NEW SOLVABILITY CONDITIONS FOR CONGRUENCE ax b (mod n)

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1 m Mahemaical Publicaios DOI: /mmp Tara M. Mah. Publ ), NEW SOLVABILITY CONDITIONS FOR CONGRUENCE ax b mo ) Šefa Porubský Deicae o he memory of a uforgeable frie Kaz Szymiczek ) Absrac. K. Bibak e al. [arxiv: v1 [mah.nt], March ] prove ha cogruece ax b mo ) has a soluio x 0 wih =gcx 0,)ifa oly if gc a, ) ) =gc b, hereby geeralizig he resul for =1prove by B. Alomair e al. [J.Mah.Crypol ), ] a O. Grošek e al. [ibi ), ]. We show ha his geeralize resul for arbirary follows from ha for = 1 prove i he laer papers. The we shall aalyze his resul from he poi of view of a weaker coiio ha gc a, ) ) oly ivies gc b,. We prove ha give iegers a, b, 1a 1, cogruece ax b mo ) has a soluio x 0 wih iviig gcx 0,) if a oly if gc a, ) ) ivies gc b,. G a u ß revoluioize he umber heory wih he iea of he cogruece i his D. A. He irouce cogruece i he very firs aricle of D. A., a he followig basic resul o he solvabiliy of liear cogruece 1 ax b mo ) 1) which belogs o saar requisies of elemeary umber heory ca be fou i Ars. 29, 30 of D. A. cf. [4]): 2 Lemma 1. If a, b, Z, agca, ) =, he he cogruece 1) is solvable if a oly if b. c 2015 Mahemaical Isiue, Slovak Acaemy of Scieces M a h e m a i c s Subjec Classificaio: Primary 11A07;Secoary 11D04,11D45, 11A25, 11B50. K e y w o r s: liear cogruece, he greaes commo ivisor, umber of soluios. The auhor was suppore by he Gra Agecy of he Czech Republic, Gra # P201/12/2351, he Mobiliy gra 7AMB14SKXXX a he sraegic evelopme fiacig RVO I wha follows Z will eoe he rig of iegers. To simplify he worig a oaio all mouli a ivisors will be always assume o be posiive i wha follows. 2 For he hisory of he relae liear Diophaie equaio ax + y = b cosul, e.g., [3, Chaper II]. 93 Uauheicae Dowloa Dae 7/4/18 5:04 PM

2 ŠTEFAN PORUBSKÝ I is a bi surprisig ha A l o m a i r e a l. [1, Lemma 3.1] oly recely oice he resul give i followig Proposiio 1 which, as i seems, has o appeare explicily i he lieraure before, a which hey use i a cosrucio of a hash fucio. Neverheless, foreruers of his resul coul be alreay fou i various hie forms earlier. Oe such resul ca be fou i Lemma 2 which we shall use i wha follows. Proposiio 1. Give a, b Z, a 0, such ha gca, ) = b, hereexiss asoluioocogruece1) which is coprime o if a oly if b gc, ) =1, or equivalely, if a oly if gcb, ) =gca, ). I [5] a shor proof a a quaiaive exesio of Proposiio 1 is give. I [7] is geeralizaio base o a iempoe aalysis of he semigroup of he resiue class rig moulo ca be fou. I [2, Theorem 3.1] he resul of Proposiio 1 was geeralize o Proposiio 2. Le a, b, 1 a 1 be give iegers. The cogruece 1) has a soluio x 0 wih gcx 0,)= if a oly if gc a, ) b =gc, ). 2) I his oe we shall shorly aalyze he valiiy of Proposiio 2 uer a weaker coiio ha gc a, ) b gc, ) 3) for some iviig gcb, ). The we show ha Proposiio 2 acually follows from Proposiio 1 hereby givig a shorer proof ha he origial oe i [2]. The followig elemeary resul cf. [8, Lemma 2.1], [6, Lemma2.1] or [7, Corollary 4]) will be use i wha follows: Lemma 2. If, x Z a =gc, x), he here exiss a ieger a coprime o such ha x a mo ). Noice ha ecomposiio x = x oes o yiel a represeaio give i previous Lemma 2 i geeral. Take for isace, = 12 a x = Uauheicae Dowloa Dae 7/4/18 5:04 PM

3 NEW SOLVABILITY CONDITIONS FOR CONGRUENCE ax b mo ) The gc12, 9) = 3. Sice gc 9 3, 12) 1,prouc9=3 3isoherepreseaio of x = 9 i he spiri of Lemma 2. From icogrue mo 12 soluios 3, 7, 11 o cogruece 3 a mo 4) oly 7, 11 are coprime o 12. Thus oly represeaios 3 7or3 11 fulfil he saeme of Lemma 2. The ex reformulaio of he Gauß solvabiliy coiio give i Lemma 1 ca be euce i ur Lemma 3. If a, b, Z, he cogruece 1) is solvable if a oly if gca, ) gcb, ). The ecessary coiio of Proposiio 2 ca be moifie i he spiri of he previous Lemma 3 as follows Proposiio 3. Le a, b, Z. Ifcogruece1) has a soluio x 0,he3) hols wih =gcx 0,). P r o o f. Suppose ha x 0 is a soluio o 1) a x 1 wih gcx 1,)=1isa represeaio of x 0 as i is give i Lemma 2. The b a x 1 solves he cogruece ax 1 b mo ). 4) Lemma 3 fiishes he proof. Noice ha a solvabiliy of 1) implies more ha simple ivisibiliy relaio 3). Iee, if x 0 is a soluio of 1) a x 0 = x 1 is a represeaio of his x 0 i he spiri of Lemma 2 wih gcx 1,)=1,hex 1 solves 4) a 4) ogeher wih gcx 1,) = 1 imply ha gc b, ) ivies a, a cosequely gc b, ) also ivies gc ) a,. This shows ha eve he reverse ivisibiliy b gc, ) gc a, ) o ha of 3) is also rue for =gcx 0,)if1)hasasoluiox 0.Ioher wors, if 1) is solvable, he 2) hols wih =gcx 0,). If 1) is solvable, he =gcx 0,) ivies b for every soluio x 0 o 1). However he ecessary coiio gcb, ) for possible caiaes wih =gcx 0,) is oo geerous. For isace, cogruece 18x 12 mo 24) has o soluio ivisible by = 4. The se of soluios o his cogruece is {2, 6, 10, 14, 18, 22}, a eiher of hem is ivisible by 4. Aoher example is cogruece x 2 mo 4) o possessig soluios coprime o 4 which woul correspo o ivisor =1. Now we show ha relaio 3) is also sufficie for he solvabiliy of 1), however wih a weaker biig bewee he s a soluios, as he ex resul shows: 95 Uauheicae Dowloa Dae 7/4/18 5:04 PM

4 ŠTEFAN PORUBSKÝ Proposiio 4. Le a, b, Z. If 3) hols for a gcb, ), he cogruece 1) has a soluio x 0 wih x 0. P r o o f. Coiio 3) implies ha cogruece ax b mo ) is solvable. If x 1 is oe of is soluios, he x 1 solves he origial cogruece 1). I ca be also oe ha a mere solvabiliy of 1) provie 3) hols for a arbirary iviig gcb, ) ca be prove via Lemma 3 i several iffere ways.herearewoofhem: T h e f i r s m e h o. We prove ha if for a iviig gcb, ) coiio 3) is saisfie, he always gca, ) gcb, ). Suppose o he corary ha here is a prime p a a posiive ieger α such ha p α gca, ),p α+1 gca, ) while p α b. Le a = p α a 1, = p α 1,wherep a 1 or p 1.Leb = p β b 1,wherep b 1, β 0 a α>β.thegcb, ) =p β gcb 1,p α β 1 ) a 3) ca be rewrie i he form ) gc p α a 1, pα 1 p β ) b 1 gc, pα 1. If p, hep α ivies he LHS of 3) bu o is RHS. Thus = p γ 1 wih p 1 a 0 <γ<α.the ) gc p α a 1, pα 1 = p α γ gc p γ a p γ 1, ) 1, 1 1 a p β ) b 1 gc, pα 1 = p β γ b1 gc,p α β ) Sice p b 1, p β γ is he highes power of p which ivies he RHS of 3). To fiish he proof cosier he followig wo cases: p 1 : The p α γ is he highes power of p which ivies he LHS of 3), a 3) implies a γ b γ, i.e., α β, wha is impossible. p 1 : I his case he highes power of p iviig he LHS of 3) is p α γ+ω for some posiive ieger ω. The 3) implies α γ + ω β γ, orα + ω β, wha is agai impossible a he solvabiliy coiio gca, ) gcb, ) follows. 96 Uauheicae Dowloa Dae 7/4/18 5:04 PM

5 NEW SOLVABILITY CONDITIONS FOR CONGRUENCE ax b mo ) T h e s e c o m e h o. The greaes commo ivisor possesses he followig muliplicaive propery gcah, bk) = gca, b)gch, k)gc a gca, b), Cosequely for every we have gca, ) =gc a 1, ) ) k gc gch, k) =gc a, ) gc b gca, b), ) h. 5) gch, k) ) a gc ), a,. 6) Sice also ivies b, he he firs gc o he RHS ivies gc b, ) ue o 3) while he seco oe ivies. Cosequely he RHS of 6) ivies heir muual prouc gcb, ), ha is gca, ) gcb, ). There follows from he proofs above ha parameer ivies gcx 0,), where x 0 is a soluio o 1). This gives he followig compaio o Proposiio 2. Theorem 1. Le a, b, 1 a 1 be give iegers. The cogruece 1) has a soluio x 0 wih gcx 0,) if a oly if 3) hols wih his. We show ow ha Proposiio 1 implies Proposiio 2. Cogruece 1) is solvable a a x 0 wih gcx 0,)=, x 0 = x 1 wih gc ) x 1, = 1 is is soluio, if a oly if 4) has a soluio x1 coprime o is moulus. Proposiio 1 shows ha his ca happe if a oly if gc a, ) b =gc, ) as Proposiio 2 claims. All above resuls remai rue verbaim wihou ay chage of argumes i a arbirary commuaive pricipal ieal omai. Typical example besies he rig of raioal iegers is he rig of Gaußia iegers a + bi wih a, b Z, ormore geerally, he rigs of algebraic iegers wih he class umber 1. Fially, le us a ha i he case of coprime soluios i is prove i [5] ha he umber of icogrue coprime soluios is give by he followig rule: If gca, ) =gcb, ) =, he here are exacly δ ϕδ) icogrue soluios of 1) coprime o, whereδ is he larges ivisor of wih gcδ, )=1,aϕm) is he umber of iegers k, 1 k m, coprime o m. O he oher sie, i [2] i is prove ha he umber of icogrue soluios x 0 moulo o 1) wih =gcx 0,)isgiveby ϕ ) ϕ ) = 1 1 p gc b, ) p p ). 7) 97 Uauheicae Dowloa Dae 7/4/18 5:04 PM

6 ŠTEFAN PORUBSKÝ This gives for he umber of coprime icogrue soluios moulo o 1) he formula ϕ) ϕ ). 8) Tha he umbers for coprime soluios give by hese wo iffere formulae coicie, i.e., ha ϕ) ϕδ) = δ ϕ ) ca be show as follows: The equaliy above reuces o Here, δ = ϕ) ϕ ). ϕδ) a δ are coprime, a herefore ) ) δ ϕ ϕδ) =ϕ. Sice a δ have he same prime ivisors, he formula ϕm) =m 1 1 ) ϕ) implies ha p ϕ ) = = p m ϕδ) δ δ, as i is claime. Fially, oice ha also he umber of soluios x 0 o 1) give i [2] follows from he formula givig he umber of coprime soluios. Really, as we have meioe above here is oe o oe correspoece bewee icogrue soluios x 0 moulo o 1) wih = gcx 0,)a icogrue soluios x 1 moulo o 4) saisfyig coiio gc ) x 1, =1. Relaio 8) implies ha he umber of he laer oes is ϕ ) ϕ ) which is jus 7). gc b, ) Ackowlegeme. The auhor haks Professor O. G r o š e k for callig his aeio o mauscrip [2] a Professor O. S r a u c h for simulaig iscussios. refereces [1] ALOMAIR, B. CLARK, A. POOVENDRAN, R.: The power of primes: securiy of auheicaio base o a uiversal hash-fucio family, J.Mah.Crypol ), [2] BIBAK, K. KAPRON, B. M. SRINIVASAN, V. TAURASO, R. TÓTH, L.: Resrice liear cogrueces a a auheicae ecrypio scheme, arxiv: v1 [mah.nt], March 5, Uauheicae Dowloa Dae 7/4/18 5:04 PM

7 NEW SOLVABILITY CONDITIONS FOR CONGRUENCE ax b mo ) [3] DICKSON, L. E.: Hisory of he Theory of Numbers. Vol. II. Diophaie Aalysis. Caregie Isiuio of Washigo, New York, [4] GAUSS, C.-F.: Disquisiioes Arihmeicae. Trasl. from he Lai by Arhur A. Clarke, Rev. by William C. Waerhouse, wih he help of Corelius Greiher a A. W. Grooeors. Repri of he 1966 e.). Spriger-Verlag, New York, [5] GROŠEK, O. PORUBSKÝ, Š.: Coprime soluios o ax b mo ), J. Mah. Crypol ), [6] LAŠŠÁK, M. PORUBSKÝ, Š.: Ferma-Euler heorem i algebraic umber fiels, J. Number Theory ), [7] PORUBSKÝ, Š.: Iempoes a Cogruece ax b mo ). i: From Arihmeic o Zea-Fucios. Number Theory i Memory of Wolfgag Schwarz. Jürge Saer, Jör Seuig a Rasa Seuig, Es.), Spriger Verlag, 2016 o appear). [8] SCHWARZ, Š.: The role of semigroups i he elemeary heory of umbers, Mah. Slovaca ), Receive November 22, 2015 Isiue of Compuer Sciece Acaemy of Scieces of he Czech Republic Po Voáreskou věží Praha 8 Libeň CZECH REPUBLIC sporubsky@homail.com 99 Uauheicae Dowloa Dae 7/4/18 5:04 PM