On the introductory notes on Artin s Conjecture

Transcription

1 O the troductory otes o Art s Cojecture The urose of ths ote s to make the surveys [5 ad [6 more accessble to bachelor studets. We rovde some further relmares ad some exercses. We also reset the calculatos whch lead to the desty aearg Hooley s formulas ad a roof for the corollares of the result by Heath-Brow. Cotets. What we are seakg about 2. The cyclcty of the grou (Z/Z 3. Exercses o rmtve roots 2 4. The modfed desty of Hooley s formulas 3 5. Corollares of the result by Heath-Brow 5 6. Datas 5 Refereces 6. What we are seakg about Oe of the most famous oe cojectures umber theory s Art s rmtve root cojecture, whch s due to Eml Art ad dates back to 927. Let deote a rme umber ad cosder the reducto of the tegers modulo, so the rg homomorhsm Z Z/Z whch seds ay teger to ts resdue class modulo. Let a be a teger ot dvsble by. The Fermat s Lttle Theorem tells us that a (mod. Ths meas that the teger a s dvsble by. Do there exst ftely may rmes such that the smallest such that a (mod s fact? The classcal verso of Art s rmtve root cojecture states that the aswer to ths questo s affrmatve, rovded that a s ot a erfect square or. 2. The cyclcty of the grou (Z/Z The set of o-zero resdue classes modulo s a grou for the multlcato duced by Z/Z. It s deoted by (Z/Z. I outle two roofs of the fact that the grou (Z/Z s cyclc. Lemma 2.. A olyomal of degree d wth coeffcets a feld ca have at most d dstct solutos. Proof. Let α be a soluto for f(x. The we ca wrte f(x g(x(x α + r by eucldea dvso ad the costat r s 0 (evaluate f(α. So f(x g(x(x α. A soluto β of f s such that ether g(β 0 or (β α 0. So f β α the β s a soluto of g ad (by ducto o the degree there are at most d solutos for g. Lemma 2.2. For the Euler-φ fucto, the followg relato holds for every : d φ( Proof. Oe ca rove the formula by ducto o the umber of rme factors of. See [3, Proosto For ay grou, the order of the sum of two elemets clearly dvdes the least commo multle of the orders of the elemets. Lemma 2.3. I a grou, f two elemets have orders whch are corme, the the order of the sum s exactly the roduct of the orders.

2 2 Proof. Suose that a ad b have order ad m resectvely ad (, m. Let d, d. The m d (a b d (a 0 hece the order does ot dvde m d hece the order s a multle of (aalogously for m. We kow that the order of the grou (Z/Z s hece t suffces to show that there s at least oe elemet of order exactly. Frst roof We frst rove that there are at most φ(d elemets of order d. Suose that there s a elemet a of order d. The the owers a, a 2..., a d are all dstct. They are d dstct solutos of the olyomal x d so they are all the solutos of that olyomal. Thus every elemet whch has order d, beg a soluto of x d, s a ower of a. The order of the ower of a elemet s gve by the formula: ord(a h ord(a (h, ord (a The the owers of a whose order s d corresod to the tegers 0 < h < d that are corme to d: they are exactly φ(d. So we have show that f there s a elemet of order d, there are exactly φ(d such elemets. We have a set of elemets, whch s the dsjot uo, for d, of subsets of order at most φ(d. Sce φ(d d the t must be that every such subset has order φ(d. I artcular, we have rove that for every d (cludg, whch s what we eed there are exactly φ(d elemets of order d. Secod roof Wrte as roduct of rme owers e. By Lemma 2.3, t suffces to fd, for every, a elemet whch has order a multle of e (otce that a ower of t wll have order exactly e. Wthout loss of geeralty, suose that o elemet has order a multle of e, equvaletly that every elemet has order dvdg. The there would be dstct solutos of the olyomal x, whch has degree strctly smaller tha. Cotradcto. 3. Exercses o rmtve roots ( Prove that (2 mod 29 s a rmtve root. Ht: mmze your calculatos by thkg of the grou structure. (2 Lst all the rmtve roots modulo 9. Ht: mmze your calculatos by thkg of the grou structure; you wll ossbly ecouter the same elemets whe you do the calculatos, so kee track of the formatos you have already foud. (3 Prove that (a mod ad ts verse [(b mod such that (ab mod ( mod have the same order. (4 Let q be a rme ad defe ord q (a mod as the hghest ower of q dvdg the order of (a mod. Show that f ord q (a mod >ord q (b mod the ord q (ab mod ord q (a mod. (5 Let be a odd rme. Prove that (a mod ad (-a mod have the same order f 4 dvdes the order of (a mod. Prove that (-a mod has order twce the order of (a mod, f the order of (a mod s odd. Prove that the order of (-a mod s half of the order of (a mod, f the order of (a mod s eve but ot dvsble by 4. Ht: Use exercse 4. Show that ord q (a mod ord q (-a mod for every odd rme q hece restrct your atteto to ord 2. For the last asserto, cosder that f you rase (a mod to the

3 odd art of ts order, you must ecessarly get (- mod [whch s the oly elemet of order 2. (6 Let G be a fte grou ad let g be a elemet of G. Show that the order of g s Ht: wrte (ord(g,, wth (ord(g,. 3 ord(g (ord(g,. (7 Let d be a dvsor of. Let (g mod be a rmtve root. Show that (g /d mod has order exactly d. Ht: use exercse 6. (8 Let be a odd rme. Show that (a mod s a rmtve root modulo f ad oly f (a /d mod s ot ( mod, for every rme dvsor d of. Ht: use exercse 7. (9 Show that the d-th owers [(a mod (b d mod for some b form a subgrou. Show that f d ( the the order of ths subgrou s ( /d. Show that f (, d the every elemet s a d-th ower. Ht: Let (g mod be a rmtve root. The show that (a mod s a d-th ower f ad oly f oe ca wrte (a mod (g dk mod for some teger k. For the last asserto, use the fact that d has a verse d modulo (, hece (g k mod (g d(d k mod for every teger k. (0 Show that there are φ( rmtve roots. Let a be as Art s cojecture, ad ot a ower. Prove that f > 3 the exected robablty that (a mod s a rmtve root s at most. Check that ths s the exected robablty f q 2, where q s rme. [For 2,3 the values are ad /2 resectvely. ( Exla why we exclude 0, ± ad squares from Art s cojecture. Exla also why a ror we do ot exclude other owers. Ht: 2 dvdes for every odd rme; see the ht of exercse 9. Recall Drchlet s theorem o rmes arthmetc rogresso to argumet that there are may rmes for whch s ot dvsble by some fxed odd umber. 4. The modfed desty of Hooley s formulas I ths secto, I show how oe arrves from Art s heurstcs to the formula of the desty whch aears Hooley s result. Ths calculato s take from [2, Secto 6. Followg Art s heurstcs, to calculate the desty we smly have to evaluate: squarefree µ( [K : Q where µ s the Moebus fucto ad K s the comostum of Q(ζ q, a /q for every rme umber q whch dvdes. So we are takg a sum of the verses of the degrees over Q of the umber felds K (wth a sg ± accordg to the Moebus fucto. Euler roducts. Let a( be a fucto defed over the atural umbers ( > 0 whch s multlcatve, whch meas a( ad a(m a(ma( wheever m, are corme. The the Euler roduct formula says: a( ( + a( + a( 2 + If a( 0 for ot squarefree the the followg smlfed verso clearly holds: squarefree a( a( ( + a( + a( 2 + ( + a( The case of lear dsjot extesos. Suose that we are the case where the extesos Q(ζ q, a /q are learly dsjot for every rme umber q (so, cludg q 2. Ths meas that a bc 2 for some

4 4 tegers b, c ad that b (mod 4. We are the the case where [K : Q (h, where h s the bggest teger such that a has a h-th root the tegers. The requested desty s the µ( [K : Q ( + µ( ( ( A(h [K squarefree : Q ( ( h h The umber A(h s obvously a ratoal multle of Art s costat A ( ( The case of o-lear dsjot extesos. b (mod 4. I artcular, b s odd. The [K : Q 2(h, (h, Suose that a bc 2 for some tegers b, c ad that f s eve ad b otherwse where h s the bggest teger such that a has a h-th root the tegers. Recall that h s odd sce we exclude squares from Art s cojecture. We have: µ( [K : Q 0(mod 2 b µ((h, + 2 0(mod 2 b µ((h, A(h + 0(mod 2 b µ((h, If m s squarefree ad m 0 (mod 2 b we ca wrte m 2 b wth corme to 2 b. The we have: 0(2 b µ((h, :(,2 b µ(2 b (h, 2 b (2 b φ(2 b µ(2 b (h, 2 b (2 b φ(2 b :(,2 b µ((h, Let µ( be equal to µ( wheever (, 2 b ad equal to 0 otherwse. roduct we get: By takg the Euler µ((h, A(h ( + : (2 b µ((h, ( + φ( : (2 b ( (h, A(h φ( : (2 b µ((h, ( φ( : (2 b φ( φ( (h, (h, φ( Puttg thgs together: [ A(h µ( b (h, 2 b [ A(h µ( b µ( [ [K : Q A(h µ(2 b (h, 2 b + 2 b φ(2 b : b; h : (2 b φ( : (2 b [ A(h µ( b φ( (h, : b; h φ( φ( (h, : (2 b (h, φ( (h, A(h [ µ( b φ( [K : Q b

5 5. Corollares of the result by Heath-Brow A -tule of teger umbers x,..., x s sad to be multlcatve deedet (the umbers are the called multlcatve deedet f there exst a,..., a tegers, ot all zero, such that xa. Multlcatve deedet meas ot multlcatvely deedet the sese above. Heath-Brow roved 967 ([ the followg result: Theorem 5.. If q, r, s are three ozero multlcatvely deedet tegers such that oe of q, r, s, 3qr, 3qs, 3rs, qrs s a square, the there are ftely may rme umbers for whch at least oe betwee q, r, s s a rmtve root modulo. We deduce the followg: Corollary 5.2. Cosder the classcal verso of Art s cojecture (we look for a fte set of rmes : ( There are at most two rme umbers for whch Art s cojecture fals. (2 There are at most three ostve squarefree umbers ( for whch Art s cojecture fals. Lemma 5.3. Let x, y, z be three dstct ostve squarefree tegers ( ad suose that x a y b z c for some tegers a, b, c. The (u to a reorderg we have z xy ad (x, y. Proof. It s easy to see that oe or two dstct ostve squarefree tegers ( are always multlcatve deedet. The a, b, c are all o-zero. Sce x, y, z are tegers, a, b, c caot be all > 0 or all < 0. So, wthout loss of geeralty, we ca wrte x a y b z c for some a, b, c > 0. Sce x ad y are dstct squarefree tegers there exsts rme (w.l.o.g. dvdg x but ot y. Hece c must dvde a (the left-had sde s a c-th ower. So we have x ca y b z c. By comarg the exoets the rme factorsatos, c b. The we have x a y b z [we could have x a ζc y b ζc m z but sce x, y, z are ostve tegers t must be that ζc ζc m s also a ostve teger, hece t s. So we have x a y b z. Sce a > 0, b > 0 ad z s square-free t follows that a b ad (x, y. So z xy ad (x, y. Proof of Corollary 5.2. ( Suose that there are three dstct rme umbers q, r, s for whch Art s cojecture fals. They are multlcatve deedet ad qrs s ot a square. Sce egatve umbers are ot squares, we aly the result by Heath-Brow ad deduce a cotradcto. (2 It suffces to show that every set of four dstct ostve squarefree tegers (, there s at least oe whch satsfes Art s cojecture. We wll show that for every four dstct ostve squarefree tegers (, there are three of them whch are multlcatvely deedet ad are such that the roduct of them s ot a square. It s the clear (sce egatve tegers are ever squares that the assumtos of the result of Heath-Brow hold. Hece at least oe of the four umbers satsfes Art s cojecture. Let {x, y, z, w} be our set of four dstct ostve squarefree tegers (. Frst case: there are three elemets our set whch are deedet. So w.l.o.g. we have z xy ad (x, y. Sce w xy ad (x, y [hece x y or vceversa Lemma 5.3 mles that {x, y, w} are deedet. We have that xyw s ot a square: sce (x, y ad w s squarefree, that would mly w xy. Secod case: o three elemets our set are deedet. What f xyz s a square? Let δ (x, y so that x δx ad y δy. Because z s square-free the that would mly z x y. Sce w z we the coclude that xyw s ot a square. I artcular, Art s cojecture s true for at least oe betwee 2,3,5. 6. Datas I collect some datas whch are avalable at The O-Le Ecycloeda of Iteger Sequeces (OEIS. Datas for (2 mod : The rmes u to.000 for whch (2 mod s a rmtve root, see: htt://oes.org/classc/a0022 5

6 6 (PARI forrme(3, 000, f(zrmroot(2, rt(. 3,5,,3,9,29,37,53,59,6,67,83,0,07,3, 39,49,63,73,79,8,97,2,227,269,293,37, 347,349,373,379,389,49,42,443,46,467,49,509, 523,54,547,557,563,587,63,69,653,659,66,677, 70,709,757,773,787,797 Let a( be the -th term of the sequece above. There s a grahc for (, a( at the age: htt://oes.org/classc/table?a22&fmt5 Ths grahc vsually shows the desty of the set of rmes such that (2 mod s a rmtve root. Art s costat: You ca look at the frst cyhers of the decmal exaso of Art s costat (wth grahc, by followg the lks at the age: htt://oes.org/classc/a Datas for other elemets: Ufortuately, the rogram (PARI forrme(3, 000, f(zrmroot(3, rt(. seems to rt the rmes for whch 3 s the smallest rmtve root, so t excludes those for whch both 2 ad 3 are rmtve roots. See htt://oes.org/classc/a0023 But for ay a we ca wrte: (PARI f(zorder(mod(a,-, rt(. For 6, see: htt://oes.org/classc/a09336 You get the lst of rmes for whch 6 s a rmtve root: (MATEMATICA r6; Select[Prme[Rage[200, MultlcatveOrder[r, # #- & By relacg 6 wth 8, see: htt://oes.org/classc/a09338 The same rogram le commad wth (MATEMATICA, by relacg 6 by 3, should gve you the rmes (betwee the frst 200 rmes such that (3 mod s a rmtve root. Possble comutatoal exercses: Study the reductos modulo the frst ratoal rmes (200 rmes, or rmes u to 000 ad comare the (aroxmated destes for a2 wth the aroxmated Art s costat [decmal exaso. Study the reductos modulo the frst ratoal rmes (200 rmes, or rmes u to 000 ad comare the (aroxmated destes for a8 wth the modfed Art s costat A(3 [you ca deduce the decmal exaso of A(3 from that of A. Refereces [ D. R. Heath-Brow, Arts cojecture for rmtve roots, Quart. J. Math. Oxford Ser. (2 37 (986, o. 45, [2 C. Hooley, O Arts cojecture, J. Ree Agew. Math. 225 (967, [3 K. Irelad ad M. Rose, A classcal Itroducto to moder umber theory, Secod edto. Graduate Texts Mathematcs, 84. Srger-Verlag, New York, 990. xv+389. [4 S. Lag, Algebra, Revsed thrd edto. Graduate Texts Mathematcs, 2. Srger-Verlag, New York, xv+94. [5 R. Murty, Art s cojecture for rmtve roots, Math. Itellgecer 0 (988 o. 4, 59 67, htt:// murty/dex2.html. [6 P. Moree, Art s rmtve root cojecture -a survey - (2004 arxv:math.nt, htt://arxv.org/abs/math/042262, 30 ages.