CeRmIslMh; KNvTü Phgors Augus Lous Cuh PK
GrmÖf sysþi RmþEdlMBugEGesovePA PK CTIeKrBr;G. esovepaehruv)eroerogeligñúgekl MNgpþl;CÉsrsRm;CMYdl;rsSRsvRCvdl;GñsSCBess KW s½ VsSEmþg. esovepaehruv)eroerogeligedmcmybiesovepaknvtü détetoceri esovepaknvtüñúgrets g erarets. elis BIeHeTAeTo eig Mú)MeRCIserIslMh;mYMYdUC lmh;fø; ejrlgñúgrets g NþReTsepSg²eTomd;C MeNTBð RBmTMgmUvdMeNHRsEdleIgJMú)eFIVrRs:ge,H, edimi,egmþgñggrsyll;. CugeReIgJMúmRwmEBüCUBr dl;gñssrk;rueg CYRTHE esþ IsueRmI supbl g eckc½ñúgrss ¾dUC rgrpgedr. eigjmú¾sum GrKuNTuCmupgEdrrl;mrHK;rs;Gñ ssrk;ruñúg½sß. eroeroged C Bsdæ PñMeBj/ éf TI E mfu qñm
lmh;! gðjf 6 8 5. srmb ; gðjf 6 8 5 eigm 6 8 5 6 8 5 6 dueh 6 8 5 lmh; @ ekeg, CBIrMYMeBjlçNÐ RsB ;f. srmb ;
RsB ;f eigm * reoti! ed MeBH / IR eig) * B ln dueh reoti @ eigm
MeG 8 6 6 6 5 5 8 8 5 8 6 6 6 5 RK; IR MeG ed MeG RK;, IR eig) * B ln dueh lmh; # rémø m edimi,egplueré srs;smir m m mémøumpu. emøi rémø m edimi,egplueré srs;smirmémøumpu reoti!, m m ek) C srs;smir m m m m
UGgÁ g GgÁeK) m m ed m MeG m m MeG m m m m m 6 m m 9 m 9 9 mémøumpuesµi 9 ebl m dueh mmémøumpuesµi 9 ebl m reoti @ eigm smir m m m m / m MeG m m m m m m 6 m 9 m 9 9 mémøumpuesµi 9 ebl m dueh mmémøumpuesµi 9 ebl m
lmh; $ edhrssmir > >. emøi edhrssmir >! g / MeG m! eig) 5 ln rni MeG m s, rni MeG m s, rni eig) dueh smirm s ln,,,
6 > eigm g MeG ln rw ei MeG m 5 smirm s 5 ei MeG m smirkµ s dueh smirm s 5 lmh; % ekeg,, C srs;smir. KNémøøé. emøi KNémøøé f eigm
ed,, C srs; f MeG,, C srs; f Lemm: ei,, C srs;smir MeG. srm mrtwsþitevü,, C srs;smir MeG ek) MeG eig) dueh
lmh; ^ RsB ;f³ el 5 CerR)d. 8 99 998 srmb ; RsB ;f³ el 5 CerR)d 99 998 g N 5 dueh 99 9 9 9 998 999 5 99 998 99 999 998 5 99 999 998 996 999 999 9 9 996 999 996 9 5 5 998 5 5 998 5 998 5 99 5 5 CerR)d 99 5 5 99 998 99 5 CerR)d
9 lmh; & / / CIMYepSgKñ. RsB ;f ln. Guvþ_ > edhrssmir. > r CMYBEdlMeBjlçNÐ r r. KN r r. srmb ; RsB ;f ln reoti! eigm ed erbh eig)
ln dueh ln reoti @ BüGuKm_dWeRTIIm s / / KW P MeBH / / P eig)! @ # U! / @ / # eig) ek) ed MeG ed erbh eig)
ln dueh ln Guvþ_ > edhrssmir! ed ln MeBH / / m! eliggátmgbirckueig) dueh smirm s >MN;émøé r r eigm r r smmul r r MeG r r r r
eig) r 9 r r r 8 r r 8 8 r r r r 8 5 58 r dueh r 58 r lmh; *³ rémøumpuéesm A. emøi rémøumpuéesm A eigm erbh smpbebl, MeBH / / >>> / 5 eig) 99 99 9 98 95 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 5 5 5 UGgÁ g GgÁ A 99 9 95 99 5
smpbebl 5,5, 5 dueh 5 m A ebl 5,5 lmh; ( KN w eddwgf 5 w 5 w 6 5 6 6 6 w 8 5 8 8 8 w. ( AIME 98 ) emøi KN w 5 w p MeBH / 6 / 6 / 6 MeG p smµmµ MeG / 6 / 6 / 6 C srs; p ek) 6 6 6! ei p eig) 5 w 9 5 9 w
9 5 9 9 5 9 5 9 9 9 9 5,9,5,9 MeBH MeG 9 5 9 w CemKuN w 8 @ m! g @ eig) w 8 MeG w 6 dueh w 6 lmh;! ekeg / CBIrMYMeBjlçNÐ 99 / 99 KN. emøi KN eigm 99! 99 @ U! / @ eig). 99 99
5 99 99 99 998 segá 998 99 99 RK;, IR MeG 998 ln dueh lmh;!! edhrsvsmir MeBH IN. emøi ei eig) / - rni KU rw Z rgsbø
6 mrgsbø ln - rni ess rw rgsbø Z / / mrgsbø ln srueta / / rni / rni lmh;!@ ekegymyk;vc m,,, d epþógpþ; log KN,log d. 5 d / Z eiekdwgf 9 emøi.
KN eigm d log log 5 / d / smmul ed / / / d IN sµ / eig) 5 d MeBH / IN MeG 9 * ed / IN MeG BI * MeG / 9 Uu! / @ eig) 5! MeG 5 ek) 5, 6 ehi 5 d MeG d 9 dueh d 9 () () 5 / 5
lmh;!# rémøumpué 8 A log log log,,,,,. eiekdwgf rémøumpué emøi A log log log eigm RsdUKñ eig) log log log log log log >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> log log MeG A log log log
9 mvsmipbusiu ( Cuh ) eig) log log log log log log ed log log log MeG A dueh émøumpué A KW A m 6 6 lmh;!$ ryerymfmmpué 5 A MeBH // / >>> / 999. (Juor Bl 998) emøi ryerymfmmpué A 6 6 eigm A 5 ei MeG A 5 5 5 6 ehi 9 ei A mod 5 MeG A 9 mod 5 ekf YErYmFMMpué A / A / A / >>> / A 999 GesIµ rw ed A 8 9 9 5 5 mod 8 6 MeG A Ed;wg & MeBH // / >>> / 999 dueh GCD A, A, A,, A
lmh;!% MY / / / >>> / RUv)eRCIsejBI / ehi / 6 9. RsB ;f Ed;wg. emøi RsB ;f Ed;wg eigm / MeG (Juor Bl 998) 6 8 MeBH, MeG 6 8 smpbebl, MeBH, ed 6 9 MeG 6 8 6 9 8 6 eig) / MeBH, MeG MeBH Z MeG ed GCD, dueh Ed;wg
lmh;!^ MN;émøUMpué eiekdwgf log log. emøi MN;émøUMpué eigm log log m / MeBjlçNÐ ln eig) g MeG MeG! mü:geto log log @ m! / @ eig) m ' m ln ' eig) 8 8 MeG
émøumpué KW ebl m! MeG smmul dueh m ebl / lmh;!& ekeg A g 99 998 B. 998 99 998 A RsB ;f CMYK;. B srmb ; A RsB ;f B CMYK; ed eig) A 99 998 99 998 99 998 998 99 998 999 999 999 998
998 99 996 ek) 99 998 A 998 99 998 998 998 998 998 99 998 998 998 99 998 998B MeG A 99 CMYK; B dueh A 99 CMYK; lmh;!* KN KN E eigm! B E.!!!!!! 999!!! srmb ;!!!!!
ei!!!!!!!!!!!! / / >>> / 999 eig) >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>! 999!!!!! UGgÁ g GgÁ E!! dueh E! lmh;!( f!!!!! f : IN IN IN MeBjlçNÐ f, m, f m, m / f m, f m, MeBH m / p, q IN. rrk;ku q p, edimi,eg f. ( Kore ) srmb ; rrk;ku p, q eigm f m, f m, m MeG f p, q f p, q p / f m, f m,
5 f p, q p p >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> p f, q f p, q q p p >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> q pp q f, q pp p p q q p, q q erbh f, ed f ek) 999 p q p q 999 999 CMYm / p q p q MeBH p / q IN eig)!> p q p p q 998 @> p q p p q 999 / q 999 / q 999 dueh p, q,999 rw,999 lmh; @³KN eiekdwgf o o o o 5 emøi.
6 KN reoti! eigm 5 s 5 os5 os s 5 os5 s 9,,,, 5 eig) s 6 os 5 s 89 s os5 s 88 s 6 os5 s 8 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> s 9 5 os5 s 5 eig) 5 s 6 s s 9 s 9 5 os5 s 5 s 6s 89 5 6 o o o o ed 5 MeG smmul
dueh reoti @ eigm 5 5! 5 ed 5 5 5 5 5 5 MeG 5 5 m! MeG 5,,,, eig) >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> KuNGgÁ g GgÁeIg) 5
8 o o o o ed 5 MeG smmul dueh lmh; @!³ > RsB ;f Ar Ar Ar Ar Ar. eiekdwgf > RsB ;f Ar Ar Ar. 5 8 Ar Ar. K> edhrssmir X> srmylesm Ar os os Ar os s os s > RsB ;f Ar mrumþ MeBH MeBH emøi MeBH Ar / Ar MeG Ar Ar /. Ar Ar
9 Ar Ar Ar Ar MeG Ar Ar Ar Ar ed Ar Ar dueh Ar Ar Ar > RsB ;f 8 5 Ar Ar Ar eigm 8 5 Ar Ar Ar 8 5 5 Ar Ar 8 9 Ar Ar 8 9 Ar Ar 8 9 8 9 Ar
65 Ar Ar Ar 65 Ar Ar Ar 5 8 dueh K> edhrssmir eigm Ar Ar ek) Ar Ar Ar MeBH m 9 8 dueh X> srmylesm Ar os MeBH os /
eigm os s os os Ar Ar Ar Ar ei ek) os os Ar Ar ei ek) os os Ar Ar Ar ed os os Ar dueh os os Ar ei os os Ar ei Ar s os s os MeBH eigm Ar Ar os s os os os s os os s os s os
Ar Ar Ar ed erbh MeG Ar s os s os dueh Ar s os s os lmh; @@ RsB ;f os os os. srmb ; RsB ;f os os os reoti! g S os os os ek)³ os s os s os s s S ed s s os s s s
eig) s s s S s s s s s s s s s s s s s s s s s MeG S dueh os os os reoti @ GuKm_ f mrws s os / 6,,,, ed ek) os os 8 os 6 os os os erbh Re
MeG os os os os os os smmul os os os smmul os os os dueh os os os reoti # g os os os S os os os os os os os os os os q p q p q p os s os s os s s os s s ed s s s
5 MeG os s s s S s s s s os os os dueh lmh; @# KNlImIgeRm KNlImI > lm > lm 5. > lm emøi eigm
6 MeG lm lm erbh lm dueh lm > 5 lm eigm / / / MeG... MeG 5
eig) dueh lm 5 lm erbh lm lm 5. lmh; @$ edhrssmir emøi edhrssmir eigm ei MeG ekf mü:geto g f CemøIésmIr 65 65 B / g / ed / MeG f CGuKm_uH
8 ehi MeG g CGuKm_eI RéGuKm_uH g GuKm_eIRsBVKñEmYMNuK; MeG Ré f g g RsBVKñRg;mYMNuE:ueNÑH dueh CrwsEmYK;rs;smIr lmh; @% ekeg,,,,, CsVúIéMYBEdl KN eigg ehi. KN. ReTs @% srmb ; 6 8 Edl,,,,.. MeG 6 8 ekf eig) 6 CsVúIFrNImRm q ed
9 MeG MeG dueh lmh; @^ svúi / / >>> g / / / >>> MN;ed / / MeBH / / / >>> RsB ;f MeBH / / / >>>. emøi RsB ;f g MN;ed / MeBH / / / >>> ek)
eig) ekf CsVúIFrNImRm / q MeG! ehi ei MeG MeG ]mf Bdl; KW RsB ;f B m½f³ @ m! / @ smmul
smmul dueh lmh; @& ekeg / / / / 5 MeBjlçNÐ KN 5 5 5 / / / / 5. KN. 8 9 APMO 9 emøi 5 8 9 5 5 R / R R / R 9 6 R 5 5 R Edl 5 5 ehi R6 8 9 g P 5 Q MeBH / / / / 5 eig) Q R P MeG P Q MeBH / RP P
ed P Q CBhuFdWeRTI!m s 5 eig) P / / / / Q A 5 MeBH ek) P MeG A 9 6 5 5 9 6 5 P Q 9 6 5! efivplefo! wg P ek) Q 9 6 5 5 9 6 5 5 P R MeBH 6 eig) 6R 6 MeG dueh 5 8 9 9 699 865 R 6 658 5 865 8 9 658
lmh; @* ekegymy,,, Edl. RsB ;f. srmb ; RsB ;f eig)! eig) @ eig) # eig)
$ U! / @ / # g $ eig) dueh lmh; @( ekg CMYK;FMMpuUCg esiµ. ]ThrN_.6. CEpñTsPKé. Guvþ_ MeBH IR. > ei. rémøé > edhrsrb½ smir. 9. 8.8 5. K> edhrssmir emøi > rémøé eigm ed.
5 ek) dueh > edhrsrb½ smir U! / @ / # 9. 8.8 568. 9 9 568. 8.5 U! / @ 9. 9. 9. 8.5 5.65 ek) 5 /. 65 U @ / # 68. 9 68. 9 68.9 8.5 8.5 ek) 8/. 5 U! / # 8. 8
6 8. 8 8.8 8.5 9.5 ek) 9 /. 5 srum 9.65 9. 65 5.5 5. 5 8.5 8. 5 dueh 9. 65 / 5. 5 / 8. 5 K> edhrssmir 5! eigm 5 5 ed 5 ek) / / >>> / 8 smir! CMYs dueh 5 5 @ / / >>> / 8 ñúg @ ek) 9 89 / / 9 / 9 / 69 89 / / 9 69
lmh; # ekeg / IR / Z / IN. RsB ;f > > K> X>. srmb ; RsB ;f > mm½ / Z ek) dueh > mm½ / erbh CMYK; ed
8 eig) dueh K> eigm³ / BI! BI @ m! / @ dueh X> eigm / ehi
9 RsB ;f edvsmpbmlçn³siuemrti eigsµ ek) dueh lmh; #! CMYB/ CMYK;FmµC. RsB ;f. smpb Herme Guvþ_ CMYB. RsB ;f. IMO 968 srmb ; RsB ;f ei IN / ed MeBH,
B ei IR Z ³ IN ek) 5 MeBjlçNÐ /! @ m! g m @ / Z eig) dueh Guvþ_ RsB ;f msmpb Herme
5 ei eig) erbh m½f ei ek) dueh
5 lmh; #@ ekeg MYB / / / >>> / g / / / >>> /. RsB ;f. vsmpb Cuh-Shwr smpbeieligebl Guvþ_ > r sk;ésmir. > ekeg,,, ehi os os os. rémøgrmé os os os os K> / / CIMYBvC m. RsB ;f reoti! BüGuKm_ f T os os.. srmb ;
5 f / IR MeG DIsRKImINg;é f KW ' C erbh eigm ' ei ' RK; / IR ei ' smir f mrwsdu IR m½f ek) MeG Rsmvj f mrwsdu KW IR dueh smpbeieligebl reoti @ Lemm: ei / CMYB g / ek)
5 srm eigm B smpbeielig ln eri Lemm BIrdg eig) mvrgumrymknvtü ek) MeBH / / >> / CMYB g / / >>> / smpbeielig ln eig)
55 dueh smpbeieligebl Guvþ_ > r sk;ésmir mvsmpb Cuh Shwr eigm smpbeieligebl dueh, /, > rémøgrmé T os os os os os os eigm,,, MeG os,os,os mvsmpb Cuh Shwr eig) os os os os os os os os os T os os ed os os os! MeG os os os T MeG T T os
56 MeG T smpbeieligebl os os os m! eig)³ os os os dueh émøgrmé T KW T eaebledl m os os os K>RsB ;f eigm m Lemm geli eig) RsB ;f eigm smmul B dueh smpbeieligln
lmh; ## ekeg / / >>> / C MYBmGvC m. gðjf 5 / Guvþ_ > / / >>> / C MYBmGvC m. RsB ;f > RsB ;f K> MeBH / IN RsB ;f ei. AM-GM Iequl /... RsB ;f. srmb ; MeG ]mf Bdl; B ek)
58 g M MeG M RsB ;f Bdl; KW eigm M sµ / / / >>> / MeG M M / eig) M M M M MeG M M M M M
59 MeG B dueh smpbebl Guvþ_ > RsB ;f mvsmpb GM AM ek) dueh > RsB ;f eigm MeG
6 dueh K> RsB ;f m AM-GM³ dueh lmh; #% ekeg / / / >>> / lçnð eigm Edl 998 CMYBvC mmebj 998 998 998 RsB ;f 998!. evonm!((* srmb ; 998 998 998 998 998 998 998 ek) 998 998 998! 998 998 998 998 998 MeBH,.
6 B edehuf ek) dueh 998 lmh; #% ]mf / / CIMYBvC medl. RsB ;f. rusiús @@ srmb ; eigm 9! erbh mvsmpb AM-GM eig)
6 UGgÁ g GgÁ ek) smmul 9! RUv)RsB ; dueh lmh; #^ ekeg / / MeBjlçNÐ. RsB ;f. APMO srmb ; RsB ;f reoti! eigm! RsdUKñeIg) @
6 # UGgÁ g GgÁé! / @ / # gðjf sµf eig) ehi MeG ed
6 MeG ek) dueh reoti @ mvsmpb Cuh Shwr! ehi @ U! / @ eig)
65 gðjf mvsmpb AM GM # RsdUKñeIg) $ UGgÁ g GgÁ # / $ / % ek) % dueh lmh; #& ekeg A CEpñmTeTé IR EdlmlçN³ ei / CMYBEdl A ek) A. RsB ;f A IR. srmb ; ed A CEpñmTeTé IR ]mf A
66 eigm A A MeBH IR ek) A ek), A ehi MeG A ek) A srum A IR lmh; #* > rmyb / / edimi,eg 5 MeBH h /. > KN hd eddwgfrg ; mmnu M,. qmsti fñ;ti!@ qñm @!@ emøi > rmyb / / 5 eigm h ed
6 MeG 5 ln 5! MeG 5 @ MeG 5 dueh 5 / / > KN d h ei 5 / / MeG 5 h eig) d 5 d 5 l l 5 ed Rg ;mmnu, M MeG l 5l MeG 5l MeG 5l l l 5
68 dueh 5l l 5l lmh; #( s os edimi,kn F d eksresr F CTRmg; s os A s os B os s eigmn; d s os d s os A d B s os A Bl s os MeBH IR. s os B os s s os A eig) A Bs A Bos A B A B Guvþ_ KNGMgeRl s os s os > d > d s os. s os KNGMgeRl > s os d s os. emøi ]mf s os As os Bos s A Bs A Bos
69 ln eig) A B A B A / B s os s os os s d d d s os s os s os d s os d s os dueh l s os / IR s os d l s os / IR s os > s os d s os ]mf s os As os Bs os ln eig) dueh A Bs A Bos A B 5 B, A A B s os d s os 5 s os s os d d s os s os 5 s os d d s os 5 l s os / IR s os 5 d l s os / IR s os
Ésreg ( Referees ). Alfred Posmeer d Chrles T. Sld, 9, Chllegg Prolems I Geomer d Alger, Dover Pulos, INC. New Yor.. D Bráe, Io Serde d Vsle Serde,, Juor Bl Mheml Olmpds, Plus Pulshg House.. Tu Adreesu d Bogd Eesu, Mheml Olmpd Tresures, Brhuser.. Tu Adreesu d Zumg Feg,, Prolems Alger, AMT Pulshg. 5. Numer Theor Prolems,, Brhuser. g ÉsrCeRIepSgeTo. NþjsgÁm Feoo Pge: Mhems For Sholrshps Feoo Aou: Khmer Mhs Group: eromrlgghrurn_ sumgrkun