DOCSLIB.ORG

As (1) W Ill Fail Only for T H Ose a [1"Η % 1] Wh Ich H a V E a C O Mm on Fa C Tor

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
As (1) W Ill Fail Only for T H Ose a [1 ¢¡¤£¦¥¨§ © ¨£¨¡¨© ¡¨ !§" $#&%¨§(' £¦¡¨§*)+)+§") ,"- .0/213/547698;:=<?>A@547B2>2<?CEDGFH<?I=B¦J*K96L69CEM"N!@5I=B2>POQI=<?6SRT:=UVCW<?I=B2XECEM ¡5£¦ZS©¨¨E£¦ $[¡ Y \CE<?UVI=]+^`_baL69KL]?]?69C^*]?c2CE:=<dCEUeI=_?_?CE<?]d_]?c2I=] f?gWh i=j5kmlbn0g UV:">Hoqp i f?gWh DQc2CEB2CEJ=CE<ro K9_sI3t2<?K9UVCu]?c2I=]s>2:"CE_vB2:=]v>2K9J*KL>2C / wx8 c2:=6L>2_v8;:=<yI3XE:=UVt[:=_?KL]?CuKLB¦]?CEz=CE<¨o i , ]?c2CEB{DvCQXEI=6L6o|I}a~2E2 =~2?9 u =+ ^L/wx8IXE:=U{t:=_?K9]?CbB"2U CE<¨o|KL_qI}t2_dCE2>2:=t2<?KLU{C i fi hg -P, ]?:¢CEJ=CE< I=_?C @8;:=<7DQc2KLXc pdo @[]?c2CWBDvCXEI=696SoI¢aT=?V2¨ ¡¢[£Z^9/¥¤B2C - , - -¢,"- XEI=BPK9>2CEB¦]?KL8 OQI=<?UVK9X¦c2I=CW6AB¦2U CW<?_78;I=KL<?6 CEI=_?K96 2_?K9B2z !)+§"£¦¨$)VZ £¦§"! $[¡y© , , - § 4ªXW:=UVt:=_dKL]?CB"2U CE<o«KL_QIVOQI=<?UVK9X¦c2I=CW6AB¦2U CW<¬KL8I=B2>P:=B26 KL8o­K9_ g g - _?®"2I=<?CE8;<?CECHI=B2>¯3° >2KLJ"KL>2CE_soP° 8;:=<CWJ=CE< t2<?KLU{C¬¯¢>2KLJ*K9>2KLB2zHob/ g3f´¥µ g=gsµ|gW¶=h , ,"- c2C¬_?UVI=696LCE_d]vOuI=<dUVKLXc2I=CE6[B"2U CE<@*²=³ @DvI=_v8;:=2B2> OuI=<dUVKLXc2I=CE6 ± gW·gW¸ - - - , KLB /1uCEXECEB¦]?6 DvCVt2<d:!J=CE>¹]dc2I=]}]?c2CW<?CVI=<?C{KLB2º2B2K9]?CE6 U{I=B OuI=<dUVKLXc2I=CE6qB¦2U CE<?_+» , KLB8;I=XE]@Q]?c2I=]¢]?c2CE<?C¼I=<?C¼UV:=<?C|]?c2I=B¾½T¿xÀ+ÁÂOuI=<dUVKLXc2I=CE6¥B¦2U CE<d_Ã2t]d: ½y@Q:=B2XEC½0KL_ f h - _?2Ä{XEKLCEB¦]?6 69I=<?z=C _?CECÅ`4QFHRsÆ / , wx8roKL_}B2CEKL]dc2CE<3t2<?KLU{CVB2:=<ÇI OQI=<?U{KLXc2I=CE6qB"2U CE<@¨]?c2CEB¹]?c2CW<?CI=<?CU{:=<?CV]dc2I=B|oqÈ!É i g g f?gWh KLB¦]?CEz=CW<?_ K9B¹Å pdo¢° ÆS8;:=<bDuc2K9XcP]?c2CXE:=B2z=<?2CEB2XEC >2:*CE_QB2:=]bc2:=6L>A/ c¦2_K98DsC}t2K9X¦Ê ± i g g , I=BËK9B"]?CWz=CE< I=]H<?I=B2>2:=UÌKLBË]dc2CVKLB¦]?CE<dJ!I=6Å pdo¼° Æy]?c2CWB¹]?c2CE<?C{KL_HI CE]?]dCE<H]?c2I=B¹CWJ=CEB fdgWh Xc2I=B2XECÃ]dc2I=] DQKL696v8;I=KL6bI=B2>­_?:DsCÂDQKL696c2I!J=C¢It2<d:*:=8Q]?c2I=]HoGKL_XE:=U{t:=_?K9]?C/ wx8DsC gW¸=¸ - <?CEt[CEI=]s]dc2KL_qaL]?CE_d]^Í_?I ]dKLUVCW_@]dc2CEBÎ]?c2CE<?CuKL_qIÇUVK9B"2_dXE26LC¬Xc2I=B2XEC]?c2I=]yDsCDQKL6L6&8;I=K96]d: f , <?CEXW:=z=B2KLÏECÇ_d2X¦c I=BÃKLB¦]?CEz=CE<so¾I=_XE:=U{t:=_?K9]?C I=B2>A@&K9BP8;I=XE]@2DvC}CEÐ*t[CEXE]]d:: ]?I=KLBÂ_?2Xc|I i h - , aLDQKL]dB2CE_?_^ KLB{B2:HUV:=<dCu]?c2I=B{]xDv:H_?2XcÎ]?CW_?]?_ / c2KL_yI=6Lz=:=<?K9]?c2UÑKL_qJ=CE< CWÄVXEK9CEB"] CEXWI=2_?C ± - - :=B2CHXWI=BÃ>2CE]?CW<?UVK9B2CHt:ZDvCE<d_¬UV:">226L:Ço«CEÐ*]d<?CEUVCW6 <?I=t2K9>26 / - , - , Ò7B28;:=<?]?2B2I=]dCE6 ]?c2C{]?CE_?]Ó2_?]H>2CW_?XE<?K CE>¹<?I=<dCE6 <?CEXW:=z=B2KLÏECE_HOuI=<?U{KLXc2I=CE6rB¦2U CE<?_HI=_ l , - - , CWKLB2zXE:=U{t:=_?K9]?C /rÔ2KLB2XEC}]?c2CE<?CHI=<?C}K9B2º2B2KL]?CW6 UVI=B OQI=<?U{KLXc2I=CE6mB"2U CE<?_@&DvCHXWI=B2B2:=] -Ã,¦- _?Ê"KL<?]u]?c2KL_¬>2K9ÄVXE269] KLB2_d]?<?2XE]?K9B2zÎ:=2<¬I=6Lz=:=<dKL]?c2Ue]?:HÓ2_?]6L:*:=Ê¢:=2]8;:=<Iκ2B2KL]?CÇ69KL_?]u:=8 - , - ,¦- CEÐ"XECEt2]?K9:=B2I=62KLB¦]?CEz=CE<?_+/TÕ7:ZDvCEJ=CW<q]dc2KL_q>2KLÄVXW26L] XEI=B CuB2CWI=]?6 <dCE_?:=6LJ=CW> <dCEt26LI=XEK9B2z}]dc2C fdgWh ,¦- - © - , I _?69KLz=c¦]?6 _?]?<?:=B2z=CE<H]?CE_?] \:=<ÇI=B z=KLJ=CWB¹:*>2>¹KLB¦]?CEz=CW< aL\CE<?U{I=]H]?CW_?]^ I=_dCE>¤:=B g g g , o Ö @6LCE]r× C7]dc2C76LI=<?z=CE_d]v:*>2>¢8;I=XE]?:=<:=8o° @"_?:DvC7XEI=B¢DQ<?KL]dCbo° ÉÍØÍ×b8;:=<_?:=U{C i t:=_dKL]?K9J=C}K9B"]?CWz=CE<sÙy/Tw8¡o­KL_QIVt2<dKLUVC}Duc2K9XcP>2:*CE_QB2:=]Q>2KLJ*K9>2C @*]?c2CEB f h Ú iÍÛqn0g i Ûqn g ¸ g gâ ¿ÝÜ £¦¥¨§" ¢ É UV:*>Hoqp ° UV:">Ho«8;:=<_?:=U{C¬ÞsßËà p p+á+á+áÝpdÙ¢° á , wx8¨]?c2K9_uK9_u]?<d2CHDQc2CEBÎo­K9_uI{XE:=UVt[:=_?KL]dCHB¦2U CE<u]?c2CEBPDvC}XEI=6L6o«IVa9+? = 2ãV~2+E2 =~2?LP i ¶=¸ f h , - - «= ^L/ÎwxB¹]?c2CÎUVK9>2ä ^å_ÇÔ2CE6L8;<dKL>2z=C2_?CW>­I ]?CW_?] I=_?CE>¤:=B É ]?: <?I=t2K9>26 KL>2CEB¦]?KL8 , - , XE:=U{t:=_?K9]?C}B¦2U CW<?_@DQc2KLXcPDs:=<?Ê*_QDuc2CW]?c2CE<:=<QB2:=]]dc2C I=<?C}OQI=<?U{KLXc2I=CE6mB"2U CE<?_/ æçÝèÝé!éÍê¦ëíìîdï{ðòñsé*óxëìmô+õ3ó¦ñö÷!ø¥ùë?óñdì?ú ñ ëîdï¡ ú÷ êó¦ñ£¢&î£çíî+ó륤§¦7øî ꩨyú ûýü üþ ÿ þ þòþ ç ¥ ¨yð ó¦ð êÍñ õ ê¦ëm쩦Ýêçî § ¨¦Z𠤧¦¦!ó¥ îQó!¤dê#"$"¨êñ ó%¤£ìíê¦ë&¨yð ì'¦(ú ü þ þ+ÿ þ þ ÿ ÿ õ+éÍî£çíî£ìAô+õ+*-,/.10 ) )3254 6 i i f h ¿ 47B¼KLB¦]?CEz=CE< K9_XWI=6L6LCW>I¢a87T A?Z^ 8;:=<uoKL8o>2:"CE_¥B2:=]>2K9J*K9>2C I=B2>¼KL8 É 8;I=K96L_§9/ , - Ô2CE698;<?KL>2z=C{c2I=>Ë: _?CE<?J=CW>Ë]?c2I=]]?c2CE<?C{I=<?C{I=69DvI _IÃ6L:=]¥:=8vDQKL]dB2CE_?_?CE_}8;:=<= ;:XE:=UVt[:=_?KL]?C - , - KLB¦]?CEz=CW<ob»&U{:=<?CÇt2<?CWXEKL_?CW6 @=<|:=B2K9CE<3Å><¼ÆI=B2>P1I K9B«Å`1uÆ KLB2>2CWtCEB2>2CEB¦]?6 t2<?:ZJ=CE> ]?c2I=]I=] i g g 6LCWI=_?]}]dc2<?CECEä®*2I=<?]?CW<?_}:=8y]?c2CVK9B"]dCEz=CE<?_ KLB¼]?c2CVK9B"]?CW<?J!I=6QÅ pdo|° ÆTI=<dC{DQKL]?B2CW_?_?CE_}8;:=<ob/ i , wxUVK9]?I=]?KLB2z7]dc2Ct2<?:*XECW>22<?CuI :!J=C=@DvCB2:=]?C]?c2I=]TB2:ZD¾K98DsCQ_dCE6LCEXW]TI=B{K9B"]?CWz=CE< I=]¨<?I=B2>2:=U g g , , KLBÃ]?c2CÇKLB¦]?CE<dJ!I=6vÅ pdo° Æ]?c2CWB ]?c2CE<?CÇK9_I8;I=< CE]?]dCE<¬]?c2I=B CEJ=CWB X¦c2I=B2XWC3]?c2I=]K9]DQKL6L6 C , - - IPDQKL]?B2CW_?_Ç8;:=<7ob» I=B2>¹_?: DsCXEI=B CVI=6LUV:=_d]}XECW<?]?I=KLB¹]dc2I=]HDvCK9>2CEB¦]?KL8 I=B XE:=UVt[:=_?KL]?C , B¦2U CE<¬K9B(Ó2_?]uIV8;CEDG_?2Xc ]?CW_?]?_%?2/ f h -*, , Ô2:¼UVI C¢DvC¢XEI=B 2_?C É ]?:¼]?CE_?](DQc2CE]?c2CE<{IB¦2U CE<VKL_t2<?KLU{CA@ wxB2>2CECE>A@<|KL696LCE< - t2<?:ZJ=CE>A@ I=_?_?2U{KLB2z|]?c2C]d<?2]?c¹:=8bI=B¹I=t2t2<?:=t2<?K9I=]?Cz=CEB2CE<dI=6LKLÏWI=]?KL:=B¹:=8]?c2CV1KLCWUVI=B2B¤Õ ä f h , t:=]dc2CE_?KL_ FH1Õ @A]?c2I=]ÇK98ro KL_3XE:=U{t:=_?K9]?C]?c2CWB«]dc2CE<?CÎU32_?] CÎ_?:=UVCÎa9_?UVI=696^SJ!I=692C:=8 i f h - - 8;:=<Duc2K9X¦c É 8;I=KL69_@¡]?c"2_}z=KLJ"KLB2z¢IÂa9t:=6 B2:=UVK9I=6¨]?K9UVC^>2CE]dCE<?UVK9B2KL_?]dKLX(t2<?KLU{I=6LKL] ]dCE_?]/ f h BACE]DC o >2CEB2:=]?C7]?c2C6LCEI=_?]yt:=_?K9]?KLJ=CuDuK9]?B2CE_?_8;:=< ob/¨\[:=696L:ZDuK9B2zÇDv:=<?Ê(:=8E<|KL6L69CE<@=¤}CW_?]?CE<?6GFC f h I=B2>IHbI=Xc _dCECÎÅ>HQÆ @*DvCB2:!DGÊ"B2:!D ]?c2I=] ¿ f´=h f h K w8]?c2CHF}1Õ KL_]d<?2C}]dc2CEBJC o Éq6L:=z oqá f h .ËCÇI=<dC3XE:=B2XECE<?B2CW>K9B¼]dc2KL_¬t2I=t[CE<7DuK9]?c >2CE]?CW<?UVK9B2KLB2z¢c2:!D 6LI=<?z=CC o XWI=B¼z=CW]/Qwx]7KL_ f h , Ê*B2:ZDQBÂ]?c2I=]É(KL_I(DQKL]?B2CE_d_Q8;:=<bU{:=_?]:*>2>¢XE:=UVt[:=_?KL]dCB¦2U CW<?_ _?CECVÅMLsÆ[I=B2>ËÅåRsÆ /"Õ7:ZDuä - - , CEJ=CE<K9]3KL_I=6L_?:|Ê"B2:!DQB­]dc2I=]3]?c2CE<?C¢I=<?C¢KLB2º2B2K9]?CE6 UVI=B _?]?<d:=B2zt2_?CW2>2:=t2<?KLU{CE_ I=_?C¢É@ ´ , _?:¹]?c2I=]V]?c2CP6LCWI=_?]DQKL]?B2CE_d_KL_]dc2CEB I=]V6LCWI=_?] /Ô2tCEXEK9º2X CEÐ*I=U{t26LCE_Vc2I!J=C CECEB 8;:=2B2> f h f´ g ¸=´gW¶ gWh¹ g=g f h - © KLB DQc2KLXcNC o KL_ 8;I=K9<?6 69I=<?z=C \:=< KLB2_d]?I=B2XECOC É ² ² Å`RyÔ2. Æ I=B2> f´1Pg ¸=¸=¶gW¶ ´ gWh ´ f h§S ´=´=¶ - C ²=² É1Q É É Å>R¦Æ /mÅ`47<dÆAt2<d:!J"KL>2CE_uI=BÃCEÐ"]?<?I=:=<?>2K9B2I=< CEÐ*I=U{t26LC:=8¨I ä ¸=¸ >2KLz=K9]b:*>2>PXE:=U{t:=_?K9]?C@5DQc2:=_?CH69CEI=_?]b~2?LPQDQKL]?B2CW_?_CEÐ"XECECE>2_É / f h ´ , - - Rs<dKL:=<Q]?:V]?c2K9_t2I=t[CE<¬KL]Qc2I=> B2:=] CECEB t2<?:ZJ=CE>P]dc2I=](C o Ö 8;:=<¬K9B2º2B2KL]?CE6 UVI=B ob@ f h , , - CEJ=CEB¢]?c2:=2z=cÎK9]vc2I=_v69:=B2z CECEB¢CEÐ"tCEXW]?CE>¢]?c2I=]DC o XEI=B¢z=CE]vI=< KL]?<?I=<dKL6 69I=<?z=C/¨Õ7CE<?C¬DsC © t2<?:ZJ=CH]?c2K9_uI=B2>ÃUV:=<?C ¥¨§"[Z§VU WYXNZY[Y\']A\^A]A\`_badcea_bf©\'gihkjl^Aahnmo^G]jp_rq©[Y^s\'ga=tjlu\']v [YyGvz\`g\5^Gv5f _bf¥a{\'v5v oxw w T f h l jV n l l¥©© _bv|g^A]~}\©]|f¥[Y^Aa aOA^sq©f+f[\']\l^A]\p v5tEq'[ 6L:=zo ÀGi? © © © ÀG © © © xAyA]v5tq'_r\©afgihg^A]r}\ o ´ f h , - wxB}_dCEXE]?K9:=B DvCrDuK96L6ÍI=<?z=2Cq]?c2I=]S]?c2CqUVI=Ð"KLU{I=6Í:=<?>2CE<m:=8dC o KL_mt2<?CE_d2UVI 6 69:=z o6L:=zq6L:=zob@ ¸ - , 8;:=<r_?:=UVCbXE:=B2_d]?I=B"] Ö @Í]?c2:=2z=c(]?c2CE<?CbI=<?CbUVI=B : _?]dI=XE6LCE_q]?:¥]?2<?B2K9B2z}:=2<ra9I=<?z=2UVCEB¦]^ , KLB¦]?:ÎI¢t2<?:":=8!/uÕ7:ZDvCEJ=CW<72B2>2CE<¥]?c2C3I=_?_?2U{t2]?KL:=B|:=8rI¢_?2KL]?I 69CÇ2B2KL8;:=<dU J=CE<?_?K9:=B :=8q]dc2C f h , t2<?K9UVC|"äí]d2t26LCE]?_7XW:=BÓCEXE]?2<?CDvC3I=<dC3I 6LC3]?:Î_?c2:ZDª]dc2I=]¬]?c2C3UVI=Ð"KLU{I=6:=<?>2CW<7:=8{C o KL_ ¸ f i1 g â I=]s69CEI=_?] 69:=z o8;:=<v_?:=UVCXE:=B2_?]?I=B¦] Ö / 4 _?CE]s:=8S6LKLB2CWI=<v8;:=<?UV_bà ½ p Þ ¡ g `K i1 - KL_XEI=696LCE>¼a9{L??+^¡K98 ¢ 8;:=<CEI=Xc Þ+@¡I=B2>Ë8;:=<¥CEJ=CW< t2<?KLU{CH¯m@¡]?c2CE<dCVCEÐ"KL_?]?_ id - - I=B«KLB¦]?CEz=CE<}oe£P_?2Xc­]?c2I=]¯«>2:"CE_{B2:=]>2K9J*KL>2C¢I=B :=8u]dc2C oe£ /¼Õ7I=<d> I=B2>¤BAKL]?ä - , ]?69CEDv:*:">A^`_baLt2<dKLUVCk"äí]d2t26LCE]?_QXE:=BÓCEXW]?2<?C^AÅ`Õ£BAÆmXE:=B"]dCEB2>2_]?c2I=]b8;:=<QI=B I=>2U{KL_?_?K 6LC_?CE]Q:=8 i1 h - - 6LK9B2CEI=<b8;:=<?UV_+@*]?c2CE<?CI=<?C¥KLB2º2B2K9]?CE6 U{I=B K9B"]dCEz=CE<?_yo«8;:=<bDQc2KLXcÃCWI=X¦c o¥¦ KL_bt2<?KLU{C/ § ìíêb쩦Ýî ó%¤£ìm쩦Zóxì{¢ð ç¤dê#"¨éÍêç?ð ìíî ÿ ¨ 쩦Zóxì7ð ç©© ð çªz«¬bóbçíìíëêÍñÝùÇéZçíîEèïZêéZëÝð­"TîìíêHô*óçíî£ú Íî£ë©¦ZóxéZçvô*óxçîdço3ìíê¨ ¦Z𠤧¦® ð ç7óQçíìëíêñù éZçíî+èÝïZêéZëÝð¯"¨îr穦Ýêè ï7ôîqëî îdëëíî£ï¬ìíê}óç(°²±³¯´bµ©´¯¶8·òú þ ÿ ¸ ¤£ìÝèZó õ+¹îG¦3"¨îdëº ¹ZîG¦©ó¦ñï7÷ê ê© ZóíõuóñÝï¥÷Ýìíë?óxççíî+ñ ÷!÷©ñÝêìîdïvì'¦!óì2êÍñÝî¤Eóñê¦ôZìÝó¦ðòñçÝ詤§¦só[ç?èëî¼»Zëíî ü þòþ þ ¤dê#"¨éÍê¦çÝð ìî+ñÝîdçíçAìíî£çíìyèÝçÝðòñÝùó¨é!ëêA¤dî£ï5èÝëíîuðòñdìîdë©"¨î£ï5ðòóìíîvðòñsçíìëíîEñù¦ì©¦vôÍî£ì¨îdî+ñ+ ¥ Aó¦ñÝï¥ ú 6 ½ ¦î£çíîuñ+èz"AôÍîdëçsóëíîA©"ð ñ ó%¤£ì©©;°²¾%¿z±À!Á-´¯«ª3¶·¯©=ðòñsì'¦!óì{Â( Ã=mð çmç©"bó î£ë ê¦ësó ç'"Qó î£ë&[ú ÿ þòþ ÿ þòþ þòþ ) Ä ÷î£îuó çê¥ ÅEÆ=;ú þ Ç g ¡ zU É ! U¹§ £¦!ɨ$§Z£¦)([¡1˧*E£¦¨Z§VXÍÌeyA]J\5^sq'[l_baf©\8}\'] ºf¥[Y\']A\p\'Î_v5f®q5yAaÏ È #Ê ¸ i1 g v5f©^Aaf¥vÐ$Ñ Ñ v5tEq'[Íf[^AfeAyA]^Aah¤Ói^sÔAjp_bvv5_ru'g\Õ-vz\'f®yr£gi_ba{\5^G]$GyA]jpv p¼Ò Ö à ½JÖ p â fi i i h¼×ÙØ id f¥[Y\']A\£\'Î_bv5f¥v^AaJ_ba=f'\8}V\'] Ñ Ñ v5tEq'[Úf[^Af(\^sq'[ _bv{ÜY]§_bjl\ Þ ¡ á+á+á oÛl o ¤Ò l ¿ ÌgPf h Ð ²"á² Ô22Xc¤IP<?CE_?269]ÇKL_HÊ*B2:ZDuB¹8;:=<¡ BAKLB2B2KLÊ¡^`_ c2CE:=<?CWU I=B2>¹CEJ=CWB¹DuK9]?c ± l f h -|, , _?CWC Å`Õ£HbÆ »I=B2>¹KL]KL_}DuK9>2CE6 CW6LKLCWJ=CE>¹]?c2I=]]?c2CVI :!J=C{2B2KL8;:=<?UÌJ=CE<?_?K9:=BË:=8v]?c2C{t2<?KLU{C ´ "äí]d2t26LCE]?_uXE:=BÓCEXE]d2<?CÇKL_Q]?<?2C=/rwxBP_?CWXE]?KL:=B DvCt2<?:ZJ=CH]?c2C}8;:=6L6L:ZDQKLB2z(<?CE_?269]/ ¥¨§"[Z§VU Ý=X/Þt3ÜsÜyAv#\®f¥[Y^Af`f¥[Y\/Óbßa_¯AyA]§jàÜY]§_bjl\f]§_¯ÜYg\'f¥vJq5yAadáA\q'ftV]\VÕ_bvf]§t\¥â8f¥[Y^Af`_bv T ´ã ¸ AyA] Z[Y\']\\'Î_v5fv ^xqyAav5f©^AafÚä vtq'[¤f¥[Y^AfÚf¥[Y\']\^A]A\_ba1c{a_bf©\'gihåjl^Aa=h
Load more

Recommended publications
  • Related Distribution of Prime Numbers
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CSCanada.net: E-Journals (Canadian Academy of Oriental and... ISSN 1923-8444 [Print] Studies in Mathematical Sciences ISSN 1923-8452 [Online] Vol. 6, No. 2, 2013, pp. [96{103] www.cscanada.net DOI: 10.3968/j.sms.1923845220130602.478 www.cscanada.org Related Distribution of Prime Numbers Kaifu SONG[a],* [a] East Lake High School, Yichang, Hubei, China. * Corresponding author. Address: East Lake High School, Yichang 443100, Hubei, China; E-Mail: skf08@ sina.com Received: March 6, 2013/ Accepted: May 10, 2013/ Published: May 31, 2013 Abstract: The surplus model is established, and the distribution of prime numbers are solved by using the surplus model. Key words: Surplus model; Distribution of prime numbers Song, K. (2013). Related Distribution of Prime Numbers. Studies in Mathematical Sci- ences, 6 (2), 96{103. Available from http://www.cscanada.net/index.php/sms/article/ view/j.sms.1923845220130602.478 DOI: 10.3968/j.sms.1923845220130602.478 1. SURPLUS MODEL In the range of 1 ∼ N, r1 represents the quantity of numbers which do not contain 1 a factor 2, r = [N · ] ([x] is the largest integer not greater than x); r represents 1 2 2 2 the quantity of numbers which do not contain factor 3, r = [r · ]; r represents 2 1 3 3 4 the quantity of numbers do not contain factor 5, r = [r · ]; ... 3 2 5 In the range of 1 ∼ N, ri represents the quantity of numbers which do not contain factor pi , the method of the surplus numbers ri−1 to ri is called the residual model of related distribution of prime numbers.
    [Show full text]
  • Generator and Verification Algorithms for Prime K−Tuples Using Binomial
    International Mathematical Forum, Vol. 6, 2011, no. 44, 2169 - 2178 Generator and Verification Algorithms for Prime k−Tuples Using Binomial Expressions Zvi Retchkiman K¨onigsberg Instituto Polit´ecnico Nacional CIC Mineria 17-2, Col. Escandon Mexico D.F 11800, Mexico [email protected] Abstract In this paper generator and verification algorithms for prime k-tuples based on the the divisibility properties of binomial coefficients are intro- duced. The mathematical foundation lies in the connection that exists between binomial coefficients and the number of carries that result in the sum in different bases of the variables that form the binomial co- efficent and, characterizations of k-tuple primes in terms of binomial coefficients. Mathematics Subject Classification: 11A41, 11B65, 11Y05, 11Y11, 11Y16 Keywords: Prime k-tuples, Binomial coefficients, Algorithm 1 Introduction In this paper generator and verification algorithms fo prime k-tuples based on the divisibility properties of binomial coefficients are presented. Their math- ematical justification results from the work done by Kummer in 1852 [1] in relation to the connection that exists between binomial coefficients and the number of carries that result in the sum in different bases of the variables that form the binomial coefficient. The necessary and sufficient conditions provided for k-tuple primes verification in terms of binomial coefficients were inspired in the work presented in [2], however its proof is based on generating func- tions which is distinct to the argument provided here to prove it, for another characterization of this type see [3]. The mathematical approach applied to prove the presented results is novice, and the algorithms are new.
    [Show full text]
  • 25 Primes in Arithmetic Progression
    b2530 International Strategic Relations and China’s National Security: World at the Crossroads This page intentionally left blank b2530_FM.indd 6 01-Sep-16 11:03:06 AM Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 SA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 K office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Cataloging-in-Publication Data Names: Ribenboim, Paulo. Title: Prime numbers, friends who give problems : a trialogue with Papa Paulo / by Paulo Ribenboim (Queen’s niversity, Canada). Description: New Jersey : World Scientific, 2016. | Includes indexes. Identifiers: LCCN 2016020705| ISBN 9789814725804 (hardcover : alk. paper) | ISBN 9789814725811 (softcover : alk. paper) Subjects: LCSH: Numbers, Prime. Classification: LCC QA246 .R474 2016 | DDC 512.7/23--dc23 LC record available at https://lccn.loc.gov/2016020705 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Copyright © 2017 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, SA. In this case permission to photocopy is not required from the publisher. Typeset by Stallion Press Email: [email protected] Printed in Singapore YingOi - Prime Numbers, Friends Who Give Problems.indd 1 22-08-16 9:11:29 AM October 4, 2016 8:36 Prime Numbers, Friends Who Give Problems 9in x 6in b2394-fm page v Qu’on ne me dise pas que je n’ai rien dit de nouveau; la disposition des mati`eres est nouvelle.
    [Show full text]
  • Exceptional Prime Number Twins, Triplets and Multiplets
    Exceptional Prime Number Twins, Triplets and Multiplets H. J. Weber Department of Physics University of Virginia Charlottesville, VA 22904, U.S.A. November 11, 2018 Abstract A classification of twin primes implies special twin primes. When applied to triplets, it yields exceptional prime num- ber triplets. These generalize yielding exceptional prime number multiplets. MSC: 11N05, 11N32, 11N80 Keywords: Twin primes, triplets, regular multiplets. 1 Introduction Prime numbers have long fascinated people. Much is known about arXiv:1102.3075v1 [math.NT] 15 Feb 2011 ordinary primes [1], much less about prime twins and mostly from sieve methods, and almost nothing about longer prime number multiplets. If primes and their multiplets are considered as the elements of the integers, then the aspects to be discussed in the following might be seen as part of a “periodic table” for them. When dealing with prime number twins, triplets and/or mul- tiplets, it is standard practice to ignore as trivial the prime pairs 1 (2,p) of odd distance p − 2 with p any odd prime. In the following prime number multiplets will consist of odd primes only. Definition 1 A generalized twin prime consists of a pair of primes pi,pf at distance pf − pi = 2D ≥ 2. Ordinary twins are those at distance 2D =2. 2 A classification of twin primes A triplet pi,pm = pi +2d1,pf = pm +2d2 of prime numbers with pi <pm <pf is characterized by two distances d1,d2. Each triplet consists of three generalized twin primes (pi,pm), (pm,pf ), (pi,pf ). Empirical laws governing triplets therefore are intimately tied to those of the generalized twin primes.
    [Show full text]
  • Prime Number Research Program (Prp) by Theo Kortekaas
    Prime number Research Program (PrP) by Theo Kortekaas Description and user guide PrP version 2.0 Purpose of the program is to generate and to manipulate prime numbers in the number domain from 0 to 264. The program is designed to operate in a 32-bit Windows environment on a typical desktop or laptop computer with a main memory of at least one gigabyte. Manipulation can be defined as: count primes; generate twin primes; count prime gaps and manufacture statistics about prime gaps; search for prime k-tuple constellations. The method used to generate prime numbers is the sieve of Eratosthenes. See website: en.wikipedia.org/wiki/Sieve_of_Eratosthenes The sieve of Eratosthenes To generate prime numbers up to 264 with the sieve of Eratosthenes you need prime numbers up to 232 (root 264 is 232). Furthermore, a sequence of consecutive numbers is required from 0 to 264. We will see that this is not possible (with the currently available computers) and what alternative possibilities there are. The first action of the PrP program is to set up a table with all the prime numbers up to 232. (This is also done using the sieve of Eratosthenes). That are 203,280,221 prime numbers. These primes can be up to 32 bits long. So four bytes are required to store each prime. A complete table would cover more than 800 megabytes. Because this table is to be used frequently, it should be in computer memory permanently. That puts too much pressure on the computer memory. It was decided not to store prime numbers itself, but to store the gaps between primes in a table.
    [Show full text]
  • Arxiv:1110.3465V19 [Math.GM]
    Acta Mathematica Sinica, English Series Sep., 201x, Vol. x, No. x, pp. 1–265 Published online: August 15, 201x DOI: 0000000000000000 Http://www.ActaMath.com On the representation of even numbers as the sum and difference of two primes and the representation of odd numbers as the sum of an odd prime and an even semiprime and the distribution of primes in short intervals Shan-Guang Tan Zhejiang University, Hangzhou, 310027, China [email protected] Abstract Part 1. The representation of even numbers as the sum of two odd primes and the distribution of primes in short intervals were investigated in this paper. A main theorem was proved. It states: There exists a finite positive number n0 such that for every number n greater than n0, the even number 2n can be represented as the sum of two odd primes where one is smaller than √2n and another is greater than 2n √2n. − The proof of the main theorem is based on proving the main assumption ”at least one even number greater than 2n0 can not be expressed as the sum of two odd primes” false with the theory of linear algebra. Its key ideas are as follows (1) For every number n greater than a positive number n0, let Qr = q1, q2, , qr be the group of all odd primes smaller than √2n and gcd(qi, n) = 1 for each { · · · } qi Qr where qr < √2n < qr+1. Then the even number 2n can be represented as the sum of an ∈ odd prime qi Qr and an odd number di = 2n qi.
    [Show full text]
  • Maximal Gaps Between Prime K-Tuples: a Statistical Approach
    Maximal Gaps Between Prime k-Tuples: A Statistical Approach Alexei Kourbatov JavaScripter.net/math [email protected] Abstract Combining the Hardy-Littlewood k-tuple conjecture with a heuristic application of extreme-value statistics, we propose a family of estimator formulas for predicting max- imal gaps between prime k-tuples. Extensive computations show that the estimator a log(x/a) ba satisfactorily predicts the maximal gaps below x, in most cases within − an error of 2a, where a = C logk x is the expected average gap between the same ± k type of k-tuples. Heuristics suggest that maximal gaps between prime k-tuples near x are asymptotically equal to a log(x/a), and thus have the order O(logk+1 x). The dis- tribution of maximal gaps around the “trend” curve a log(x/a) is close to the Gumbel distribution. We explore two implications of this model of gaps: record gaps between primes and Legendre-type conjectures for prime k-tuples. 1 Introduction Gaps between consecutive primes have been extensively studied. The prime number theorem arXiv:1301.2242v3 [math.NT] 14 Apr 2013 [15, p. 10] suggests that “typical” prime gaps near p have the size about log p. On the other hand, maximal prime gaps grow no faster than O(p0.525) [15, p. 13]. Cram´er [4] conjectured 2 that gaps between consecutive primes pn pn−1 are at most about as large as log p, that is, 2 − lim sup(pn pn−1)/ log pn = 1 when pn . Moreover, Shanks [28] stated that maximal prime gaps−G(p) satisfy the asymptotic equality→ ∞ G(p) log p.
    [Show full text]
  • Prime Numbers
    Prime numbers Frans Oort Department of Mathematics University of Utrecht Talk at Dept. of Math., National Taiwan University (NTUmath) Institute of Mathematics, Academia Sinica (IoMAS) 17-XII-2012 ··· prime numbers “grow like weeds among the natural numbers, seeming to obey no other law than that of chance, and nobody can predict where the next one will sprout.” Don Zagier Abstract. In my talk I will pose several questions about prime numbers. We will see that on the one hand some of them allow an answer with a proof of just a few lines, on the other hand, some of them lead to deep questions and conjectures not yet understood. This seems to represent a general pattern in mathematics: your curiosity leads to a study of “easy” questions related with quite deep structures. I will give examples, suggestions and references for further study. This elementary talk was meant for freshman students; it is not an introduction to number theory, but it can be considered as an introduction: “what is mathematics about, and how can you enjoy the fascination of questions and insights?” Introduction In this talk we study prime numbers. Especially we are interested in the question “how many prime numbers are there, and where are they located”? We will make such questions more precise. For notations and definitions see § 2. The essential message of this talk on an elementary level: mathematicians are curious; we like to state problems, find structures and enjoy marvelous new insights. Below we ask several questions, and we will see that for some of them there is an easy an obvious answer, but for others, equally innocent looking, we are completely at loss what the answer should be, what kind of methods should be developed in order to understand possible approaches: fascinating open problems.
    [Show full text]
  • MAT 311: Number Theory Spring 2006
    MAT 311: Number Theory Spring 2006 HW2 - Solutions 1. (Davenport, pp.215-216, ex. 1.04) In general, given any positive integer n, then {(n + 1)! + m : 2 ≤ m ≤ n + 1} is a set of n consecutive composite numbers, because m divides (n + 1)! (and hence (n + 1)! + m) whenever 2 ≤ m ≤ n + 1. 2. (Davenport, pp.215-216, ex.1.05) If we evaluate n2 + n + 41 for first few n = 0, 1, 2,... we see that they turn out to be primes. However, for n = 40, we have n2 + n + 1 = 412 which is composite. Alternatively, n = 41 actually divides n2 + n + 41 since each term is divisible by 41. It is an interesting fact that for n = 0, 1,..., 39 this expression gives prime numbers. This can be checked easily by writing a simple program (for instance in pari) for(n=0,40, if(isprime(n)=0, print(n))) which will print 40 as output. 3. (Davenport, pp.215-216, ex. 1.11) Assume that n is a composite number, say n = ab, where a, b ≥ 2. We want to show that 2n − 1 cannot be prime. Indeed, 2n − 1 = 2ab − 1 = (2a)b − 1 = (2a − 1)((2a)b−1 + (2a)b−2 + ··· + (2a)1 + 1). Now, since a ≥ 2, we have 2a − 1 ≥ 3. Moreover, 2a − 1 is strictly less than 2n − 1 since a < n. Hence 2n − 1 is a product of two numbers both of which are > 1. Therefore, 2n − 1 cannot be a prime. The converse does not hold, as for n = 11, we have 211 − 1 = 23 · 89.
    [Show full text]
  • Proof of Infinite Number of Triplet Primes
    Proof of Infinite Number of Triplet Primes Stephen Marshall 28 May 2014 Abstract This paper presents a complete and exhaustive proof that an Infinite Number of Triplet Primes exist. The approach to this proof uses same logic that Euclid used to prove there are an infinite number of prime numbers. Then we prove that if p > 1 and d > 0 are integers, that p and p + d are both primes if and only if for integer n (see reference 1 and 2): 1 ( 1)dd! 1 1 = ( 1)! + + + p + d p p + d − 푛 푝 − � � We use this proof and Euclid logic to prove푝 only an infinite number of Triplet Primes exist. However we shall begin by assuming that a finite number of Triplet Primes exist, we shall prove a contradiction to the assumption of a finite number, which will prove that an infinite number of Triplet Primes exist. Introduction In number theory, a Prime Triplet is: A set of three prime numbers of the form (p, p + 2, p + 6) or (p, p + 4, p + 6), see reference 3. Another way of describing prime triples is that they are three consecutive primes, such that the first and the last differ by six. For example: (5, 7, 11), (7, 11, 13), (11, 13, 17), (13, 17, 19), (17, 19, 23), (37, 41, 43), (41, 43, 47), (67, 71, 73), (97, 101, 103), and (101, 103, 107). 1 It is conjectured that there are infinitely many such primes, but has not been proven yet, this paper will provide the proof. In fact the Hardy-Littlewood prime k-tuple conjecture (see reference 4) suggests that the number less than x of each of the forms (p, p+2, p+6) and (p, p+4, p+6) is approximately: The actual numbers less than 100,000,000 are 55,600 and 55,556 respectively.
    [Show full text]
  • Prime Numbers– Things Long-Known and Things New- Found
    Karl-Heinz Kuhl PRIME NUMBERS– THINGS LONG-KNOWN AND THINGS NEW- FOUND A JOURNEY THROUGH THE LANDSCAPE OF THE PRIME NUMBERS Amazing properties and insights – not from the perspective of a mathematician, but from that of a voyager who, pausing here and there in the landscape of the prime numbers, approaches their secrets in a spirit of playful adventure, eager to experiment and share their fascination with others who may be interested. Third, revised and updated edition (2020) 0 Prime Numbers – things long- known and things new-found A journey through the landscape of the prime numbers Amazing properties and insights – not from the perspective of a mathematician, but from that of a voyager who, pausing here and there in the landscape of the prime numbers, approaches their secrets in a spirit of playful adventure, eager to experiment and share their fascination with others who may be interested. Dipl.-Phys. Karl-Heinz Kuhl Parkstein, December 2020 1 1 + 2 + 3 + 4 + ⋯ = − 12 (Ramanujan) Web: https://yapps-arrgh.de (Yet another promising prime number source: amazing recent results from a guerrilla hobbyist) Link to the latest online version https://yapps-arrgh.de/primes_Online.pdf Some of the text and Mathematica programs have been removed from the free online version. The printed and e-book versions, however, contain both the text and the programs in their entirety. Recent supple- ments to the book can be found here: https://yapps-arrgh.de/data/Primenumbers_supplement.pdf Please feel free to contact the author if you would like a deeper insight into the many Mathematica programs.
    [Show full text]
  • A78 Integers 18 (2018)
    #A78 INTEGERS 18 (2018) NEW CONGRUENCES AND FINITE DIFFERENCE EQUATIONS FOR GENERALIZED FACTORIAL FUNCTIONS Maxie D. Schmidt School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia [email protected], [email protected] Received: 2/11/17, Revised: 5/8/18, Accepted: 9/24/18, Published: 10/5/18 Abstract The generalized factorial product sequences we study in this article are defined symbolically by p (↵, R) = R(R + ↵) (R + (n 1)↵) for any indeterminate R. This definition of the generalized n · · · − factorial functions pn(↵, R) includes variants of the classical single and double factorial functions, multiple ↵-factorial functions, n!(↵), rising and falling factorial polynomials such as the Pochhammer symbol, and other well-known sequences as special cases. We use the rationality of the generalized th h convergent functions, Convh(↵, R; z), to the infinite J-fraction expansions enumerating these generalized factorial product sequences first proved by the author in 2017 to construct new con- gruences and h-order finite di↵erence equations for generalized factorial functions. Applications of the new results we prove in this article include new finite sums and congruences for the ↵-factorial functions, restatements of classical necessary and sufficient conditions of the primality of special integer subsequences and tuples, and new finite sums for the single and double factorial functions modulo integers h 2. ≥ 1. Notation and Other Conventions in the Article 1.1. Notation and Special Sequences Most of the special conventions in the article are consistent with the notation employed within the Concrete Mathematics reference, and the conventions defined in the introduction to the first articles [11, 12].
    [Show full text]