On the Infinitude of Covering Systems with Least Modulus Equal to 2

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1 Annals of Pure and Applied Mahemaics Vol. 4, No. 2, 207, ISSN: X (P), (online) Published on 23 Sepember DOI: hp://dx.doi.org/ /apam.v4n2a3 Annals of On he Infiniude of Covering Sysems wih Leas Modulus Equal o 2 7 Arlozorov Sree, Tel Aviv , Israel anb7@nevision.ne.il Received 9 Sepember 207; acceped 22 Sepember 207 Dedicaed o he memory of he lae Prof. Paul Erdös friend and menor Absrac. A finie se of residue classes a i (mod n i ) wih < n < n 2 < < n s is called a covering sysem of congruences if every ineger saisfies a leas one of he congruences x a i (mod n i ). An example is he se {0 (mod 2), (mod 3), 3 (mod 4), 5 (mod 6), 9 (mod 2)}. A covering sysem all of whose moduli are odd called an odd covering sysem is a famous unsolved conjecure of Erdös and Selfridge. In his paper, we esablish ha here exis infiniely many even covering sysems in which he leas modulus is 2 and all oher moduli are even. In each such even covering sysem, he number of he moduli and heir prime facors are deermined. Moreover, we consruc a covering sysem wih nine moduli, he smalles modulus is 2, and he lcm of he moduli is divisible by only he primes 2 and 5. Wih he smalles modulus 2, his is an aemp in he direcion of consrucing covering sysems none of whose moduli is a produc of he prime 3. Keywords: Congruences, covering sysems, exacly covering sysems AMS Mahemaics Subjec Classificaion (200): B25, A07. Inroducion The concep of Covering sysems was inroduced and developed by Erdös. Many auhors and numerous aricles have been wrien on his ineresing and non-rivial opic, and quie a wide lieraure exiss on he subjec. We begin wih some definiions and noaions. Definiion.. The sysem of congruences x a i (mod n i ), < n < n 2 < < n, 0 a i < n i () is called a Covering Sysem (abbreviaed CS), if every ineger saisfies a leas one of he congruences (). 307

2 Definiion.2. The sysem of congruences x a i (mod n i ), < n n 2 n k, 0 a i < n i (2) is called an Exacly Covering Sysem (abbreviaed ECS), if every ineger saisfies exacly one of he congruences (2). A sysem of congruences corresponding o any of he above wo definiions, will be denoed as a se of ordered pairs of inegers of he form (a, n), where a is he residue class and n is he modulus. The leas common muliple of he moduli in a CS/ECS will be denoed by LCM. The smalles and simples example of a CS is he following se of ordered pairs {(0, 2), (0, 3), (, 4), (5, 6), (7, 2)}. (3) Hereafer, we shall refer o he se (3) as he basic CS. A necessary condiion [7] on he moduli n, n 2,..., n of a CS is = i n i >. Definiion.3. An ECS, in which for every value m here are a mos M moduli which are equal o m, will be called an ECS(M). I is well known [8, 7], ha he moduli n n 2 n k of any ECS saisfy: = ( n i, n j ) > (4) k = i n i and n k- = n k. (5) Condiion (5) implies moreover ha M = is impossible. The simples examples of ECS's are he following ses {(0, 2), (, 2)}, {(0, 2), (, 4), (3, 4)}, {(0, 2), (, 4), (3, 8), (7, 8)}. (6) Hereafer, we shall refer o ses described in (6) as basic ECS's. The moduli of each se in (6) saisfy he condiions in (4) and (5). The basic ECS's in (6) are also known in he lieraure as Naural ECS's (abbreviaed NECS's). As for ECS's(M) we also have accordingly NECS's(M). Many resuls on ECS's and NECS's may also be found in [, 2, 3, 4, 0, 6, 7, 8, 20, 2]. We shall now cie some known resuls, in paricular hose concerned wih he leas modulus n of a CS. Erdös asked wheher here are CS's in which n R for arbirary R. Over he las 60 years he value R increased. In 968, Churchhouse [6] using a compuer found CS's wih n = 2, 3,, 9. In 97, Krukenberg [3] gave examples of CS's for all values n up o 8 inclusive. 308

3 On he Infiniude of Covering Sysems wih Leas Modulus Equal o 2 Choi [5] found a CS wih n = 20. The curren record belongs o Owens [5] wih n = 42. Hough showed he amazing resul [2] ha n is a mos 0 6. For his achievemen he won he 207 David P. Robbins prize. Oher resuls may also be found for insance in [9, 3, 6, 9, 2]. 2. On he infiniude of even CS's when he leas modulus is 2 In his secion, we exhibi a connecion beween an ECS and a CS in each of which he smalles modulus is 2. Firs, his is illusraed in wo self-conained examples, namely Example 2. and Example 2.2 using wo of he basic ECS's in (6) and he basic CS in (3). Secondly, he paern described in hese examples enables us o esablish he general case, i.e., he exisence of infiniely many even CS's wih smalles modulus 2. This is done in Theorem 2.. Example 2.. The basic ECS in (6) whose moduli are {2, 4, 4} combined wih he basic CS in (3) whose moduli are {2, 3, 4, 6, 2} yield he se of seven moduli { 2, 4, 4 2, 4 3, 4 4, 4 6, 4 2 }, 7 which are even, disinc and i= >. The respecive CS is ni {(0, 2), (3, 4), (5, 8), (5, 2), (9, 6), (9, 24), (, 48)}. Example 2.2. The basic ECS in (6) whose moduli are {2, 4, 8, 8} combined wih he basic CS in (3) whose moduli are {2, 3, 4, 6, 2} yield he se of eigh moduli { 2, 4, 8, 8 2, 8 3, 8 4, 8 6, 8 2 }, 8 which are even, disinc and i= >. The respecive CS is ni {(0, 2), (3, 4), (, 8), (5, 6), (3, 24), (29, 32), (45, 48), (77, 96)}. Example 2. and Example 2.2, clearly show he paern which connecs a basic ECS wih he basic CS. The general case will now be esablished for all basic ECS's of he form described in (6) combined wih he basic CS in (3). Theorem 2.. For each and every value, he se of ordered pairs { (0, 2), (, 4), (3, 8), (7, 6),..., (2 - -, 2 ), (2, 2 ) } (7) represens a basic ECS. Le be any fixed value. Then: (i) The se { 2, 4, 8, 6,..., 2, 2 2, 2 3, 2 4, 2 6, 2 2 } (8) is a se of moduli of a CS. (ii) There are infiniely many CS's whose moduli saisfy (8). (iii) All he moduli in (8) are even and disinc. (iv) In (8), he LCM is 2 2 whose prime divisors are 2 and

4 (v) The number k of he moduli in ( 8) is equal o k = + 5. Proof: For (i), recall from (3) he moduli { 2, 3, 4, 6, 2 } of he basic CS whose LCM = 2. I is noed ha if he inegers 2 are covered, hen all he inegers are covered, and he se of moduli is a CS. As for he moduli 2, 4 in (3), he moduli 2 2, 4 2 in (8) and heir values a i yield up o he new LCM = 2 2 respecively six and hree inegers all of which are disinc wih no overlaps. The oher hree moduli in (8), namely 3 2, 6 2 and 2 2 and heir appropriae values a i, each yields exacly one ineger up o 2 2, whereas all oher obained inegers are overlaps. Thus, all welve inegers are covered up o 2 2 as required, and (8) is a se of moduli of a CS as assered. As for (ii), each value in (7) yields a basic ECS. Hence, here are infiniely many basic ECS's. Then, for each and every value here exiss a CS. Thus, here exis infiniely many CS's whose moduli saisfy (8). The saemens in (iii), (iv) and (v) follow direcly from (8). This complees he proof of Theorem 2.. The following Corollary 2. follows from Theorem 2.. Corollary 2.. The sum of he reciprocals of he five moduli in (3) is 5 4 i= =. The sum of he reciprocals of he ( + 5) moduli in (8) is equal o n 3 i = + = = + i= ni For arbirarily large, i follows ha he reciprocals of he moduli of he CS in (8) have a sum which is as close o as we wish, bu is never equal o. Remark 2.. In Theorem 2., we have used he basic CS having he five moduli 2, 3, 4, 6, 2. As he leas modulus increases, he larger are he number of he moduli and so are heir prime facors. Evidenly, for he purposes of Theorem 2., any known CS will suffice insead of he basic CS. However, consrucing a CS union of a basic ECS and he basic CS, has is advanages primarily in (i) he smalles number of obained moduli, and (ii) heir larges prime divisor which is p = On a CS whose leas modulus is 2 and he LCM is In he lieraure, all CS's wih leas modulus 2, have LCM's which are divisible by p = 3. Hough and Nielsen have shown ha every CS has a modulus divisible by eiher 2 or 3. In his secion, we exhibi Example 3. i.e., a CS of nine moduli conaining he leas modulus 2, he modulus 5, and he LCM = 80. This is he firs CS none of whose moduli is divisible by p =

5 On he Infiniude of Covering Sysems wih Leas Modulus Equal o 2 Example 3.. The following se of nine ordered pairs is a CS in which excep for, he nine moduli are all he divisors of 80. {(, 2), (0, 4), (0, 5), (2, 8), (6, 0), (6, 6), (4, 20), (22, 40), (78, 80)}. A CS wih leas modulus 2 and he only odd modulus 5 does no exis when k < 9. Hence, Example 3. is a CS whose number of mosuli k = 9 is minimal. For any oher CS of he same naure, i follows ha k > 9. Moreover, as menioned earlier, Example 3. now implies ha here exiss a leas one CS which does no conain a modulus divisible by 3, bu raher, moduli divisible by 2. Final Remark. In view of Example 3., we may now presume ha for primes p 7, here exiss a CS conaining he leas modulus 2 and he LCM = 2 r p. REFERENCES. N.Burshein, Exacly covering sysems of congruences, Ph.D. hesis, Universiy of Tel Aviv, N.Burshein, On naural exacly covering sysems of congruences having moduli occurring a mos M imes, Discree Mahemaics, 4 (3) (976) N.Burshein, On naural exacly covering sysems of congruences having moduli occurring a mos wice, Journal of Number Theory, 8 (3) (976) N.Burshein and J.Schönheim, On exacly covering sysems of congruences having moduli occurring a mos wice, Czechoslovak Mahemaical Journal, 24 (3) (974) S.L.G.Choi, Covering he se of inegers by congruence classes of disinc moduli, Mah. Comp., 25 (97) R.F.Churchhouse, Covering ses and sysems of congruences, Compuers in Mahemaical Research, Norh-Holland, Amserdam, 968, pp P.Erdös, On a problem concerning congruence sysems, Ma. Lapok 3 (952) P.Erdös, Számelmélei megjegyzések iv, Ma. Lapok 7 (962) P.Erdös and R.L.Graham, Old and new problems and resuls in Combinaorial Number Theory, Monographie n o 28 de ĽEnseignemen Mahémaique Universié de Genève, Imprimerie Kundig Genève, A.S.Fraenkel, A characerizaion of exacly covering congruences, Discree Mahemaics 4 (973) D.J.Gibson, A covering sysem wih leas modulus 25, Mah. Comp., 78 (266) (2009) B.Hough, Soluion of he minimum modulus problem for covering sysems, Annals of Mahemaics, 8 () (205)

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