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1 Chapte 8: Pimitive Roots ad Idices 6 SECTION C Theoy of Idices By the ed of this sectio you will be able to what is meat by pimitive oot detemie the C Pimitive Roots Fist we defie what is meat by a pimitive oot. Popositio (8.0. If gcd, ad a has ode φ the itege a is called the pimitive oot of the ( a = itege. What does this mea? Meas the followig: ( a/ mod, a / mod, L, a φ mod idex The fist time a is coguet to coguet modulo is whe the idex is φ. Fo example is a pimitive oot of modulo because 5 mod, mod, 8 mod 5 mod, 0 mod,, 7 8 0, 7 ( mod, ( mod, 6 ( mod ad ( mod 6 mod Example Show that is a pimitive oot of 7. φ 7 = 7 = 6. Evaluatig the idices of we have 7 is pime so 5 6 ( mod 7, 5 ( mod 7, ( mod 7 mod 7, mod 7, 6 mod 7, Hece is a pimitive oot of 7. Example Show that is ot a pimitive oot of 7. Sice mod7 so the ode of modulo 7 is ad ot 6 which meas is ot a pimitive oot of 7. Popositio (8.. Let gcd ( a, = ad a, a, a, L, a φ be positive iteges less tha ad elatively pime to. If a is pimitive oot of, the ( a, a, a, L, a φ ae coguet modulo to a, a, a, L, a φ i some ode. Poof. See page 50 of Buto Elemetay Numbe Theoy. C Theoy of Idices
2 Chapte 8: Pimitive Roots ad Idices 7 Let be a itege that has a pimitive oot. By the above Popositio (8. we have that the fist φ powes of :,,, L, φ ( ae coguet modulo i some ode to those iteges less tha ad elatively pime to it. By the above Example we have = is a pimitive oot of 7 ad mod 7, mod 7, 6 mod 7, ( 5 6 ( mod7, 5( mod7, ( mod7 The set {,,,, 5, 6 } is elatively pime to 7 ad each of these elemets ae coguet dex to modulo 7. We coclude that if a is abitay itege which is elatively pime to, the this itege a ca be expessed as a mod fo a idex whee φ. Example 5 Detemie the value of such that 5( mod7 Fist we ca wite 5 as 5 ( mod 7 By ( we have ( mod7 which meas Hece =. 5 mod 7 = is called the idex of 5 elative to ad omally deoted by id 5 =. We wite the geeal defiitio of idex as follows. Defiitio (8.. Let be a pimitive oot of. If the gcd ( a, = the the smallest positive such that a (mod is called the idex of a elative to. This is deoted by id ( a φ ( id a o just id a. Clealy Example 6 is a pimitive oot of 5. Usig this fid the followig: (a id ( (b id (c id ( (d id ( (e id ( 7 Well evaluatig the powes of we have ( mod 5, ( mod 5, ( mod 5, ( mod 5 Usig these esults we have (a id ( = (b id ( = (c id ( = (d id ( = (e 7 is ot amogst this list of powes of. How do we evaluate id 7? Well 7 ( mod 5 ad by pat (b we have id = theefoe id 7 =
3 Chapte 8: Pimitive Roots ad Idices 8 What does this esult id 7 = mea? Meas that to the idex of is coguet to 7 modulo 5: 7 mod 5 Example 7 is a pimitive oot of 5. Usig this fid the followig: (a id ( (b id ( (c id ( (d id ( (e id ( 8 Evaluatig the powes of we have mod 5, mod 5, mod 5, mod 5 Usig these esults gives: (a id ( = (b id ( (e S imilaly to the pevious example we have 8 mod 5 Hece id ( 8 =. = (c id = (d ad b y pat (c id = id = C Elemetay Popeties of Idices I this subsectio we pove some of the ules of idices ad you will fid them to be aalogous to the ules of logs i algeba expect that the base is the pimitive oot. Fom the above two examples we have id id = Ca we coclude that id =? ( = ad Yes ot because these t wo esults. We eed to pove this fo the geeal case. Popositio (8.. Let be a pimitive oot of. The id = Poof. Execise. ( is aalogous to m ( m Note that id = log =. Popositio (8.. Let be a pimitive oot of. If a b the ( mod id ( b id a = Poof. We use the defiitio of the idex give above i (8.: Let be a pimitive oot of. The smallest positive such that a mod is called the idex of a elative to. This is deoted by id ( a φ ( Applyig this we have ( a ( mod ad id ( b ( id a id a o just b mod id a. Clealy
4 Chapte 8: Pimitive Roots ad Idices We ae give that a b ( mod theefoe ( b id a id ( mod Next we use Popositio (8.6 o page 8 of sectio A o this esult: Let the itege a modulo have ode. The j m a a mod j m ( mod f φ ( We ae give that is a pimitive oot so ode o is ( mod gives id a id b ( mod φ id a id b We ow by defiitio of idex (8. that ( meas that id ( a = id ( b which is ou equied esult.. Applyig Popositio (8.6 to { φ } id a, id b,,,, L,. This Popositio (8.5. Let be a pimitive oot of ad id a be the idex of a elative to. The we have the followig esults: id ab id a + id b mod (a ( φ (b id( a id( a ( mod φ ( (c id ( 0 ( mod φ ( ad id ( ( mod φ ( Note the aalogy of these esults with logaithms. How do we pove these esults? Use the defiitio of the idex (8. o page 7: Let be a pimitive oot of. The smallest positive such that a ( mod is called the idex of a elative to. Poof. (a Usig the odiay ules of idices we have id( a + id( b id( a id( b = Applyig the defiitio (8. o this: id( a a ( mod id( b ad b ( mod Combiig these two esults gives id( a + id( b id( a id( b ab mod Usig the defiitio (8. o id ab gives ( ab ( id ab mod Hece we have id( a + id( b id( ab ab ( mod Usig (8.6 o page 8 of sectio A: Let the itege a modulo have ode. The j m a a mod j m mod
5 Chapte 8: Pimitive Roots ad Idices 0 whee j, m ae positive iteges. Applyig this to the last esult: we have ( a + id ( b id ( ab id ( mod + ( b id( ab ( mod φ ( id a id This is ou equied esult. (b We eed to pove id ( mod φ a id a. Agai applyig the defiitio of the idex (8. o page 7: Let be a pimitive oot of. The smallest positive such that a mod is called the idex of a elative to. o id a gives We have id ( a id ( a a mod ( mo a d. Taig this to the powe : id ( a ( ( mod a a mod Usig x = x Equatig the last two esults we have id ( a id( a mod Agai usig (8.6 o page 8 of sectio A: id a m m Let the itege a modulo have ode. The j m a a mod o id ( a id( a mod gives j m ( mod id ( a ( mod φ id a (c Need to pove id ( 0 ( mod φ ( ad id ( ( mod φ ( id( ( mod 0 Also ( mod. Equatig these we have id( 0 ( mod Applyig Popositio (8.6 to these id See Popositio (8.. ( 0 ( mod φ ( : C Solvig No-liea Equatios The above theoy of idices ca be used to solve cogueces of the type
6 Chapte 8: Pimitive Roots ad Idices ( mod d x a (* We covet this ito a liea fom. How? By usig the above ules of idices: d id x id a mod Let g gcd ( d, φ ( If id( a the the =. Ude what coditios do we have a solutio(s? g e ae exactly g solutios. If g does ot divide id ( a the (* has o solutios. Example 8 Solve x 7( mod is a pimitive oot modulo. mod mod 8 mod mod 5 6 mod 6 mod 7 mod 8 mod 5 mod 0 0 mod 7 mod mod We ceate the table of idices: a id a Applyig the ules of idices give i Popositio (8.5 o page : Let be a pimitive oot of ad id ( a be the idex of a elative to. The id ab id a + id b mod φ (a ( (b id( a id ( a ( mod φ ( (c id ( 0 ( mod φ ( ad id ( ( mod φ ( to the give coguece x 7( mod yields:
7 Chapte 8: Pimitive Roots ad Idices ( + ( 7 ( mod id id x id ( + ( 7( mod + id ( x ( mod id ( x ( mod id ( x ( mod id ( x ( mod id id x id Covetig to module gives id x, 5, mod Usig the table i evese diectio we have x, 6, 5 mod Example Solve the followig by usig the pimitive oot 7: x 7( mod The above poblem was solved by usig pimitive oot i Example 8. We ca solve this equatio by usig othe pimitive oots. I this case we use ( mod 7 0 ( mod ( mod 7 5 ( mod ( mod ( mod ( mod 8 7 ( mod 7 8 ( mod ( mod 7 8 ( mod 7 ( mod The table fo pimitive oot 7 is: a id a Similaly fo x 7 mod we have
8 Chapte 8: Pimitive Roots ad Idices id7( + id7( x id7( 7 ( mod 0 + id7 ( x ( mod id7 ( x ( mod id7( x ( mod id7( x ( mod Hece id7 ( x, 7, ( mod x 5, 6, ( mod Of couse we will have the same solutios.. Usig the table i evese diectio we have C5 Solvig x a ( mod We ca use the followig esult fo solvig x a ( mod Popositio (8.6. Let have a pimitive oot ad ( a has a solutio x. gcd, =. The coguece (mod a ( g a φ ( mod whee g = gcd (, φ (. Additioally thee ae exactly g solutios. Poof. See Page 66 of Buto. Example 0 Solve the followig: x ( mod Sice is pime we have φ ( = ad g = gcd (, =. Puttig a = ito the above Popositio (8.6 we have / ( mod Hece x ( mod has o solutio. Example Solve the followig: x 5 ( mod (* As above we have φ ( = ad g = gcd (, =. Puttig a = 5 ito the above Popositio (8.6 we have / 5 5 mod Thee ae exactly solutios to (*. By taig idices of the give equatio x 5 ( mod we have id x id5 m We use the table established o page : ( od
9 Chapte 8: Pimitive Roots ad Idices a id ( a Fom this table we have id 5 ( mod id ( x ( mod id ( x ( mod id ( x, 7, ( mod. Substitutig this gives Usig the above table i evese diectio we have x 8,, 7 mod SUMMARY We use the followig ules of idices to solve o-liea cogueces: Popositio (8.5. Let be a pimitive oot of ad id ( a be the idex of a elative to. The id ab id a + id b mod φ (a ( (b id( a id( a ( mod φ ( (c id ( 0 ( mod φ ( ad id ( ( mod φ (
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