About k-perfect numbers

Transcription

1 DOI: 0.47/auom A. Şt. Uv. Ovdus Costaţa Vol.,04, About k-perfect umbers Mhály Becze Abstract ABSTRACT. I ths paper we preset some results about k-perfect umbers, ad geeralze two equaltes due to M. Persastr see []. Itroducto Defto. A postve teger s k-perfect f σ = k, whe k >, k Q. The specal case k = correspods to perfect umbers, whch are tmately coected wth Mersee prmes. We have the followg smallest k-perfect umbers. For k =,, 49,,..., for k = 3 0, 7, 5377, 45940,..., for k = , 370, 7540,..., for k = , ,..., for k = ,.... For a gve prme umber p, f s p-perfect ad p does ot dvde, the d p + perfect. Ths mples that a teger s a 3 perfect umber dvsble by but ot by 4, f ad oly f s a odd perfect umber, of whch oe are kow. If 3 s 4k perfect ad 3 does ot dvde, the s 3k perfect. A k perfect umber s a postve teger such that ts harmoc sum of dvsors s k. For the perfect umbers we have the followgs: = , 49 = , = etc. We posted the followg cojecture: Cojecture. Becze, M., 97 If s k-perfect, the exst odd postve tegers u =,,..., r such that Key Words: perfect umbers 00 Mathematcs Subject Classfcato: A5 Receved: May, 03. Revsed: September, 03. Accepted: November,

2 4 Mhály Becze = r u k+ MAIN RESULTS Theorem. If f : R R s covex ad creasg, N = p α pα...pα wrtte caocal form s k-perfect, the: 3 f f 3 f k Proof. If N s eve the t follows + > 3 f N s eve For x 3 holds x+ x 3 x x see [9], therefore the yelds + > p 3 > 3 k because = k p α+ p α+ Usg the AM-GM equalty we obta: Fally + > > k + = k p f N s eve Because f s covex ad creasg from Jese s equalty we get

3 About k-perfect umbers 47 f f p 3 f 3 f k f N s eve Theorem. If g : R R s covex ad creasg, N = p α pα...pα wrtte caocal form s k-perfect, the: g kπ f N s eve g g Proof. We have the followg: p = p α+ p α p α+ p α+ = k p α + = k j j=0 k j=0 p j k = k =0 f N s eve + = { f N s eve From AM-GM equalty yelds therefore Accordg to Jese s equalty yelds = f N s eve g g p g g f N s eve

4 4 Mhály Becze Corolloary. If N = p α pα...pα wrtte caocal form s k-perfect the: 3 3 k f N s eve f N s eve Theorem 3. If x, t > 0 the x + t x+ xt x Proof. For t = we have the equalty. Let 0 t. Sce the fucto u x = xt x s cotuous ad dfferetable we ca apply the Lagrage s theorem ad we obta x + t x+ xt x x + x = u x + u x x + x = u z whe x z x + hece we have the equalty t z z l t or z l t t z. Developg t z to McLaure s seres t results or z l t!z l t +!z l t 3!z 3 l3 t +... r= r l r t r!z r > 0 or r= l r t r!z t > 0 that s obvous because l t > 0 due to t >. Let be t >. The s eough to show that the fucto V x = x t x s decreasg. Dfferetable V we get V x = t x t x x l t = l r t x r 0 r! r Sce V s decreasg ad we may say that V x + V x hece ad from t follows the equalty of the eucato. Corollary. If N = p α pα...pα s a k-perfect umber wrtte caocal form, the: r=

5 About k-perfect umbers 49 { l 3 f N s eve 3 l k { l kπ l kπ f N s eve Proof. Usg the Theorem 3 t s proved that the seres 3 3 ad k N are decreasg, ad the seres N N ad N are creasg. It meas that the mmum ad maxmum are reached oly the. Sce we have 0. That s why L Hosptal rule ad so we fd the results of the eucato. Remark. For k = we reobta the M.Persastr s equalty l π see []. Corollary 3. Let N = p α pα...pα be a k perfect umber wrtte caocal form ad P max = {p, p,..., p } ad P m = m {p, p,..., p }, the { 3 f N s eve ad P m P max > Proof. Cosderg that P max 3 k f N s eve respectve from the theorem f follows the affrmato. P m

6 50 Mhály Becze be a k-perfect umber wrtte ca- Remark. Let N = p α pα...pα ocal form, the P m see the method of M. Persastr s k + Ackowledgemets. The author wshes to express hs grattude to the Orgazg Commttee of the workshop Workshop o Algebrac ad Aalytc Number Theory ad ther Applcatos PN-II-ID-WE The publcato of ths paper s supported by the grat of CNCS-UEFISCDI Romaa Natoal Authorty for Scetfc Research: PN-II-ID-WE Refereces [] B. Apostol, Extremal orders of some fuctos coected to regular tegers modulo, A. St. Uv. Ovdus Costata, Vol.,03, 5-9. [] M. Becze, O perfect umbers, Studa Mathematca, Uv. Babes- Bolya, Nr. 4, 9, 4-. [3] G. Hardy, D.E. Lttlewood, G. Polya, Iequaltes, Cambrdge, Uversty Press, 94. [4] H.J. Kaold, Über mehrfach volkommee Zahle, II. J. Ree Agew. Math., 957, 97, -9. [5] Octogo Mathematcal Magaze [] M. Persastr, A ote o odd perfect umbers, The Mathematcs Studet 95, 79-. [7] J. Sádor, B. Crstc, eds.: Hadbook of umber theory II, Dordrecht, Kluwer Academc, 004. [] W. Serpsk, Elemetary theory of umbers, Warsawa, 94. [9] The Amerca Mathematcal Mothly, E.3097, E.99 Smeo Rech s ote. Mhály Becze, Str. Hărmaulu,, Săcele-Négyfalu, Jud. Braşov, Romaa Emal: beczemhaly@gmal.com