On The Brocard-ramanujan Diophantine Equation


      
 

 !
"#$%
&('*),+-/.0 132547658:9 ;<138=2>@?BADC@>@?FEHGJIKI*GLCNMPOQ;(R C@ITSQCVU WYX +[Z]\_^a`=b X=cdfe=g[X b_h]bji X i X +[ZjZjka`mlnaoqpfrVs!b_p ct +Tuv` txw uyr]z g[cdfe=g y=pfrj{5| g[} rj~ X=c i X h bj|€|i X*dg hjp‚bj|@zfr]|€ƒ d i„r X z d r … +‡† &('ˆ)‰+Š-‹. 0jŒ WX +[ŽV+[a`@ƒ X b‡qb_p g rj~‘oqpfrVs!b_p c’ z3“*ƒ g pf”j`•–n–—Šb } b X ƒ˜fb X™t Zˆuv` t Ža`y@n5 w \‡u(~šrjp } ƒ›|€b dg[c$dfemg y=pfrj{5| g[} i XDdfemg ~Jrjp } `œž emgX ƒ } { g pŸ+ )¡&('@i€zby g pf~ g s d zf“aƒ=b_p g ~šrjp dfemgŸ¢ bj|€ƒ g z%£m`
ô


ô  
 

è         

W d i€z X b d ƒmp‚bj| d rs¾r X zfi c=g p dfemg} rjp g h g[X=g p‚bj| c i„rjy e b XKd i XmgQg “*ƒ=b d i€r X … ]† &('ˆ)Ÿ-‹. 0 Œ ©n5µQb_{=pfrˆŠz·Vi t ˆu g bjzfiL|„”z e r‡ g[c$dfe b d `›~šrjp g bjs e  Ê g[c  dfe b d i€z X r d bŸzf“aƒ=b_p g ` dfemg p g iLz r X |„”Db$ X i dgX ƒ } { g p rj~ zr]|€ƒ d i„r X z!nÕrjp g y=p g s!i€z g |€”j`7i„~»&ДVi g | c z bzr]|€ƒ d i„r X `N&¡i€z| g zfz - + Œ ž e ƒ=z!` … ]†qi€z­r X |„”Ÿi X*dg p g z d i X h emg[X iLz dfe b Xdfemg | g bjz d y=p¬i }g Å ~šrjp» e i€s e¡ ¿ Á  ! bzf“aƒ=b_p g n ¥ rj"p - ä 0 ° w æ ä æ k_¨V°K g z g b_p‚s emg[c ~Jrjpqzr]|€ƒ d i€r X z7î 65 8 î ' ì (ö  )*+
-,/.1ï 032(,54$ì 6 78+9;:=?@(ACBEDEDF6GH.EIKJ$+.L,/,   2 M ONCPí3ö QLR(SET=6G(QEë RK K )*+
-,/.1032(,54$6 78+9;:=?@(ACBLULVLWE6GH.EIKJ$  XYX   2 M KZ[Pí3ö QLQL\LT=6Y]L^(ö  ]Y  _ 2 ë ,/.a` bKc 6edAf?@;jGgGk(A+hA_9l95mn8+k(?@(Aporq(sutwvyxYz16{ c IE`  ,/0 M O c b -|EZuPí3ö ^L^LSET=6 ]Eˆö~î }K]EFG î ë î HXGX G  €‚ ,5ê‡2(î ,54 c X 6 ƒ @(A+?…n8K?@(A†kˆ‡/‰‹;Š9 ?@(AŸî ê ö ŽEŽë L]ŽFë eŽnŽnŽ~‘’s±ë "ö v“xYzE6”H.LIYJO   2 M {é P•1T–fPí3ö ^L—(SET=6YLLa}KL(SY \K )*+’I ë 2Y6 dî Aˆkl˜„5@a™=k(„5iLšL›k(œlk(A+8=3k(A"jO„5@E…3‡ž9;œŸ6  2 M +  IG4 X ¡ Pí3ö ^L]E\LT=î 6GRˆFö  SY ’O’I 6 ¢A+:;@L‡ £(9~i"¤-„/@E…a‡ž9;œŸ: >hAˆ¥H8Yœ"…;9;„¦{g+9~@(„=§(6+  , c XK¨ ,©}1ª ,5«¬2 ¨ 6G `  .L, b 6K3ö ^L^(G î X Xp RK  G­ J , M .L« 6-¦{g+9i(>?@;jGgGk(AG<®>¯A_995mn8+k(?@(Aoq3s ë ìíë ë ö v±° z 6O
IY«¬«•G{. 4Y. 2 M + 1.10(GNO²CPí3ö ^L^L]ET=6 3ö —(Y QYH $G 2L³"2 X IF´/2 X 6”78+9n:~?@(AŒµ$DL¶(6O·YG X 4 c 2 Xp 2 M + K.10(+²[Pínö ^3ö ]ET=6Y\(^K ^YH $G 2L³"2 X IF´/2 X 6”ƒ @L‡h‡ž95™3<©95il¤k;jG9n„=:=6+ M « 2K6O `  .L, b 6K3ö ^LSEK ¸¹Kº»_¼E½$¾¹K¿$½ÁÀ_ÂyÃf»1½OÄ$¹G¾»1½OÅhÆOÇLȟɏ¿$ŞÊ{¹K¼$Çnů½+ËÌÀ{ÂÎÍ5ÏLÏ(ů¿_À_ÅhÇLÈÑÐaÒYÓYÔÖÕw¹YÇn½Á×ؼO¹Y¹K¿ÙO½O¼O¹K¹Y½”È ɒ¼$ÚG»$¿_»ÈÍ~Ï(Ï(ů¿_À{ůÇØÛ_ÐFÜKÓ_ÐKȀɐÙ$Ý Þ-ß œlk(>•‡{kEiEi(„~9;:~:(àá1âFã(äEåLæEç3èYé(æ(êrë®ìGínìYîë©âå(ì ïLéEðñ1é(òLç3èGé(æ(êë®ì+ínìYî{ë©â(å(ì