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#1 Find a primitive root modulo 17. It is always reasonable to test 2 as a candidate first. By Fermat's little theorem and by Lagrange's theorem, ...

http://www.math.umn.edu/~garrett/m/intro_algebra/s3.pdf

Efficiently finding multiplicative inverses. Equality mod m, integers mod m. Sun-Ze's theorem. Fermat's Little Theorem. Definition of order. Primitive roots ...

http://www.math.umn.edu/~garrett/coding/Overheads/11_sunze_proots.pdf

We'd like to find primitive roots mod N, if possible. There are really two problems: Question 1. For which N does there exist a primitive root modulo N? ...

http://math.uga.edu/~pete/4400primitiveroots.pdf

4) For each prime in the table, we can find nonzero integers a that are not primitive roots mod p. In each case, if k is the smallest positive ...

http://www.math.uic.edu/~leon/mcs425-s08/handouts/PrimitiveElements.pdf

So the (unique modulo 18) solution is x = 13. 2. Find a primitive root modulo 250, and prove that it is primitive. 250 = 2 · 53. Following the proof of the ...

http://www-rohan.sdsu.edu/~vadim/teaching/522/f07/Exam 10 Solutions.pdf

§9.2 Primitive Roots for Primes. #2 Find the number of incongruent roots modulo 13 of each of the following polynomials. (a) x2 + 1. Problem Set 10 ...

http://satoss.uni.lu/members/sasa/teaching/Math356/problems10.pdf

Find a set of this many incongruent primitive roots modlo 13. 10. Find the number of incongruent roots modulo 13 of each of the following polynomials. ...

http://satoss.uni.lu/members/sasa/teaching/Math356.2003/problems13.pdf

(b) Find a primitive root modulo 4. Find a primitive root modulo 9. Find a primitive root modulo. 25. Find a primitive root modulo 27. ...

http://people.brandeis.edu/~jbellaic/nt/ex4sol.pdf

21 Jul 2009 ... (ii) either a is primitive mod p2 or a + p is primitive mod p2. Example. Find a primitive root mod 112. Solution: 2 is primitive mod 11 (we ...

http://www.ualberta.ca/~jies/Math 324/324NumLec09-7-21.pdf

20 Jul 2009 ... Finding All Primitive Roots mod p. We saw in the proof of Lemma 1.7 ... this to find all primitive roots mod 19. For each primitive root b ...

http://www.ualberta.ca/~jies/Math 324/324NumLec09-7-20.pdf

Let h(p) be the least positive primitive root modulo p2. ... If we can find at least p* bad primitive roots in the range [2,pi+e], then we can form at least ...

http://blms.oxfordjournals.org/cgi/reprint/6/1/42.pdf

Exercises. 1. Use the ring structure of Zn and the software PascGaloisJE or one of the supporting Java applets to find a primitive root modulo each of ...

http://faculty.salisbury.edu/~despickler/pascgalois/classroom/NumberTheory/Fisher/PrimitiveRoots.pdf

31 Oct 2008 ... Given a primitive root (mod p), it is quite easy to find a primitive root (mod pk). This is due to an old theorem by V. A. Lebesque which ...

http://www.ams.org/mcom/2009-78-266/S0025-5718-08-02090-5/S0025-5718-08-02090-5.pdf

P . We also find a bound for the least degree of a monic polynomial irreducible in A which is a primitive root modulo P . 1. Introduction ...

http://www.ams.org/proc/1998-126-03/S0002-9939-98-04582-1/S0002-9939-98-04582-1.pdf

Looking ahead to problem 2, we see that to find a primitive root modulo 5, it suffices to find an integer a such that a2 /= 1 (mod 5). ...

http://math.arizona.edu/~savitt/teaching/nt/solution_sets/sol5.pdf

tive root modulo p. What can one say about the integer α if gα is (a) a quadratic non-residue modulo p, (b) a primitive root modulo p? (a) find, modulo p, ...

http://math.arizona.edu/~tsutton/teaching/446/IV.pdf

Determine a primitive root modulo 19, and use it to find all the primitive ... Since 27 = 33, we first find a primitive root modulo 3, namely ...

http://www.math.ucsd.edu/~okikiolu/104b/hws1.pdf

First we need is to find the primitive roots modulo 5, namely 2 and 3. Then we find all the integers a from 1 to 30 such that (a, 30) = 1 and a ≡ 2, 3 ...

http://www.math.ucsd.edu/~okikiolu/104b/hws2.pdf

In particular, we find that there are exactly 5 solutions modulo. 114. Problem 2. (Short answer.) (a) How many primitive roots are there modulo the prime ...

http://www.cems.uvm.edu/~voight/255/255-EX02-solns.pdf

probability that 2 is a primitive root modulo p for p a large prime. 9.2: Primitive Roots for Primes. Problem 9.2.1(a)–(b). Find the number of incongruent ...

http://www.cems.uvm.edu/~voight/255/255-HW10.pdf

(ii) It is known that 2 is a primitive root modulo 101. (Note that 101 is prime.) Find: (a) The total number of incongruent integers with order exactly 25 ...

http://www.math.uiuc.edu/~hildebr/453/exam2sol.pdf

x5 ≡ 3 (mod 13), and find the solutions, if there are any. Solution. We know from a previous example that 2 is a primitive root modulo ...

http://www.math.uwaterloo.ca/~pingram/pmath340/notes_week4.pdf

These solutions are 1,3,5,7 mod 8. • Problem 1, page 106: Find a primitive root for the prime 3; for the prime 5; for the prime 7; the prime 13. ...

http://eiu.edu/~cfdmb/5335/ps05solution.pdf

by F Douglas - 2008 - Related articles

http://filer.case.edu/bej2/shared/primitiveroots.pdf

16 Nov 2005 ... Find all primitive root modulo 37. Solution. φ(37) = 36 = 22 · 32. Suppose that a is an integer which is not divisible by 37. ...

http://www.math.lsa.umich.edu/~hderksen/math475.f05/PS7sol.pdf

Since none of these are 1 mod 31, we find that the order of 3 mod 31 must be 30. (b) [5] How many primitive roots are there mod 961? (961 = 312) ...

http://oyeat.com/courses/math3070/solns/math3070midterm2s.pdf

(g) By parts (c)–(f), for every odd prime p and every positive integer k we can find a primitive root a mod pk. Now, 1 is a primitive root mod 2. ...

http://oyeat.com/courses/math3070/solns/math3070hw5s.pdf

a primitive root modulo p. Show that also -r is a primitive root modulo p. 44. Let p be a prime. Find the remainder after division by p of the product ...

http://math.ucsc.edu/~boltje/courses/f07/f07m110ex.pdf

ROO~ER, on the set of primes p such that 2 is a primitive root mod p, is .... prescribed primitive root, which already included finding Euclidian fields and ...

http://www.springerlink.com/index/2476L7663673225H.pdf

then 10 is a primitive root modulo n, and in such cases, the length of the repeating portion .... modulo 41 is 5, or 1/8 of 40; so we expect to find 8 such ...

http://www.springerlink.com/index/E5062830071M478M.pdf

by FE KARADAYI - 2009 - Related articles

http://mcs.cankaya.edu.tr/ogrenciler/proje2009Yaz/200422026elvankaradayi/sunum.pdf

(9.1, 4f) Find a primitive root mod 18. Proof: φ(18) = φ(2)φ(9) = 1 · 6 = 6. So we have to find an element of order 6 (modulo. 9). This is trial and error. ...

http://math.gmu.edu/~jshapiro/m301/sol5.pdf

11.1 Show that 2 is not a primitive root modulo primes p ≡ ±1 mod 8. 11.2 Find a prime p ≡ 3 mod 8 for which 2 is not a primitive root. ...

http://www.fen.bilkent.edu.tr/~franz/nt/ch11.pdf

by S GUN - Cited by 2

http://mathsci.ucd.ie/~sahu/primitive.pdf

by C Cobeli - 1998 - Cited by 12

http://matwbn.icm.edu.pl/ksiazki/aa/aa83/aa8325.pdf

by P Moree - 1999 - Cited by 10

http://matwbn.icm.edu.pl/ksiazki/aa/aa89/aa8912.pdf

n is cyclic and g is a primitive root modulo n, then g2l−1 for l = .... find an arbitrary long sequence of consecutive quadratic residues (or non-residues) ...

http://www.emis.de/journals/MB/130.4/mb130_4_5.pdf

(b) Also find the elements of order 4 mod 61. 4). Use the fact that 6 is a primitive root mod 41 to solve 1 ≡ x4( mod 41). ElGamal Public Key scheme ...

http://www.math.ruh.ac.lk/~upeksha/mam3213-tutes/MAM3213-tute7.pdf

we know that 2 is a primitive root mod 11. 3.13.17.b Note that 23 = 8 (mod 11). Find x such that 8x = 2 (mod 11) (Hint: What is inverse of 3 (mod 10)?) ...

http://squall.cs.ntou.edu.tw/CryptoIntro/2006fallC1/hw2sol.pdf

primitive root mod 17, so we can use item (5) of the previous proposition to conclude that ind3(7x ... Now, to find the index for any x, it turns out that ...

http://www.cs.xu.edu/math/math302/08f/15_DiscreteLog.pdf

(b) Given that 2 is a primitive root modulo 29 and that 37 ≡ 12 (mod 29), find all solutions to x7 ≡ 12 (mod 29). Solution. The solutions all have the ...

http://www.math.princeton.edu/~mahlburg/teaching/handouts/18.781/Exam2Solns.pdf

0 (mod m). So every power 2 or greater for each is zero, so they are all congruent. 4. (Niven 2.8.2) Find a primitive root of 23. ...

http://www.math.princeton.edu/~mahlburg/teaching/handouts/18.781/PS5Solns.pdf

by J Dubrois - 2008 - Cited by 2

http://hal.archives-ouvertes.fr/docs/00/34/52/73/PDF/polypr.pdf

17 Apr 2007 ... This is imposible, and so it follows that the order of 2 modulo p is 2k = p − 1 = φ(p), i.e. 2 is a primitive root modulo p. (5) Find the ...

http://www.math.ucsb.edu/~agboola/teaching/2007/spring/115B/sol1.pdf

whether a number g is a primitive root mod p (Therefore, if we need to find a primitive root mod p, we can simply use this ...

http://www.math.ucsb.edu/~julius/MATH115B/Homework Section 3.13/Section 3.13.pdf

g is a primitive root modulo p iff rq ≠ 1 mod p for each distinct prime divisor q of p - 1. Arkin's Conjecture: (very messy). It is always possible to find ...

http://www.krapp.org/hydra/courses/analysis/5-Number.pdf

(a) Write out a table of indices mod 23 for the primitive root 5. (b) Use the table constructed above to find all solutions to 314x ≡ 2 mod 23 ...

http://www.cims.nyu.edu/~burr/Classes/Homework8.pdf

Find the greatest common divisor of the polynomials f(X) = X3 −2X2 +X −2, ... Let g be a primitive root modulo m. Prove that gk is a primitive root modulo ...

http://www.math.tau.ac.il/~borovoi/courses/NumberTheory/nt10hw4.pdf

8.1 #1: Find the order of the integers 2, 3, and 5 modulo 17. Solution: By Theorem 8-1, ... ord173 = 16 (Remark: so 3 is a primitive root modulo 17. ...

http://math.uh.edu/~minru/4383/hwkey.pdf

Find a primitive root mod 23. Solution. As φ(23) = 22 = 2 · 11, the order of a number mod 23 can only be 2,11, or 22. We want it to be 22, i.e we're looking ...

http://www.math.sunysb.edu/~olga/mat311-spr09/mat311soln6.pdf