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1 Chapter 5 O A Cojecture Of Erdíos Proceedigs NCUR VIII è1994è, Vol II, pp 794í798 Jeærey F Gold Departmet of Mathematics, Departmet of Physics Uiversity of Utah Do H Tucker Departmet of Mathematics Uiversity of Utah I this paper we preset some prelimiary results o a cojecture by Paul Erdíos ë1,2,5ë cocerig coverig sets of cogrueces A coverig set cosists of a æite system of cogrueces with distict moduli, such thatevery iteger satisæes as a miimum oe of the cogrueces A iterestig cosequece of this cojecture is the depedece of the solutio o abudat umbers; a abudat umber is a iteger whose sum of its proper divisors exceeds the iteger Complemetary Sets Deæitio 1 If a ad b are itegers, the a mod b = fa; a æ b; a æ 2b;:::g : Deæitio 2 If a 1,a 2,:::,a,b 2 Z, the ëa 1 ;a 2 ;:::;a ëmodb = fa 1 mod bgëfa 2 mod bgëæææëfa mod bg = ë i=1 fa i mod bg : 1
2 CHAPTER 5 ON A CONJECTURE OF ERD í OS 2 The Remodulizatio Theorem ë3ë states that if a; b; c 2 Z ad cé0, the a mod b =ëa; a + b;:::;a+ bèc, 1èë mod cb : If we use Deæitio 2, the complemetary set of fa mod bg is give by fa mod bg c = Z fa mod bg =ë0; 1; 2;:::;a, 1;a+1;:::;b, 1ë mod b: I this case, the complemetary set cosists of b, 1 cogrueces modulo b We will always refer to the size of a set ad its complemet with respect to a speciæc modulus The followig theorem ad its proof is foud i ë4ë S i=1 ëa i;1;:::;a i;æbi ëmodb i o, where Theorem 1 The complemetary set of a i;j 6= a i;k for j 6= k, ad æ bi é b i, ad the b i are pairwise relatively prime, cotais exactly Q i=1 èb i, æ bi è cogrueces modulo Q i=1 b i Coverig Sets of Cogrueces I Daveport ë1ë, a problem has bee proposed to costruct a set of cogrueces with distict moduli, such thatevery iteger is cotaied i at least oe of the cogrueces of the system All moduli are ç 2, sice modulo 1 costitutes its ow complete residue system A extesio has bee proposed by Erdíos ë5ë: If give ay iteger N ç 1, does there exist aæitecoverig set of cogrueces usig oly distict moduli greater tha N? The followig system represets a set of coverig cogrueces for N = 1: 8 é x ç 0mod2 x ç 0mod3 x ç 1mod4 x ç 1mod6 x ç 11 mod 12 Note that the moduli are all divisors of 12 Usig the remodulizatio method to remodulize each cogruece to the modulus 12, wehave x ç ë0; 2; 4; 6; 8; 10ë mod 12 8é x ç ë0; 3; 6; 9ë mod 12 x ç ë1; 5; 9ë mod 12 è51è x ç ë1; 7ë mod 12 x ç ë11ë mod 12 By ispectio, this system costitutes a coverig system, because it is equivalet to the complete residue system modulo 12, that is, ë0; 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11ë
3 CHAPTER 5 ON A CONJECTURE OF ERD í OS 3 mod12 ç Z The questio aturally arises as to the possibility of costructig a set of coverig cogrueces whose moduli are pairwise relatively prime The aswer is o, as we will ow show The proof depeds o results of Theorem 1 S Theorem 2 Ay æite system of cogrueces ëa i=1 i;1;a i;2 ;:::;a ëmodb i;æbi i where the b i are pairwise relatively prime, ad a i;k 6= a i;m for k 6= m, ad æ bi éb i,caot form a coverig set of cogrueces Proof We use Theorem S 1, for the case æ bi =1for 1 ç i ç Forasystem of cogrueces f a i=1 i mod b i g with pairwise relatively prime moduli b i to be a coverig set, it is ecessary that the system of cogrueces forms a complete residue Q Q system, that is, a uio of b i=1 i distict èicogruetè residues modulo b i=1 i However, Q Q sice the complemetary set cosists of èb i=1 i,1è cogrueces modulo Q Q b i=1 i, the system cosists Q of Q Q b i=1 i, èb i=1 i, 1è cogrueces modulo b i=1 i This meas that b Q Q i=1 i, èb i=1 i, 1è must equal b i=1 i,or èb i=1 i,1è = 0; this is a cotradictio, sice 2 ç b 1 éb 2 ::éb That is to say,ittakes iæitely may cogrueces with pairwise relatively prime moduli to costruct a coverig set The situatio is actually muchworse; if we costruct a system of cogrueces è ë è ëa i;1 ;a i;2 ;:::;a i;æbi ëmodb i ; i=1 a system of cogrueces with æ bi residues for each modulus b i, where æ bi éb i, ad a i;k 6= a i;m for k 6= Q m, the the complemetary Q Q set cosists of èb i=1 i,æ bi è cogrueces modulo b i=1 i Here, èb i=1 i, æ bi è must equal zero, which is a cotradictio because b i, æ bi ç 1 Agai, it takes iæitely may such cogrueces to costruct a coverig set I the extreme case, whe æ bi = b i, 1, we have the system 8é o, x ç ëa 1;1 ;a 1;2 ;:::;a 1;b1,1ë modb 1 x ç ëa 2;1 ;a 2;2 ;:::;a 2;b2,1ë modb 2 x ç ëa ;1 ;a ;2 ;:::;a ;b,1ë modb è52è where the b i are pairwise relatively prime, ad each set of cogrueces modulo b i cotais b i,1 cogrueces, ie, oe cogruece shy of a complete residue system for each modulus b i Q Q The complemet of the system cosists Q of èb i=1 i,èb i,1èè cogruece modulo b i=1 i, or 1 cogruece modulo b i=1 i Hece the system è52è cotais a complete residue system oly if we cap it oæ with the last
4 CHAPTER 5 ON A CONJECTURE OF ERD í OS 4 remaiig residue modulo Q i=1 b i However, by addig to the system the remaiig cogruece, we have relaxed our requiremet that all the moduli are pairwise relatively prime Upshot If a æite system of distict cogrueces è52è with pairwise relatively prime moduli forms a coverig set, it must cotai a cogruece class which itself forms a complete residue system, or coverig set Suppose p 1 is a prime such that p 1 é N, ad M = p ç1 1 pç2 2 æææpç, where p 1 Q é p 2 é ::: é p The total umber of divisors of M which are ç M is èç i=1 i + 1è; however, to form a coverig set we may oly use Q all factors greater tha 1, the total umber of useable factors is ç =,1+ èç i=1 i +1è We ow costruct a system of ç cogrueces 8 é x ç c 1 mod d 1 x ç c 2 mod d 2 x ç c ç mod d ç è53è where the d i are the various factors of M = p ç1 1 pç2 2 æææpç,add 1 éd 2 ::é d ç Note that d 1 = p 1 ad d ç = M Observatio 1 The umber of cogrueces is give by çx i=1 M d i = M d 1 + M d 2 + æææ+ M d ç =1+d 1 + d 2 + æææ+ d ç,1 = ç 0 èmè ; after remodulizig all cogrueces of system è53è to the modulus M = p ç1 1 pç2 2 æææpç Here, ç 0 deotes the sum of all proper divisors, ie, all positive divisors less tha M Up to this poitwehave ot made ay claims about these residues modulo M, that is, we have ot yet determied how may repetitios exist, ad equivaletly, if the total umber of distict residues is suæciet to create a complete residue system modulo M We ca see, however, that the total umberofresiduesmust be at least M to form a complete residue system modulo M Therefore, M must be a abudat or perfect umber, that is, the sum of all proper divisors çx i=1 M d i =1+d 1 + d 2 + æææ+ d ç,1 = ç 0 èmè ç M i order for this system to cotai a complete residue system modulo M We have proved the followig theorem
5 CHAPTER 5 ON A CONJECTURE OF ERD í OS 5 Theorem 3 I order for the proper divisors of a umber M to costitute the moduli of a coverig set it is ecessary that M be perfect or abudat, ie, ç 0 èmè ç M We will prove i Theorem 5 that if M is a perfect umber, ie, ç 0 èmè =M, the a system è53è caot comprise a coverig set Observatio 2 It may ot be ecessary to use all divisors of a abudat umber M to form a coverig set; however, accordig to our eumeratio of the residues modulo M of system è53è, we must remove all residues associated with the divisors that are removed For example, suppose we do't use the divisor d k, the we must remove a total of M=d k residues from the set of ç 0 èmè residues couted i Observatio 1 However, the divisor d k caot cotai the greatest multiple of ay oe prime appearig i the prime decompositio M = p ç1 1 pç2 2 æææpç, for i that case, lcmèd 1 ;d 2 ;:::;d k,1;d k+1 ;:::;d ç è 6= M, that is to say, we would ot have remodulized the system to the modulus M, but istead to some modulus ém I the origial set 8 é x ç ë0; 2; 4; 6; 8; 10ë mod 12 x ç ë0; 3; 6; 9ë mod 12 x ç ë1; 5; 9ë mod 12 x ç ë1; 7ë mod 12 x ç ë11ë mod 12 we æd that the itegers 0,1,6, ad 9 represet 4 repetitios, sice ç 0 è12è,12 = 4 The total umber of repetitios that occur i a system è53è which forms a coverig set is ç 0 èmè,m The followig theorem eumerates the total umber of repetitios that occur i two cogrueces Theorem 4 If two cogrueces a 1 mod b 1 ad a 2 mod b 2, where gcdèb 1 ;b 2 è= 1, are remodulized to the modulus pb 1 b 2, where p 2 Z, the the solutio set èitersectioè cosists of p residues modulo pb 1 b 2 Proof If we obtai a pair of cogrueces ç x ç a1 mod b 1 x ç a 2 mod b 2 è54è where b 1 ad b 2 are relatively prime, the remodulizig each tomodulo b 1 b 2, the itersectio by the Chiese Remaider Theorem ë3ë is determied to be the uique cogruece x ç a 0 mod b 1 b 2, where a 1 ç a 0 ç a 1 + b 1 èb 2, 1è ad a 2 ç a 0 ç a 2 +b 2 èb 1,1è If the pair is remodulized, ot to the smallest modulus,
6 CHAPTER 5 ON A CONJECTURE OF ERD í OS 6 b 1 b 2, but istead to some multiple of it, say pb 1 b 2, where p is a positive iteger, the ç x ç ëa1 ;a 1 + b 1 ;:::;a 1 + b 1 èpb 2, 1èë mod pb 1 b 2 x ç ëa 2 ;a 2 + b 2 ;:::;a 2 + b 2 èpb 1, 1èë mod pb 1 b 2 è55è We show that the itersectio of è55è is ëa 0 ;a 0 + b 1 b 2 ;:::;a 0 + b 1 b 2 èp, 1èë mod pb 1 b 2 ; which is equivalet to the solutio a 0 mod b 1 b 2 of the origial pair after a 0 mod b 1 b 2 has bee remodulized by the factor p By writig the ærst cogruece of è55è as a 1 mod b 1 =ë a 1 ; a 1 + b 1 ; ::: a 1 + b 1 èb 2, 1è; a 1 + b 1 b 2 ; a 1 + b 1 + b 1 b 2 ; ::: a 1 + b 1 è2b 2, 1è; a 1 +èp, 1èb 1 b 2 ; a 1 + b 1 +èp, 1èb 1 b 2 ; ::: a 1 + b 1 èpb 2, 1èë mod pb 1 b 2 we ote that the ærst row cotais the solutio, a 0 mod pb 1 b 2 Moreover, by addig multiples of b 1 b 2 to the residue a 0, we æd the subsequet solutios withi the same colum; hece there are p solutios èmodulo pb 1 b 2 è Costructig the secod cogruece of è55è i the same maer, we extract the same p solutios Therefore, if a pair of cogrueces è54è is remodulized to the modulus pb 1 b 2, the they share exactly p simultaeous residues As a example, if we have the pair of cogrueces ç x ç a1 mod p 1 x ç a 2 mod p 2 which are remodulized to the modulus M = p ç1 1 pç2 2 æææpç, the they share residues modulo M M=p 1 p 2 = p èç1,1è 1 p èç2,1è 2 p ç3 3 æææpç Theorem 5 If M is a perfect umber, the a system of cogrueces whose moduli cosist of all divisors é 1 of M caot form a coverig set Proof Suppose M is a eve perfect umber ë6ë; the it is of the form 2 k p, where p is a odd prime of the form 2 k+1, 1 Suppose we form a system of cogrueces è53è where the d i are all divisors of M greater tha 1, ad remodulize all cogrueces modulo d i to the modulus M By Observatio 1, ç 0 èmè = M; a complete residue system modulo M must cotai M distict residues modulo M Sice p is prime, the cogrueces modulo 2 ad modulo p
7 CHAPTER 5 ON A CONJECTURE OF ERD í OS 7 share 1 residue modulo 2p, or2 k,1 residues modulo 2 k p by Theorem 4 These represet 2 k,1 repetitios, ad M, 2 k,1 ém= ç 0 èmè; hece the total umber of distict residues is ot suæciet to form a coverig set If M is a odd perfect umber èif ay existè, the it must cotai more tha 8 distict prime factors ë7ë Sice p 1 ad p 2 are two distict prime factors, their itersectio cotais 1 cogruece modulo p 1 p 2, or M=p 1 p 2 cogrueces modulo M I that case, M, M=p 1 p 2 ém= ç 0 èmè, meaig that the total umber of distict residues modulo M is too small for a system of cogrueces è53è to form a coverig set Remark 1 Theorems 3 ad 5 combied suggest that if for each N ç 2 there exists a coverig set whose distict moduli all exceed N, the there would exist abudat umbers whose least prime factor exceeds N This is true I fact, eve more is true Deæitio 3 A umber M is said to be abudat of order k ç 1 if ad oly if ç 0 èmè=m ék Theorem 6 If K ad N are ay itegers, the there exists a iteger M, abudat of order K, whoseleast prime factor exceeds N Proof Sice the primes are such P P that 1=p i = +1, we may Q select N é p 1 é p 2 é æææ é p such that 1=p i=1 i = K Set M = p i=1 i, the ç 0 èmè = 1+p 1 + p 2 + æææ + p 1 p 2 æææp,1 ad ç 0 èmè=m = 1=èp 1 æææp è+ 1=èp 2 æææp è+æææ+1=p 1 + æææ+1=p ék Remark 2 Theorem 6 shows that there are umbers M whose divisors caot yet be excluded from formig a coverig set whose moduli all exceed N However, a settlig of this cojecture may well require ædig methods that ca accurately accout for the total umber of repetitios that occur i such systems Refereces ë1ë Harold Daveport, The Higher Arithmetic, Dover Publicatios, Ic, New York, 1983 p 57 ë2ë Erdíos, Paul, O Itegers of the Form 2 k + p ad some Related Problems, Summa Brasiliesis Mathematicae, Istituto de Mathematica Pura e Aplicada, 1950, Vol 2, p 120 ë3ë Gold, Jeærey F ad Do H Tucker, Remodulizatio of Cogrueces, Proceedigs Natioal Coferece o Udergraduate Research, èuiversity of
8 CHAPTER 5 ON A CONJECTURE OF ERD í OS 8 North Carolia Press, Asheville, North Carolia, 1992è, Vol II, pp 1036í41 ë4ë Gold, Jeærey F ad Do H Tucker, Complemetary Sets of Systems of Cogrueces, Proceedigs Natioal Coferece o Udergraduate Research, èuiversity of North Carolia Press, Asheville, North Carolia, 1993è, Vol II, pp 793í96 ë5ë Wacèlaw Sierpiçski, Elemetary Theory of Numbers, Paçstwowe Wydawictwo Naukowe, Warszawa, 1964, pp 190, 413 ë6ë Keeth H Rose, Elemetary Number Theory ad its Applicatios, Third Editio, AddisoWesley Publishig Compay, Massachusetts, 1993, pp 223í 29 ë7ë David M Burto, Elemetary Number Theory, Secod Editio, Wm C Brow Publishers, Iowa, 1989, p 167
Chapter 4 Complementary Sets Of Systems Of Congruences Proceedings NCUR VII. è1993è, Vol. II, pp. 793í796. Jeærey F. Gold Department of Mathematics, Department of Physics University of Utah Don H. Tucker
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