DETERMINING PRIMITIVE ROOTS

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DETERMININGPRIMITIVE ROOTS

• by
• Christoph and John C. Witzgall
• September 15, 2015

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Divisors/Multipliers

• Integers will be capitalized. For A, B gt 0,
• gcd(A, B) greatest common divisor
• WA and WB gt Wgcd (A, B)
• lcm(A, B) least common multiple
• AW and BW gt lcm(A, B)W
• (1)
• A, B are relatively prime ltgt gcd(A, B)1

gcd(A, B) lcm(A, B)
AB
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REDUCTION MODULO Pgt0

• For P gt 0 , any integer X may be represented
  as
• X SP r, 0 r lt P,
• with r the
• remainder of X modulo P
• and S the value of integer (long) division of
  
• X by P. We say that X is reduced to r
  modulo P.

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ARITHMETIC MODULO Pgt0

• The reduction process is used to define the
• arithmetic modulo P.
• For remainders a, b between 0 and P-1, the
  operations
• a b, a - b, ab
• are evaluated using their integer face value, if
  necessary, reducing the results modulo P.

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THE GROUP GP

• For P a prime, the remainders,
• 1, 2, 3, , P1
• under multiplication modulo P form the group,
• GP
• our subject of interest.

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CYCLES MODULO 7

• 1

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CYCLES MODULO 7

• 1 2

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CYCLES MODULO 7

• 1 2 4

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CYCLES MODULO 7

• 1 2 4 8
• -7

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CYCLES MODULO 7

• 1 2 4 1

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CYCLES MODULO 7

• 1 2 4 1 2

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CYCLES MODULO 7

• 1 2 4 1 2 4

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CYCLES MODULO 7

• 1 2 4 1 2 4 8
• -7

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CYCLES MODULO 7

• 1 2 4 1 2 4 1 . . .

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CYCLES MODULO 7

• (1 2 4)
• 1

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CYCLES MODULO 7

• (1 2 4)
• 1 3

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CYCLES MODULO 7

• (1 2 4)
• 1 3 9
• -7

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CYCLES MODULO 7

• (1 2 4)
• 1 3 2

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CYCLES MODULO 7

• (1 2 4)
• 1 3 2 6

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CYCLES MODULO 7

• (1 2 4)
• 1 3 2 6 18
• -14

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CYCLES MODULO 7

• (1 2 4)
• 1 3 2 6 4

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CYCLES MODULO 7

• (1 2 4)
• 1 3 2 6 4 12
• -7

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CYCLES MODULO 7

• (1 2 4)
• 1 3 2 6 4 5

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CYCLES MODULO 7

• (1 2 4)
• 1 3 2 6 4 5 15
• -14

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CYCLES MODULO 7

• (1 2 4)
• 1 3 2 6 4 5 1

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CYCLES MODULO 7

• C(2) (1 2 4)
• C(3) (1 3 2 6 5) 3 primitive root
• C(4) (1 4 2)
• C(5) (1 5 6 2 3) 5 primitive root
• C(6) (1 6)

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GROUPS

• Groups considered here are finite and abelian.
  The notation
• G order of G
• Means number of elements. Fundamentally,
• N G gt aN 1 for a ? G
• H ? G gt H divides G

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CYCLES

• The cycle
• C(a) (1, a, a2, , aN-1), aN 1,
• encapsulates the period of a sequence of consecu-
• tive powers of an element a ? G.
• (3) N C(a), aR 1 gt NR
• The entries in C(a) form a group. Such groups,
  generated by a single element are called
  cyclic.

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SUBCYCLES

• Suppose N C(a), KN, M N/K. Then
• C(aK ) (1 aK a2K )
• is a subcycle of C(a). Its length is given by
• (4) Proposition C(aK ) N/K M
• Proof aKM aN 1. If aKJ 1 for 0 lt J M
• then NKJ. Thus MJ, so that J M.

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SPREADS

• Subgroups H1, H2 ? G together
• generate what we call their
• spread H1 ? H2
• Spread
• H1 H2
  
• Inter section
  

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ORDERS OF SPREADS

•  

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COROLLARY

•  

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PRIMITIVE ROOTS

• (6) Primitive Root Theorem Gp is cyclic
• This means that Gp may be generated by
• a single one of its elements. Each such
  generator is a primitive root of P. We propose
• A constructive proof based on prime factorization
  of P-1
• An algorithm for computing primitive roots.

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APPROACH

• In what follows, we aim to prove the
• (7) Theorem The spread of two cycles
• C(a) and C(b) in GP is cyclic

by characterizing a generator x ? GP
__ C(a)
? C(b) C(x) Successively collapsing pairs
of cycles into single ones then yields a
primitive root.
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RELATIVELY PRIME CYCLES

• We call cycles C(a), C(b) in group G
  relatively prime if C(a), C(b) are
  relatively prime.
• (8) Theorem The spread of relatively prime
  cycles C(a) and C(b) is cyclic
• C(a) ? C(b) C(ab).
• (Proof after the next slide.)

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CONVENTION

• For what follows in this presentation,
  we are using the notation
• A C(a) , B C(b)
• for the orders of cycles C(a) and C(b).

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PROOF OF THEOREM (8)

• By (5) and (1), C(a) ? C(b) A B lcm (A,
  B).
• For M gt 0
• (ab)M 1 ltgt aM (bM ) -1 ? C(a) n C(b)
• ltgt aM b M 1 ltgt AM and BM by
  (3)
• In other words, the exponent
• M lcm(A, B) AB
• is the smallest positive exponent with (ab)M
  1.
• Thus C(a) ? C(b) C (ab).

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CYCLES MODULO 11

• C(2) (1 2 4 8 5 10 9 7 3 6)
• C(3) (1 3 9 5 4)
• C(4) (1 4 5 9 3)
• C(5) (1 5 3 4 9)
• C(6) (1 6 3 7 9 10 5 8 4 2)
• C(7) (1 7 5 2 3 10 4 6 9 8)
• C(8) (1 8 9 6 4 10 3 2 5 7)
• C(9) (1 9 4 3 5)
• C(10) ( 1 10)
• Cycles of equal lengths have the same elements.

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LAGRANGE

•  

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CONSEQUENCES

• Recall
• A C(a), B C(b).
• In GP the statements
• i. AB ltgt C(a) ? C(b)
• (10) ii. C(a) n C(b) gcd(A, B)
• iii. C(a) ? C(b) lcm(A,
  B)
• are consequences of Lagrange (9).

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SEPARATION

• We call UA and VB separators of A and
  B if
• (11) ( i) lcm(A/U, B/V) lcm(A, B)
• ( ii) gcd(A/U, B/V) 1
• Theorem If U and V separate A and B,
• C(a) ? C(b) C(aU bV).

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Proof of Theorem 12

• Proof By (4),
• C(aU )A/U, C(bV)B/V. Thus by (10)
• C(a) ? C(b) lcm(A,B)
• C(aU ) ? C(bV) lcm(A/U, B/V)
• Yielding by (11.i), C(a) ? C(b) C(aU ) ?
  C(bV).
• As C(aU) ? C(a), C(bV) ? C(b), we have
• C(a) ? C(b) C(aU ) ? C(bV )
• By (11.i), C(a) and C(b) are relatively prime. By
  (8)
• C(aU ) ? C(bV) C(aU bV ).

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SEPARATOR PRODUCT

•  

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FACTORIZATION

• A 120 B 1260
• The prime factors involved in both are
• 2, 3, 5, 7
• Prime factorizations of A and B
• A 120 8 3 5 1
• B 1260 4 9 5 7
• lcm(A, B) 8 9 5 7
• gcd(A, B) 4 3
  5 1

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THE GIST

• A 120 8 3 5 1 8 1 5
  1 40 A/U
• B1260 4 9 5 7 1 9 1
  7 63 B/V
• U 3 1 3, V 4 5 20
• Reducing the factors of lower multiplicity to 0
  
• leaves lcm(A, B) unchanged, while reducing
• A by U and B by V, effecting separation.

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Stepwise Separation

•  

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CUMULATIVE MULTIPLICATION

•  

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PARTIAL SEPARATOR

• (15) Theorem With C gcd(A, B),
• W gcd(A/C, C) gt 1
• is a partial separator of A, B.
• Proof In view of (14), it suffices to show that
  any prime
• divisor QW is a partial separator of A, B. The
  multiplicity, mult(QA), of Q in A exceeds
  mult(QC). Then
• gcd(Q, B/C) gcd(A/C, B/C) 1,
• Implies gcd(Q, B/C) 1 so that
• mult(QB) mult(QC) lt mult (QA).
• Thus gcd(A, B/Q) gcd(A, B)/Q.

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TERMINATION

•  

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EXAMPLE

• We revisit the
• A 120, B 1260
• C gcd(A, B) 60, A/C 2
• W gcd(A/C, C) gcd(2,60) 2
• A 120, B (B/2) 630
• C gcd(A, B) 30, A/C 4
• W gcd(A/C, C) gcd(4,30) 2
• A 120, B (B/2) 315
• C gcd(A, B) 15, A/C 8
• W gcd(A/C, C) gcd(8, 315) 1
• 4. U C 15, V 2 2 4

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SEPARATION ALGORITHM

• Given integer A, B gt 0 Wanted separators
  U,V.
• Step 1 1 ? V, gcd(A, B) ? C
• Step 2 If C 1 ? step 7
• Step 3 A/C ? X
• Step 4 gcd (X, C) ? W
• Step 5 If W 1 ? step 7
• Step 6 VW ? V, C/W ? C, XW ? X, ? step 4
• Step 7 C ? U, ? terminate

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NUMBERS

• 1228 primes 10,000
• primitive roots calculated
• 24 separation required
• 470 instances of primitive root 2

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/2015/2015-09-15-Witzgall.html
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