ERDO-MARANDACHE NUMBER b Tbrc* Tt Tbrc** *Trslv Uvrsty of Brsov, Computr cc Dprtmt **Uvrsty of Mchstr, Computr cc Dprtmt Th strtg pot of ths rtcl s rprstd by rct work of Fch []. Bsd o two symptotc rsults cocrg th Erdos fucto, h proposd som trstg qutos cocrg th momts of th mrdch fucto. Th m of ths ot s gv bt modfd proof d to show som computto rsults for o of th Fch quto. W wll cll th umbrs obtd from computto th Erdos-mrdch Numbrs. Th Erdos- mrdch umbr of ordr s obtd to b th Golomb-Dckm costt.. INTRODUCTION W brfly prst th rsults usd ths rtcl. Ths cocr th rltoshp btw th mrdch d th Erdos fuctos d som symptotc qutos cocrg thm. Th mrdch d Erdos fuctos r mportt fuctos Numbr Thory dfd s follows: Th mrdch fucto [mrdch, 98] s : N* N, m{ k N k! } N * Th Erdos fucto s : N* N, M. N * \{}, m{ p N M p p s prm}. Thr m proprts r: b N *,, b b m{, b}, b m{, b}. 3 N * d th qults occur f s prm. 4 A mportt quto btw ths fuctos ws foud by Erdos [99] {, < } lm, 5 whch ws tdd by Ford [999] to { } l l l, <, whr lm. 6
Equtos 5-6 r vry mportt bcus crt smlrty btw ths fuctos spclly for symptotc proprts. Morovr, ths qutos llow us to trslt covrgc proprts of th mrdch fucto to covrgc proprts o th Erdos fucto d vc vrs. Th m mportt qutos tht hv b obtd usg ths trslto r prstd th followg. Th vrg vlus O log d thr grlztos [Luc, 999], O log [Tbrc, 999] ζ O l l [Kuth d rdo 976] ζ O l l Th log-vrg vlus [Fch, ] l l lm λ [s Fch, 999] lm λ [Fch, 999] l l d thr grlztos l l lm λ [hpp, 964] λ l l Th Hrmoc rs lm [Fch, ]. lm [Luc, 999], [Tbrc, 998] lm [Tbrc, 999]. THE ERDO-MARANDACHE NUMBER From combtorl study of rdom prmutto hp d Lloyd [964] foud th followg tgrl quto lm l l! p p y dy d : λ. 7 y Fch [] strtd from Equto 7 d trsltd t from th mrdch fucto.
Thorm [Fch, ] If s postv tgr umbr th l l lm l l lm. 8 roof My trms of th dffrc l l l l r qul, thrfor thr wll b rducd. Ths dffrc s trsformd s follows: l l l l l l l l : : l l l l l l l. Th grl trm of th lst sum s suprorly boudd by l l l l bcus l l l l l d l 3. Thrfor, th ch of qults s cotud s follows: :, l l l l l l l l l l l l l. I ordr to prov tht lst rght mmbr tds to, w strt from lm. W substtut l l l d th lmt bcoms l l l lm l l l. Now, th lst rght mmbr s clcultd s follows: l l l l l lm l lm l l l l l l l l l. Thrfor, th quto l l lm l l lm holds.
Th ssc of ths proof d th proof from [Fch, ] s gv by Equto 6. But th bov proof s bt grl gvg v mor l l lm l l. l l Dfto. Th Erdos-mrdch umbr of ordr l l λ lm lm. l l N s dfd by Equto 7 gvs formul for ths umbr λ p, w obt tht th Erdos-mrdch umbr! p y dy d. For y l l λ lm lm, l l s fct th Golomb-Dckm costt. Usg smpl Mpl computto th vlus of th frst Erdos-mrdch umbrs hv b clcultd wth 5 ct dcmls. Thy r prstd blow. λ.643998854355 λ.46695764659643 3 λ 3.3363673453 4 λ 4.4387668 5 λ 5.97989344375 6 λ 6.6598556876 7 λ 7.4575545497 8 λ 8.4893444877 9 λ 9.674965 λ.9998634543 λ.99687764 λ.83337677 3 λ 3.7697489936 4 λ 4.74798599744 5 λ 5.6666647383 6 λ 6.649988748754 7 λ 7.588347988849 8 λ 8.55555533486 9 λ 9.5635688647356 λ.49999954453 3. Fl Rmrks Th umbrs provdd by Equto 7 could hv my othr ms such s th Golomb- Dckm grlzd costts or. Bcus thy r mpld Equto 8, w blv tht propr m for thm s th Erdos-mrdch umbrs. W should lso sy tht t s th Fch mjor cotrbuto rdscovrg qut old quto d coctg t wth th mrdch fucto.
Rfrcs Erdos,. 99 roblm 6674, Amr. Mth. Mothly. 98, 965. Ford, K. 999 Th orml bhvours of th mrdch fucto, mrdch Notos Jourl, Vol., No.--3, 8-86. Fch,.R. 999 Th vrg of th mrdch fucto, mrdch Notos Jourl,, 95-96. vlbl t http://www.mthsoft.com/solv/smrd/smrd.html. Fch,.R. Momts of th mrdch fucto, [to b publshd mrdch Notos Jourl], vlbl t http://www.mthsoft.com/solv/smrd/smrd.html. Kuth, D.E. d rdo, L.T. 976 Alyss of smpl fctorsto lgorthm, Thortcl Computr cc, 3, 3-348. Luc, F. 999 O th dvrgc of th mrdch hrmoc srs, mrdch Notos Jourl,, No. --3, 6-8. Luc, F. 999 Th vrg vlu of th mrdch fucto, [rsol commucto to b Tbrc]. hpp, L.A. d Lloyd,.. 966 Ordrd cycl lgths rdom prmuto, Trs. Amr. Mth. oc.,, 35-357. Tbrc,. d Tbrc, T. 998 Th covrgc of th mrdch hrmoc srs, mrdch Notos Jourl, 9, No. -, 7-35. Tbrc,. d Tbrc, T. 999 Th covrgc of th Erdos hrmoc srs, mrdch Notos Jourl, 9, No. 3-4-5. Tbrc,. d Tbrc, T. 999 Th vrg vlu of th Erdos fucto, mrdch Notos Jourl, 9, No. 3-4-5. mrdch, F. 98 A Fucto umbr thory, All Uv. Tmsor, XVIII.