examples: Input: P = 3 Output: 1 The only primitive root modulo 3 is 2. For example, if n = 14 then the elements of Z n are the congruence classes {1, 3, 5, 9, 11, 13}; there are φ(14) = 6 of them. If $$m$$ is not $$p^a$$ or $$2p^a$$, then $$m$$ does not have a primitive root. And since $$p-1\mid m$$ then we have $m=p-1 \ \ \mbox{or} \ \ m=p(p-1).$ If $$m=p(p-1)$$ and $$ord_{p^2}r=\phi(p^2)$$ then $$r$$ is a primitive root modulo $$p^2$$. 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. Determine how many primitive roots the prime 37 has. Then . On the other hand, because $p^m\mid (r^n- 1),$ we also know that $p\mid (r^n-1).$ Since $$\phi(p)=p-1$$, we see that by Theorem 54, we have $$n=l(p-1)$$. Either $$ord_{49}3=6$$ or $$ord_{49}3=42$$. Then $$s$$ is also a primitive root modulo $$p$$. Input: P = 5 Output: 2 Primitive roots modulo 5 are 2 and 3. But since $$3^6\not\equiv 1(mod \ 49)$$. From wiki... psi (25) = 20. Legal. While, if $$r$$ is even, then $$r+p^s$$ is odd. Finding Other Primitive Roots (mod p) Suppose that we have a primitive root, g. For example, 2 is a primitive root of 59. 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. Note also that $\phi(2p^s)=\phi(p^s),$ so that $p^s\mid (r^{\phi(2p^s)}-1).$. Find a primitive root of 4, 25, 18. 3. 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. $\begingroup$ Phrased differently, you say that you know that there are $\varphi(30)=8$ primitive roots. You randomly try odd numbers until the number you choose is a safe prime. Hence $2\mid ((r+p^s)^{\phi(2p^s)}-1).$. Due to this, His work was selected by the Saylor Foundation’s Open Textbook Challenge for public release under a Creative Commons Attribution (CC BY) license. Suppose you are searching for a 1024-bit safe prime. We also know from Euler’s theorem that $p^{m-1}\mid (r^{p^{m-2}(p-1)}-1).$ Thus there exists an integer $$k$$ such that $r^{p^{m-2}(p-1)}=1+kp^{m-1}.$ where $$p\nmid k$$ because $$r^{p^{m-2}(p-1)}\not\equiv 1(mod \ p^m)$$. Find all primitive roots modulo 22. Here is a table of their powers modulo 14: This power as well would also be congruent to 1 modulo $$p^s$$ contradicting that $$r$$ is a primitive root of $$p^s$$. Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. Theorem 14: If has a primitive root, then it has primitive roots. 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 .. Problem 15: For any positive integer , Solution: This is not easy unless you get the idea that . All we need to do know is calculate $\phi (36)$: (1) We start by showing that every power of an odd prime has a primitive root and to do this we start by showing that every square of an odd prime has a primitive root. Hence 3 is a primitive root of 49.