Pier Franz Roggero, Michele Nardelli, Francesco Di Noto

Transcription

1 Vrsio.0 9/06/04 Pagia di 7 O SOME EQUATIOS COCEIG THE IEMA S PIME UMBE FOMULA AD O A SECUE AD EFFICIET PIMALITY TEST. MATHEMATICAL COECTIOS WITH SOME SECTOS OF STIG THEOY Pir Fra oggro, Michl ardlli, Fracsco Di oto Abstract: I this papr w ocus atttio o som quatios cocrig th ima s prim umbr ormula ad o th bhavior o a scur primality tst. Furthrmor, w hav dscribd also som mathmatical coctios with som sctors o strig thory.

2 Vrsio.0 9/06/04 Pagia di 7 Idic:. PIMALITY TEST...3. ALGOITHM TO KOW IF A UMBE p IS PIME POOF O SOME EQUATIOS COCEIG THE IEMA S PIME UMBE FOMULA..7. EFEECES... 6

3 Vrsio.0 9/06/04 Pagia 3 di 7. PIMALITY TEST A umbr is prim i: p 6k ± with k,,... itgr or all prims cpt th umbrs, 3 Which is quivalt to say: p (mod 6) or p (mod 6) or all prims cpt th prims ad 3 I p is prim is valid o o th two cogruc rlatios. Howvr, th vic vrsa is ot tru, g, p, 9 is valid th irst rlatio but thy ar ot prim umbrs -> this is a cssary but ot suicit coditio so that p a prim umbr. Th cogruc rlatios ar quivalt to limiat all multipls o 3 ad cosidrig that th prim umbrs ar odd ar thus limiatd all odd multipls o 3. callig th rul so that a umbr is divisibl by 3, w hav: a umbr is divisibl by 3 i th sum o its digits (i o o its digits is 9, it ca b cosidrd qual to 0 i th sum) is a multipl o thr. I cas th rsult would b gratr tha 9, add up th two or mor digits o th rsult ad dtrmis whthr thy ar multipl (or dividd) by thr. Eampl: Th sum o th digits o th umbr 3 is

4 Vrsio.0 9/06/04 Pagia 4 di 7 6, so 3 is divisibl by thr. I th cas o 79, howvr, th sum is to b. Giv that 3, 79 is also divisibl by thr. Aothr cssary but ot suicit coditio so that p is prim is that th last digit ds i, 3, 7 or 9 ( is th cludd bcaus divisibl by, but uortuatly ot th 9 7 * 3). W, thror, hav that th prim umbrs th sum o thir digits umbrs ar:,, 4,, 7 ad 8 ad vr: 0, 3 ad 6 would othrwis b divisibl by 3 Atr ths iitial cosidratios w s how simpl it is possibl to crat a scur ad icit primality tst always valid ad th ida as it has dvlopd. So w choos a as th smallst compl umbr with ral ad imagiary itgr part: aj th p is prim i th ollowig rlatio o cogruc is valid:

5 Vrsio.0 9/06/04 Pagia di 7 a p- p- (mod p) or a p- must rsult i a itgr umbr or a compl umbr with oly th imagiary part, othrwis p is ot alrady a prim umbr ad is o logr cssary to apply th cogruc rlatio. Th modulo must b tak to always hav a positiv itgr valu. By choosig a w hav that p- is always a itgr or pur imagiary itgr. This is quivalt to say that: (p-)/ p- (mod p) I th rsult is qual to th othig ca b said i p is prim or ot. So should w cotiu th ivstigatio as choosig a 3, th t prim umbr. 3 (p-)/ p- (mod p) ad v i i this cas th rsult is qual to w must slct th t prim umbr a, ad so o. From th tab. w s th irst ampls:

6 Vrsio.0 9/06/04 Pagia 6 di 7 TAB. prim p (j)^(p-) mod(^(( Absolut Valu p-)/), p) mod(3^(( p-)/), p) 3 0, -4, , , , , 8 3 0, , , , , , , , , , mod(^(( p-)/), p) , , , mod(7^(( p-)/), p)

7 Vrsio.0 9/06/04 Pagia 7 di , , IS OT PIM E 6 IS OT PIM E 90 IS OT PIM E 047 IS OT PIM E 8 IS OT PIM E , thc 8 IS OT PIME 67, thc 34 IS OT PIME 44, thc 6 IS OT PIME 86, thc 90 IS OT PIME 6, thc 047 IS OT PIME

8 Vrsio.0 9/06/04 Pagia 8 di 7 W ot that i ordr to prov that th umbr p 7 is a prim umbr, w must go up to 7 (p-)/ p- (mod p). W hav: (mod 7). Istad th umbr 34 which is a psudoprim umbr that ivalidats th Frmat's littl thorm is maskd with this mthod. I act, w hav: 3 (p-)/ p- (mod p) (mod 34) ot so givig as a rsult ithr or 340, but th valu 67 w ca crtaily say that 34 is ot a prim umbr, ad as w all kow 34 * 3.

9 Vrsio.0 9/06/04 Pagia 9 di 7. ALGOITHM TO KOW IF A UMBE p IS PIME Basd o th abov cosidratios hr is th algorithm or dtrmiig i a umbr p is or is ot a prim umbr. ) W actori th umbr (p-) ) w choos as th basis a th smallst prim umbr that is ot a prim actor o th umbr (p-). So w ca vr choos as basis a. 3) w apply th ormula o cogruc: a^((p-)/) (mod p) r i th rsult is: rp- th p is a prim umbr <r<p- th p is ot a prim umbr r th w caot yt say whthr p is or is ot a prim umbr, but w must choos as th basis a th t coscutiv prim umbr that is ot obviously a prim actor o th umbr (p-). pat th stp 3.

10 Vrsio.0 9/06/04 Pagia 0 di 7 Eampl : 40? (40-)400^4*^ w caot choos as th basis a th umbrs ad. So th irst prim umbr that w ca choos as bas a is th umbr a 3 3^00 (mod 40) 400 th th umbr 40 is a prim umbr. Eampl : 6? (6-)60^4**7 w caot choos as th basis a th umbrs, ad 7. So th irst prim umbr that w ca choos as bas a is th umbr a 3 3^80 (mod 6) 44 th th umbr 6 is ot a prim umbr but is a composit umbr, 63**7. Eampl 3 : 839? (839-)838*49

11 Vrsio.0 9/06/04 Pagia di 7 w caot choos as th basis a th umbr a, 49. So th irst prim umbr that w ca choos as bas a is th umbr a 3 3^49 (mod 839) th w cotiu with th t coscutiv prim umbr util w gt ot mor as a rsult r. ^49 (mod 839) 7^49 (mod 839) ^49 (mod 839) 838 th th umbr 839 is a prim umbr.

12 Vrsio.0 9/06/04 Pagia di 7 3. POOF (a^((p-)/)-)*(a^((p-)/)) a^(p-)- sic: a^((p-)/) (mod p) p- (a^((p-)/)-)*(a^((p-)/)) (mod p) ((p-)*p (mod p) (p^-p) (mod p) 0 ad th ollows th Frmat's littl thorm: a^(p-) (mod p) From th ivariac with rspct to potiatio w ca also say: (a^((p-)/))^ (mod p) (p-)^ ad th rgais: a^(p-) (mod p)

13 Vrsio.0 9/06/04 Pagia 3 di 7 Wh p is a prim umbr th ormula o cogruc a^((p-)/) (mod p) r givs rsult always r or rp- or ay bas a chos with a ay prim umbr, v i w choos as th basis a th irst prim actor o (p-), which should ot b valid, howvr, to dtrmi i p is or is ot a prim umbr. I w choos as basis to a th th algorithm works up to th valu o p 377 ad that i ct is a psudo-prim. ^((377-)/) (mod 377) 376 ad th w would b ld to bliv that p 377 is a prim umbr. So or this raso w should ot choos as th basis a th prim actors o (377-) 376 ^*3^*7*3 ad th, 3, 7 ad 3 should ot b slctd. Obviously w do ot cosidr multipl actors, such as 9, bcaus thy ar vr prims, but w must choos oly ad oly th idividual actors. I r othig ca b said i p is or is ot a prim umbr. Wh p is ot a prim umbr w must choos as th basis a th irst umbr that is ot a prim actor o (p-). This is bcaus i w choos a prim actor o (p-) th th algorithm rvals psudoprims as i th prvious ampl. Cosqutly wh th ormula o cogruc a^((p-)/) (mod p) r givs as rsult rp- with th cosidratio that w must ot chos or th basis a th prim actors o (p-), othrwis w may hav also as rsult r p-, which is ot good. So w must choos or th umbr p 377 as a bas a, that is th irst prim umbr as a valid basis. W hav sic Eulr's thorm:

14 Vrsio.0 9/06/04 Pagia 4 di 7 a^((p-)*c) (mod p) (a^(p-))^c (mod p) with <a<p ad c ad p prim sic 3779*3 (^8)^8*^4 (mod 9) ^4 (mod 9) (^)^4*^70 (mod 3) ^70 (mod 3) 44 Th ^638 (mod 377) o logr givs as rsult r 376 ad th psudo-prim 377 is immdiatly maskd.

15 Vrsio.0 9/06/04 Pagia di 7 Wh it happs that or all th basis a to mak choics that ar co-prim with rspct (p-) w wr to gt th ormula o cogruc is always: a^((p-)/) (mod p) ad w hav as a rsult or all th basis chos, w ar i th prsc o a vry strog psudo-prim umbr, which blogs to th amily o compouds o Carmichal umbrs. Cosqutly i ths spciic cass th umbrs p ar vr prims. A ampl is th umbr 79: 79 7*3* ^6*3^3 W hav a^864 (mod 79) or a, 7,, 3, 7, 9,,... (but also or ad 3, which ar howvr ot ligibl bass) ad psudo-prim 79 is immdiatly maskd. Oly with this tst w ca say that 79 is ot th a prim umbr, i act w vr hav r78. Obviously it was ot possibl to say aythig with Frmat's littl thorm.

16 Vrsio.0 9/06/04 Pagia 6 di 7 OBSEVATIO: Oly i th ormula o cogruc: a^((p-)/) (mod p) r w hav r p- it is i th prsc o a prim umbr, ad i its cycl, or all th bass to b valid, thr is always rp- or r. This is a coditio "cssary ad suicit". Th algorithm works with th bas always qual or ampl a, but so w will obtai a probabilistic tst, as show by th tab.. I this algorithm howvr w ca vr choos as basis a. W ca choos so that th tst would work i or a i th vast majority o cass, cludig th umbrs with (p-) which hav as a actor. Obviously i w icras th bas a th tst bcoms mor scur rom th probabilistic poit o viw, but oly by choosig th appropriat basis w ca b sur to 00 % i p is or is ot a prim umbr. So w ca also mask th Carmichal umbrs ad th whol uivrs o psudo-prims (s th abov ampls that tst th algorithm) Th tst AKS Agrawal - Kayal - Saa is vry ic ad icit but crtaily vry mor complicatd tha this ad computatioally much slowr (a turtl tha a chtah...)

17 Vrsio.0 9/06/04 Pagia 7 di 7 A ampl a bit mor complicatd which shows th spd o th algorithm: p *3*33 p(p-) *3**3*3 w ca vr choos a, 3,, 3 ad 3. So th irst valid basis a7 W hav sic Eulr's thorm: (7^0)^3848 * 7^ mod 0 (7^30)^834 * 7^6 mod 3 (7^330)^7 * 7^ mod 33 7 Th 7^3848 (mod 47697) o logr givs as rsult r ad th umbr is immdiatly maskd as composit ad ot prim. 4. O SOME EQUATIOS COCEIG THE IEMA S PIME UMBE FOMULA Prim umbr Thorm W hav th ollowig ormula: ( ) lim π / log. (.)

18 Vrsio.0 9/06/04 Pagia 8 di 7 To prov this w hav to itroduc a w objct, th Mobius uctio µ ( ). This is did as µ ( ) ( ) k i is th product o k distict prims, ad µ ( ) 0 othrwis. Thus µ oscillats btw -, 0, ad. Its rlatioship to th ima ta uctio ca b s rom th ormula µ ( ) p.. prim or ( ) >. I othr words, w hav that p (.) µ ( ) ζ ( ) (.3) or ( ) >. Lmma W hav > γ ( ) d π (.4) Th lt-had sid is lss tha or qual to π sup γ > ( ). (.) Writ. Obsrv that i γ, th iy, ad hc ( ). (.6)

19 Vrsio.0 9/06/04 Pagia 9 di 7 Also w hav, whil (sic ( ) µ or all ) ( ) > > a da. (.7) Lmma W hav ( ) ( ) i d π π µ π γ. (.8) W itroduc th cotour γ which is th othr hal o th circl travrsd by γ aticlockwis, i.. ( ) it t : γ, / 3 / π π t. Obsrv that ( ) is aalytic vrywhr, ad so by th Cauchy itgral ormula: ( ) ( ) ( ) i i d γ γ µ π π. (.9) So, w hav that ( ) d π π γ. (.0) Oc agai, w stimat th lt-had sid by ( ) sup γ π. (.) As bor w hav ad, whil ( ). (.)

20 Vrsio.0 9/06/04 Pagia 0 di 7 I 0, th th itgral tst givs da a, (.3) whil i < 0 th th itgral tst givs Puttig this all togthr w obtai da a. (.4) sup ( ) γ. (.) Combiig Lmma ad Lmma, w s that γ ( ) d πi ( ) µ 4π π. (.6) With rgard th uctioal quatio or ay (ot a itgr), w hav ζ ( ) π cos Γ ζ ( π ) ( ) ( ). (.7) Th uiquss o aalytic cotiuatio it suics to prov this wh ( ) < 0 (say). I which cas th lt-had sid is just. call that Γ ( ) ζ ( ) πi( ) ( ) γ ε, r ( ) w Log w 0 dw (.8) γ ε, r whr is a clockwis cotour goig aroud th positiv ral ais. Writig πi, it thus suics to show that i( ) π as

21 Vrsio.0 9/06/04 Pagia di 7 ( ) Log0w ( ) Log0w πi ( π ) ( ) dw γ w ε, r π cos. (.9) w Th uctio is ot aalytic o th positiv ral ais, but is mromorphic vrywhr ls, with simpl pols at itgr multipls o πi, ad a rsidu o ( ) Log ( ) / ( ) 0 πi πi π wh w πi or som positiv, ad a rsidu o ( ) Log 0 πi 3πi( ) / ( π ) wh w πi. For ay > 0 γ, lt b th cotour cosistig o th horiotal ray rom ( ) πi to ( ) π ( ) πi, th vrtical li sgmt rom ( ) π ( ) πi to ( ) π ( ) πi, ad th horiotal ray rom ( ) π ( ) πi to ( ) πi γ γ. Th rgio o spac btw r ad ε, cotais th pols πi or (cludig 0), ad so by th rsidu thorm ( ) Log w ( ) 0 0 πi( ) / 3πi ( ) dw dw πi ( π) ( π) w w γ ε, r γ W ow claim that th γ li sgmt o γ Log w ( ) Log 0w i( ) Arg0 ( w) w O( ) w w is gativ ad so /. (.0) itgral gos to ro as gos to iiity. O th vrtical w w, th poit is that is vry small ad so is boudd, whil sic ( ) < 0. O ithr o th two horiotal rays, is agai boudd, ad agai w ca argu to show that ths itgrals dcay vry quickly i. Takig limits w thus hav: ( ) Log w 0 dw πi γ w ε, r πi( ) / 3πi( ) ( π) ( π) /. (.) Th atural logarithm (or pria logarithm) o a umbr is its logarithm to th bas, whr is a irratioal ad trascdtal costat approimatly qual to Th atural logarithm o is th powr to which would hav to b raisd to qual. For ampl, l(7.) is , bcaus Th atural

22 Vrsio.0 9/06/04 Pagia di 7 log o itsl, l(), is, bcaus, whil th atural logarithm o, l(), is 0, sic 0. Th atural logarithm uctio, i cosidrd as a ral-valud uctio o a ral variabl, is th ivrs uctio o th potial uctio, ladig to th idtitis: ( ) l ( ) l. i > 0 W ot that th valu o is vry ar to th rsult o th ollowig ormula: 8 8 ( ) 6,67 Φ 3/ 0, ,76074, , (a) Φ, (whr i.. th auro ratio) that is a rqucy o th auro musical systm calculatd rom C. Lag ad G. Bii (s r.). Thc, w ot that thr is a strog rlatioship, a mathmatical coctio btw ad Φ by th ormula (a) W rmmbr that it is wll-kow that th sris o Fiboacci s umbrs hibits a ractal charactr, whr th orms rpat thir similarity startig rom th rductio actor / φ 0,68033 (Pitg t al. 986). Such a actor appars also i th amous ractal amauja idtity (Hardy 97): 0,68033 / φ ( q) 3 p ( t) dt 4 / ( t / ) t q 0, (.) ad π Φ 3 0 ( q) 3 p ( t) dt 4 / ( t / ) t q 0, (.3)

23 Vrsio.0 9/06/04 Pagia 3 di 7 Φ whr. Furthrmor, w rmmbr that π ariss also rom th ollowig idtitis (amauja s papr: Modular quatios ad approimatios to π Quartrly Joural o Mathmatics, 4 (94), ): π ( )( 3 3) log 30, (.3a) ad π 4 log (.3b) From (.3) ad (.3b),w hav: 3 π Φ ( q) q 0 3 ( t) dt p / 4 / ( t ) t log (.4) With rgard th umbr 4 ( 4 / ad 3 4 8) thy ar rlatd to th mods that corrspod to th physical vibratios o th bosoic strigs by th ollowig amauja uctio:

24 Vrsio.0 9/06/04 Pagia 4 di 7 cosπtw' π w' d 0 4ati log coshπ πt w' 4 ( ) φw' itw' log t w'. Th quatio (.4) cocrig π cotais also Φ ad show th rlatioship btw ths two udamtal costat. From (A ) o th papr (), w hav that: / / ( ) ( ) ( ) ( ) u / u / u u u / t t u sih u dt du du u u t (.) or t < u ad From this quatio, w obtai also that:. du π ta u sih ( π) ( u) sih π 0 u ta sih ( π). (.6) Thc, rom (.4) w hav th ollowig rlatioship: 3 π Φ ( q) q 0 3 ( t) dt p 0 / 4 / ( t ) t ( ) log sih u 0 u ta π sih ( ). (.7)

25 Vrsio.0 9/06/04 Pagia di 7 Thc, w hav that th q. (.) ca b coctd with (.7). Idd, w hav: ( ) Log w 0 dw πi γ w ε, r πi( ) / 3πi( ) ( π) ( π) /, (.8) whr 3 π Φ ( q) q 0 3 ( t) dt p 0 / 4 / ( t ) t ( ) log sih u 0 u ta π sih ( ). Palumbo (00) ha proposd a simpl modl o th birth ad o th volutio o th Uivrs. Palumbo ad ardlli (00) hav compard this modl with th thory o th strigs, ad traslatd it i trms o th lattr obtaiig: ( G G ) ( φ) 6 µρ νσ µν d g g g Tr µν ρσ g µ φ νφ 6πG 8 Φ 0 ( ) Φ Φ ( ) / µ ~ κ0 d G 4 µ H 3 Tr F ν κ 0 0 g0, (.9) A gral rlatioship that liks bosoic ad rmioic strigs actig i all atural systms. Th itroductio o (.) ad (.3) i (.8) provids:

26 Vrsio.0 9/06/04 Pagia 6 di 7 ( ) ( ) ( ) Φ φ ρσ µν νσ µρ G G Tr g g t dt t t q G g d q 8 ) ( p 3 ) ( / / 6 ] µ φ ν φ µν g Φ q t dt t t q 0 4 / / 0 ) ( ) ( p 3 ) ( 0 3 κ ( ) [ ν µ µ κ Tr g t dt t t q H G d q / / 3 / 0 ) ( ) ( p 3 ) ( 0 3 ~ 4 Φ Φ Φ Φ ( ) ] F, (.30) which is th traslatio o (.9) i th trms o th Thory o th umbrs, spciically th possibl coctio btw th amauja idtity ad th rlatioship cocrig th Palumbo-ardlli modl.. EFEECES - Wikipdia () Pasqual Cutolo Ua ota sullo sviluppo dlla drivat di ordi ( itro positivo) dll uioi trigoomtrich P() Ta(), C() Sc(). Cosidraioi

27 Vrsio.0 9/06/04 Pagia 7 di 7 d ossrvaioi Fiuggi, Agosto 00