On quasiperfect numbers

Transcription

1 Notes on Number Theory and Dscrete Mathematcs Prnt ISSN , Onlne ISSN Vol. 23, 2017, No. 3, On quasperfect numbers V. Sva Rama Prasad 1 and C. Suntha 2 1 Nalla Malla Reddy Engneerng College, Dvyanagar Ghatakesar Mandal, Ranga Reddy Dstrct,Telangana , Inda e-mal: vangalasrp@yahoo.co.n 2 Department of Mathematcs and Statstcs, RBVRR Women s College Narayanaguda, Hyderabad, Telangana , Inda e-mal: csunthareddy1974@gmal.com Receved: 20 November 2016 Accepted: 28 Aprl 2017 Abstract: A natural number N s sad to be quasperfect f σ(n) = 2N + 1 where σ(n) s the sum of the postve dvsors of N. No quasperfect number s known. If a quasperfect number N exsts and f ω(n) s the number of dstnct prme factors of N then G. L. Cohen has proved ω(n) 7 whle H. L. Abbott et al. have shown ω(n) 10 f (N, 15) = 1. In ths paper we frst prove that every quasperfect numbers N has an odd number of specal factors (see Defnton 2.3 below) and use t to show that ω(n) 15 f (N, 15) = 1 whch refnes the result of Abbott et al. Also we provde an alternate proof of Cohen s result when (N, 15) = 5. Keywords: Quasperfect number, Specal factor. AMS Classfcaton: 11A25. 1 Introducton For any natural number N, let σ(n) denote the sum of ts postve dvsors. A number N s called abundant, perfect or defcent f σ(n) > 2N, σ(n) = 2N or σ(n) < 2N respectvely. It s well known that there are nfntely many abundant numbers and nfntely many defcent numbers. In [7] Serpnsk asks whether there s at least one abundant number satsfyng σ(n) = 2N + 1, (1.1) for whch there s no defnte answer tll date. P. Cattaneo [2] called any N satsfyng (1.1) quasperfect, and ntated a study of such numbers. Later H. L. Abbott, C. E. Aull, Ezra Brown 73

2 and D. Suryanarayana [1] contnued nvestgatons on quasperfect numbers and proved the followng: If a quasperfect number N exsts and f ω(n) s the number of dstnct prme factors of N, then ω(n) 5 ( [1], Theorem 2 ) (1.2) and ω(n) 10 f (N, 15) = 1 ( [1], Theorem 5) (1.3) M. Kshore [5] mproved (1.2) to ω(n)) 6 whle G. L. Cohen and Peter Hags Jr. [3] have obtaned a refnement to t as ω(n) 7 (1.4) Further detals of research on quasperfect numbers can be seen n the book by J. Sándor and B. Crstc ( [6], pp ). In ths paper we frst prove that f a quasperfect number N exsts then t has an odd number of specal factors (defned n Secton 2) and use t to refne (1.3) as and also provde an alternate proof of (1.4) n case (N, 15) = 5. ω(n) 15 f (N, 15) = 1 (1.5) 2 Prelmnares P. Cattaneo [2] has proved the followng: Theorem 2.1. If N s a quasperfect number then t s of the form N = p 2e 1 1 p 2e p 2et t, where p are dstnct odd prmes. Also e 0 or 1 (mod 4) f p 1 (mod 8); e 0 (mod 2) f p 3 (mod 8); e 0 or 1 (mod 4) f p 5 (mod 8) and e 1 f p 7 (mod 8). Further f M s a natural number for whch σ(m) 2M then no non-trval multple of M s quasperfect. Remark 2.2. It follows from the theorem that every quasperfect number s the square of an odd nteger whle the last part of t shows that every quasperfect number s prmtve abundant, n the sense that t s an abundant number havng no non-defcent number as a dvsor. In the canoncal representaton of a quasperfect number each factor s of the form p 2e p s an odd prme, of whch we consder the followng specal type of factors. where Defnton 2.3. If p s an odd prme and e 1 s an nteger such that ether p 1 (mod 8) and e 1 (mod 4) or p 5 (mod 8) and e 1 (mod 4)) then p 2e wll be called a specal factor. For example, 5 6, and are specal factors. Precsely the set of all specal factors s gven by S = {p 2e : [p 1 (mod 8), e 1 (mod 4)]or[p 5 (mod 8), e 1 (mod 4)]}. Observe that 5 4, 17 8 and are not specal factors. Also f 3 dvdes a quasperfect number then 3 2e s among ts non-specal factors; whle f 5 2e s a factor of N then t s a specal factor or a non-specal factor accordng as e 1 (mod 4) or e 0 (mod 4). Further any factor p 2e of a quasperfect number N s ether a specal factor or a non-specal factor but not both. 74

3 3 Man results Frst we prove the followng Defnton 3.1. If a quasperfect number N exsts, then t has an odd number of specal factors. Proof. Suppose N s a quasperfect number of the form Then p 2 1 (mod 8) for 1 t. Therefore Also for any, N = p 2e 1 1 p 2e p 2et t, where p are odd prmes. (3.2) 2N + 1 = 2.(p 2 1) e 1 (p 2 2) e 2...(p 2 t ) et (mod 8) 3 (mod 8). (3.3) σ(p 2e ) = 1 + p + p p 2e = (1 + p ) + p 2 (1 + p ) p 2(e 1) (1 + p ) + p 2e = (1 + p )(1 + p p 2(e 1) ) + p 2e (1 + p )e + 1 (mod 8) so that σ(p 2e ) 3 (mod 8) f p 2e s a specal factor 1 (mod 8) otherwse (3.4) For example, f p 5 (mod 8) and e 3 (mod 4), say p = 8u + 5 and e = 4v + 3 then (1 + p )e + 1 = (8u + 6)(4v + 3) (mod 8). Also f p 3 (mod 8) and e 0 (mod 2) then (1 + p )e + 1 = (8u + 4)(2v ) (mod 8). Hence σ(n) = t σ(p 2e ) 3 k (mod 8), (3.5) where k s the number of specal factors of N. Now (3.3) and (3.5) gve 3 k 3 (mod 8), whch holds only f k s odd, thus provng the theorem. Remark 3.6. If N s a quasperfect number of the form (3.2) then t follows from the theorem that not all e can be even showng that N cannot be the fourth power of a natural number. That s, no number of the form m 4 s quasperfect. Ths result has been proved n [3] by a slghtly dfferent method. Usng Theorem 3.1 we now mprove (1.3) as below: Theorem 3.7. If N s a quasperfect number wth (N, 15) = 1 then ω(n)

4 Proof. Suppose N s the square of an odd nteger of the form N = s P 2e. r Q 2f j j, (3.8) where P 2e are specal factors, Q 2f j j are non-specal factors, It s easy to see that where π = s (P, Q j ) = 1, P 1 < P 2 <... < P s and Q 1 < Q 2 <... < Q r. σ(n) N = s P P 1 and π = r j=1 σ(p 2e ) P 2e Q j Q j 1. r σ(q 2f j j ) Q 2f j j < π.π (3.9) Now we ntroduce a notaton: For any nteger k 1, f the k-tuples (a 1, a 2,..., a k ) and (b 1, b 2,..., b k ) of prmes are such that a b for = 1, 2,..., k then we wrte (a 1, a 2,..., a k ) (b 1, b 2,..., b k ). Clearly k a k a 1 b b 1 f (a 1, a 2,..., a k ) (b 1, b 2,..., b k ), (3.10) snce x s a decreasng functon for x > 1. x 1 If N s of the form (3.8) wth (N, 15) = 1, s odd and ω(n) 14 then we wll prove that N s defcent and hence cannot be quasperfect, so that the theorem follows. It s enough to prove n the case ω(n) = 14. That s, s + r = 14, wth s odd and (N, 15) = 1. The set E of ordered pars (s, r) of postve ntegers wth the above propertes s gven by E = {(1, 13), (3, 11), (5, 9), (7, 7), (9, 5), (11, 3), (13, 1)}. For each (s, r) E, the prmes dvdng N s a 14-tuple of the form (P 1, P 2,..., P s, Q 1, Q 2,..., Q r ) and we can fnd a 14-tuple of dstnct prmes (p 1, p 2,..., p 14 ) such that (P 1, P 2,..., P s, Q 1, Q 2,..., Q r ) (p 1, p 2,..., p 14 ), where p 1 or 5 (mod 8) for = 1, 2,..., s and p j s any prme for j = s + 1,..., 14. Then, by (3.10) π.π 14 k=1 p k p k 1. (3.11) Table A below gves the 14-tuple (p 1, p 2,..., p 14 ) for each (s, r) E and the correspondng 14 p k value of p k 1. Here each p 7 snce (N, 15) = 1. As each entry n the last column s less k=1 than 2, t follows from (3.9) and (3.11) that N s defcent. Thus ω(n) 14 s not possble for any quasperfect number N wth (N, 15) = 1, provng ω(n)

5 I II III (s, r) (p 1, p 2,..., p 14 ) 14 k=1 p k p k 1 (1,13) (13,7,11,17,19,23,29,31,37,41,43,47,53,59) (3,11) (13,17,29,7,11,19,23,31,37.41,43,47,53,59) (5,9) (13,17,29,37,41,7,11,19,23,31,43,47,53,59) (7,7) (13,17,29,37,41,53,61,7,11,19,23,31,43,47) (9,5) (13,17,29,37,41,53,61,73,89,7,11,19,23,31) (11,3) (13,17, 29,37,41,53,61,73,89,97,101,7,11,19) (13,1) (13,17,29,37,41,53,61,73,89,97,101,109,113,7) Table A Theorem If N s a quasperfect number wth (N, 15) = 5 then ω(n) 7. Proof. Suppose N s the square of an odd nteger of the form (3.8) wth (N, 15) = 5, s odd and ω(n) 6. We wll show, as n the proof of Theorem 3.1, that N s defcent and hence cannot be quasperfect. As before t suffces to prove the case ω(n) = 6. That s, s + r = 6, s odd and (N, 15) = 5. Unlke n the prevous theorem, here 5 dvdes N so that the factor 5 2e may or may not be a specal factor for N. The set F of ordered pars (s, r) of postve ntegers wth the stated condtons s F = {(1, 5), (3, 3), (5, 1)}. Now for each (s, r) F and for the 6-tuple of prmes (P 1, P 2,..., P s, Q 1, Q 2,..., Q r ) dvdng N, we fnd two 6-tuples (p 1, p 2,..., p 6 ) and (p 1, p 2,..., p 6) of prmes such that (P 1, P 2,..., P s, Q 1, Q 2,..., Q r ) (p 1, p 2,..., p 6 ) or (p 1, p 2,..., p 6) accordng as 5 2e s a specal factor or not for N. Table B gves the 6-tuples (p 1, p 2,..., p 6 ) and (p 1, p 2,..., p 6) and the correspondng products 6 p 6 p 1 and p for any gven (s, r) F. Snce each entry n columns III and V s less p 1 than 2, t follows N s defcent. Thus ω(n) 6 s not possble for a quasperfect number wth (N, 15) = 5. Hence the theorem holds. I II III IV V 6 p 6 (s, r) (p 1, p 2,..., p 6 ) (p p 1 1, p 2,..., p p 6) p 1 (1,5) (5,7,11,13,17,19) (13,5,7,11,17,19) (3,3) (5,13,17,7,11,19) (13,17,29,5,7,11) (5,1) (5,13,17,29,37,7) (13,17,29,37,41,5) Table B 77

6 Acknowledgement The second author wshes to thank the Unversty Grants Commsson, Government of Inda, New Delh for the fnancal support under the Mnor Research Project No.F MRP-5963 / 15 (SERO / UGC) References [1] Abbott, H. L., C. E. Aull, Brown, E., & D.Suryanarayana (1973) Quasperfect numbers, Acta Arthmetca, XXII, ; correcton to the paper, Acta Arthmetca, XXIX (1976), [2] Cattaneo, P. (1951), Su numer quasperfett, Boll. Un. Mat. Ital., 6(3), [3] Cohen, G. L. (1982) The non-exstence of quasperfect numbers of certan form, Fb. Quart., 20(1), [4] Cohen, G. L. & Peter Hags Jr. (1982) Some results concernng quasperfect numbers, J.Austral.Math.Soc.(Ser.A), 33, [5] Kshore, M. (1975) Quasperfect numbers are dvsble by at least sx dstnct dvsors, Notces. AMS, 22, A441. [6] Sándor, J. & Crstc, B. (2004) Hand book of Number Theory II, Kluwer Academc Publshers, Dordrecht/ Boston/ London. [7] Serpnsk, W. A Selecton of problems n the Theory of Numbers, New York, (page 110). 78