DOCSLIB.ORG

Testing the Converse of Wolstenholme's Theorem

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Testing the Converse of Wolstenholme's Theorem ¢¡¤£¥ §¦©¨ ¢ ¡ ¨¡£¡ £¥ ¢¡¨ ¡¢ £ ¢ ¡!"#¡ $&%('*),+- .+0/21¥$435)76 8 /96:6;/=<?>@ /9AB/2+ CBDFEHGJILKNMOG PQHRTSOU(U4VTQHSOR5W4XYU4ZNRT[9\O]_^`\aRbU [ XYcNd©\aRTefXgVbcihYjOkOl:U(d©\JmnUo[4dNSp[,Vq]Nris¤tuVTUvSuwNW(VTe:X cxZyez?XHW,[4d©XYc { l&rB|!h hfef\r5 J& r}|!ho~¢ [4U¥Q\acOXHW&U XO2U [4Sp[4VTc©i[4dNSp[¥SicNSp[4Z©W(SOR0rU4Sp[4VbU ]VTc©[4dyVTUQ\ac©OW(ZNXYcNQXVTU¥c©XYQ XYU4U4SpW(VbRq.SwNW(Vbe:XucZNe¥z?XHWYVTUgQ\aefe:\acNRT.z?XYRTVqXHOXY[ \z?X;[ W(Z©XO_SORT[4d©\aZ©adc©\ wNW4\\O]0dNSOU9z?XHXYcaVqOXYcBU \¥]SpWY cf[4dNVTUoc©\O[ XOmXwNW4XYU4XYc[oSOc}XYRqXYefXYc[4SpW4}wNW4\\O] \O]nSwySpW4[4VTSORW4XYU4ZyRq[H2cNSOe:XYRqO=[4dySp[u[4dNXQ\acxOXHW(U XVbU[ W(ZNXB]\OW;XHOXYccxZyez?XHW(U SOcN]\OW w5\¡mXHW&U \O]o¢:£NZ©W([4d©XHWYmoX:wNW4\¡OX:[4dNSp[Vq]9¤¥¦r5§=VTUgSwNW(VTe:Xuw?\JmoXHW [4d©XYc { { lL¤|!h l&r|!h e:\ry¨J&© ¤|!h rB|!h ~¦ ~ wNW4\NZNQHVTcN¥S;W4XYRTSp[4VTOXYRqVTcNXªw?XYcNU4VTOX«Q\acxOXHW(U X[ XYU4[,]\OW,w?\JmoXHW(U,\O]¬\ywNW(VTe:X cxZyez?XHW(UH ­B® EJ¯F°± e³W4XYU4ZyRq[4SO©\QHRY´SOU4U(VTQ\©X7^`\aRTU4[ XYcNd©\aRTe:X«e:\aU [ W(SµZ©X7U X2r´XnZNe³cg´ZNe:XHW4\ ² wNW(VTef\©XYcx[H¶Sp\ { l&rB|!h hfef\r & r}|!h ~¢ PW4XYQO´ wNW(\QHSNXYU4U Xf[ XH\OW4XYe}S¸VbcNNVTQHSOcNN\µZ©X}Zye¹cg´ZNe:XHW(\cNSp[4Z©W(SOR?rU4SL · [4VTU ]SpºHXYcN©\SiQ\acNOW(Z?»XYcNQHVTS¼´XBcNXYQXYU4U4SpW(VTSOefXYc[ XZNe½cg´ZNe:XHW(\.wNW(Vbe:\©¬XYe¥z?\OW(S SOQW4XYNVq[4SL¾U4X7OXHW(NSO©XYVTW(SSOVTcNyS c0¶Sp\ [ XYe-ZNe}SgwNW4\JpS=¿nXYU4[4Sc©\O[4SSpwNW4XYU XYcx[4SL e:\aUnZyefS:wNW4\¡LS}XYRqXYe:XYcx[4SpWNX¥ZNe W(XYU4ZNRq[4SO©\}wySpW(QHVTSORÁÀyXYU w?XYQHVqÂyQHSOe:XYcx[ XO5µZ©X S W4XYQO´ wyW4\QHS´X«OXHW(NSO©XYVTW(S wySpW(Sc´ZNe:XHW4\aU2wySpW4XYU2XnwySpW(Sw?\O[H»XYcNQHVTSOU©X«¢2PR&´XYe · § NVTU4U4\©wyW4\JpSOe:\aUnµZ©XU X¤i¥r XZNe}S;w5\O[H»´ XYcNQHVbS}©XZNewyW(VTe:\©©XYcx[H¶Sp\ { { l&r|!h lL¤|!h e:\ry¨J&© rB|!h ¤|!h ~ ~¦ µxZNX!´XZNeÃ[ XYU [ XXHÂyQHVqXYcx[ XBwÄSpW(S.[ XYU4[4SpWfS.W4XYQO´ wNW4\QHSi©Xw?\O[»XYcyQHVTS©XwyW(VTe:\aU · ´ e:wySpW(XYUH · Å9Æ}ÇfȬÉËÊJÌÍÌÏÎÑÐxÒ&ÊHÓÍÎÕÔ¡Ö¥×2ØÙnÚÛ(ÜÌ(Ý=Þ¡Þ4ß«Þ¡ÞJàÞJÞÅáaÞJàÞJÞ(âãJä ¢¡¤£ ¥§¦©¨© ¥ ¦ § ! #"%$'&)()*+ #,-$. /1032547698;:=<?>A@CBD@¤<FED:-BG2;H-@CED25I¢J)I¤KL6MB32547:=JONM4QPR:SB30T2;0 @C85UV@XWY0Z@¤J2;J[E3:-B3:S0GE32;JM\'\¢I[@C8^]_J9I¤E I¢JM8`W!Pa:-<b@CNM03:ZED>M250 47@bW!Pa:c@.>A@¤BGde6MBGI¢PM85:-4fPMNEg@¤8503IPa:-<b@CNM03:[email protected];4.6M8;: <?>A@CBD@¤<FED:-Bih 2;H-@CED25I¢Jj47@bWj85:b@¤dkE3I!@¤J):Fl7<S2;:SJOET69B32;47@¤8525EmW!ED:S0GE-]n@oE?@¤0GpeE3IdA@bWe03I¢N9\¢>[E U+25ED>)\¤B3:b@qE 2;J[E3:-B3:S0GE P[W!ED>M:T0G<-2;:SJ[ED25rM< <-I¢4.4=NMJM2`EmW¢]M:S036a:-<S2s@¤8585Wt25JtED>M:Z@¤BG:b@I¤K©JYN94=PR:SBuED>M:SI¢BGW¤v /wKx@C47I¢N90LBG:-03N985EyKzI¢B6MB32547:+JONM4=Pa:-BG0-]Y<b@¤858;:-dED>9:{u>M:SI¢B3:S4|ICK_}~I¢8;0ED:-J9>MI¢8;4.:¤]O250 ED>9:TKzI¤8;8;IXU+25JM\ Zq%OX RxS XYaxyR%' ;¡Q¢T£a¤ ¥o¦Z§A¨¥.©F¦F¤ ª«A¦F§ { ¬®­ 47IYd ²³ ­!± ~°¯ .¬®­ /6M6M@¤B3:SJOE385W¤]2`EyUV@¤0§´¢@¤47:S0Vµv´¤I¢JM:-0¶ ]Yµ%B3I¢PM85:-4¹¸§º ]»6_v%¼¤½¤¾U+>MIZrAB30EV<SI¢Jq¿G:-<h ­X· ­ EDN9B3:-deED>A@CE+E3>M:Q<SI¢J[À¢:-BG03:=ICK#ED>M250 E3>M:-I¤B3:-4Á250+E3B3NM:C]RJM@¤47:S85WtE3>A@CE@7JA@qEDNMB3@¤8JYN94=PR:SB 0D@CE32;0KW25JM\ÂED>M: <SI¢JM\¢BGNM:-J9<-:ZI¤K#µ%BGI¢6R:SBGEmW 2;0VJM:-<S:-0303@¤B32585W!69B32;4.:¤v%{u>9:Z<SI¢J[À¢:SB303:cI¤K ­ }~I¢8;0ED:-J9>MI¢8;4.:¤ÃÄ0u{u>M:SI¢B3:S4Å2;0uB3:S\[@¤BGdM:-de@¤0 @=À¢:-BW'dM25l.<SNM85E+6MBGI¢PM85:-4v JǶÉÈC¾Ê]Ë 25<?>A@¤BGdj´9vaÌ< J[EDI¢0G>jI¢PE?@¤25JM0uB3:S0GEDBG2;<FED25À¤:T<-I¢JMd925ED25I¢JM0uI¢J'Í)KzI¢Bu0GI¢8;N9E32;I¢J90 Æ Æ 47IYdÕÍRÖ @¤JMd<-I¢JM<S8;NMd9:-0QE3>A@CEQ}~I¢850GED:SJM>MI¢8547:Cû0c<-I¢J[À¤:-B30G:!2;0c6MB3I¢PYh I¤K7ÎGÏÑÐqÒAÓ ² ­± ¯ ÐCÒAÓ¤Ô @¤PM8`W EDBGNM:¤v×I¤B:Ø9@¤47698;:¤]C>M:V0G>MIXU+0©E3>A@CE©25K 2;0@Ù6MBG2;4.:JYN94=PR:SB#@¤J9d=ÍÚ 03@CED250GrA:S0 Ï { Í ¬Û­ 47IYdÝÍ ²FÞ ­7± ~ܯ Í ¬®­ U+>92;<?>'UVI¢N98;dPR:T@<-I¢N9JOE3:-BG:FØ9@¤4.6M8;: EDIÂE3>M:T<-I¢J[À¢:SB30G: ]ED>9:-J 03@CED250GrA:S0 ² ± { .¬Û­ 47IYd ²FÞ ­7± Mß ~ ¯ .¬®­ U+>M25<?>j2;0 NMJM852;pC:-85W¤v+Ìe< JOE3I¢03>k@¤8;0GIoB3:S6RI¤BGED0+ED>M@CEÙE3>M:Z<-I¢J[À¢:SB30G:=2;0 pOJMIXU+JjEDI7PR:TEDBGNM: Æ KzI¢B+@¤858à<-I¤476aI¢032`ED:gJONM4QPR:SB30+Íá v ­X·¤â I)6MBGIICKi]#>MIXUV:FÀ¢:SB-]V>M@¤0=Pa:-:-JäI¤P9E?@¤25JM:-dÇKzI¤BcED>9:t<-I¢J[À¤:-B30G:eICKu}I¢850GE3:-JM>MI¤8;4.:¤ÃÄ0 ã 6MBGI¢6R:SBGEmW¤v J70G:-<SE32;I¢J7º UV: 6A@¤BED2s@C8;85WTrA8589ED>92;0\[@¤6à]OP[W6MBGIåÀY25JM\ ED>A@CELµ%B3I¤6R:SBGEmW dMI:S0 Æ ­ JMI¤E.>MI¤8;däKzI¢B6RI¤0325E325À¤::FÀ¢:SJæJONM4=Pa:-BG0-vÛ{u>M250B3:-0GNM85E.250.69B3I¢PA@CPM85WäpOJMIXU+JÜEDIçI¤ED>M:SB ¨© y¨i¡ ¨£¢Ù ¥¤§¦u7¥# Ù9 ¨¦§© ¦§ ¨© ¢ ¦ j °¨£¢ ¦u ¤¡¢¡ @¤N9E3>MI¢BG0U+>9IçULI¢B3päI¤JÜE3>M:'03NMPY¿G:S<SE-]§PMN9EUL:j@¤BG:'NMJA@bU§@CB3:I¤K @)6MNMP98;2;0G>M:-dÜ6MBGII¤Kiv ÌeI¢BG:-IXÀ¢:SB-]AED>9:Z69B3IYI¤K©UL:Z6MB3:S03:-J[EÙNM03:S0ÙI¤JM85Wo:-8;:S47:SJ[E?@¤BWt47@CED>M:S4o@qED2;<S0-v Î ×AI¢B.@ç\¢2`À¢:SJ <SI¢476aI¢0G25ED:!JONM4QPR:SB.Í©]VEDI0G>MIXUÅED>A@CE ÏÑÐqÒAÓ 4.IYd Í ]L2`E ² ­± ¯ Ô ÐCÒAÓ Î ÏÑÐCÒAÓ 03Nl7<S:-0EDI)0G>MIXUÝED>M@CE 4.Id ]U+>M:-BG: 2;0=@¤J[WKx@¤<FEDI¢BÂI¤K Í©v ² ­± ­ ¯ Ô ÐqÒAÓ 0G2;JM\'ED>92;0T2;dM:-@9]nUV:.0GE3NMd9WED>9:.<SI¢J[À¢:SB303:oI¤KV}~I¢8;0ED:SJM>MI¢8547:Cû0gE3>M:-I¢BG:-4 KzI¢BT6aIXUV:SB30 I¤K6MB32547:S0 ]P[WdM:SE3:-BG4725JM2;JM\ÂE3>M:cÀC@¤85NM:TI¤KE3>M:QPM25JMI¢4.2s@¤8R<-IY:Sl.<-25:-J[Eg4.IYdMNM8;I Þ ¨ @¤JMd v Je03:-<FED25I¢Je½ÂUL:c69B3IXÀ¢:TED>A@qEÙ2`KÍç[email protected]:SBÙICK#º9]ME3>A@¤J25Eud9I:S0ÙJMICE+0D@CE32;0GKW Æ 6MBGI¢6R:SBGEmW ]96MBGIXÀ25JM\=ED>M@CE§ED>9:g<-I¢J[À¤:-B30G:c250LE3B3NM:gKzI¢BLÍeÚº v%/dMd925ED25I¢JA@¤8585W¤]¢UV: 6MBGIXÀ¢: ­ § ]MED>M:SJ ED>M@CE+25K 2;0+@6MB32547:gJONM4QPR:SBÙ@CJMdeÍeÚ Þ 0§ { { Í ¬Û­ .¬®­ 4.IYd ²Þ ² ± ±­ ¨ ~ä¯ ~ Í ¬®­ 7¬æ­ U+>M25<?>eBG:-dMNM<S:-0+ED>M:T0G2;H-:TI¤KED>9:Z<SI¢4.6MN9E?@qED2;I¤JA@¤8RED@¤03p7KzI¢BuED:-0ED25JM\ÂED>M:T<-I¤JOÀ¤:-BG03:¤v ×2;JM@¤8;8`W¢]nUV:o<-8s@C2;4 E3>A@CEQE3>M:o<SI¢J[À¢:SB303:!ICK§}I¤8;0GE3:-JM>9I¢8;4.:¤ÃÄ0ZE3>M:-I¤B3:-4 2;0TEDBGNM:7KzI¤B @¤858a6aIXUV:SB30§ICKn6MB32547:S0 á 0G:-:g0G:-<SE32;I¢J ]P[W7N9032;J9\QED>M:Ù<-BG25ED:SB32;@Z\¢2`À¢:SJtP[W ³ ² j­X·"!g± :$#ONA@qED2;I¤J v ² ±i­ % &' ('g"*) +b,S #,$' , J7E3>M2;00G:-<SE32;I¢J_]YUL:BG:SÀY2;:FU®03I¢4.:+UV:S8;8pOJMIXU+JtB3:S03NM8`ED0ED>A@CEU+258;8APa:ÙNM0G:-d.ED>MBGI¢NM\¢>MI¤N9E Æ ED>92;0uJMI¤E3:¤v ×2;BG0GE-]qUL:L47@¤pC:yN903:yI¤KA@uUV:S8;8OpOJMIXU+JQ:-#YNM@CED25I¢JZE3>A@CE2;00GI¢47:FED2547:S0n<b@¤858;:-d/. ¢¤§10Y¦F¤32 @ Ö>=? Ö ? Î Î Î ÚBA ¶ ]M6àv Èq¾^v KUL:Z0G:SEH©ÚJIgÚLKTÚ Í©] ¥54q§10Y¦6 ¡ 784q§9"4;: ¨Yªx <4q§ ­ Æ £ £ CEDGF @ C @ C Ô Ô Ô Ò U+>M:SB3:ZÍç250+@Â6RI¢0G25E325À¢: 25J[ED:-\¤:-B-]ED>M:SJeUL:ZI¤P9E?@¤25J { { Ï Í Í O Ð M Ú ³ ² ± ~ ~ Í CNDGF /8503I'UV:S8;8pOJMIXU+J2;0 E3>M:.KzI¤8;8;IXU+25JM\t:$#ONA@qED2;I¤Jà]E3B3NM:KzI¢BT@¤J[W6aI¢032`ED2`À¢:2;J[ED:S\¢:-BG0PK @¤JMdQISR { { K K K ¬Û­ º Ú ² ³ ± ~ ~ I I I ¬®­ /696M85WY2;J9\E3>M2;0u25dM:-J[E325EmW¢]925EuKzI¢858;IXU+0VED>M@CEuKzI¢B+@¤J[W'6aI¢032`ED25À¤: 2;J[ED:S\¢:-BuÍ©] { { Ï O Í Í ¬Û­ ­ Ð M ½ Ú ² ³ ± ~ ~ Í ¬®­ CEDGF ¼ ¢¡ ¥§¦©¨© ¥ ¦ § }:T0G:-:TKzB3I¢4|ED>M250 :-#ONA@CED25I¢J!ED>A@qE { 47IYd ²FÞ ± ~ ¯ EDBGNM:QKzI¢B6MB32547:S0 03:S:=ED>M:Q6A@¤6a:-BgP[W¡ =vR×Vv_¸§@¤2;85:SW¶ ½9]à85:-4.4o@ ]_6àv ÈC¾ ]_250 ² ~ j± ­ · :$#ONM2`ÀC@¤8;:SJOEuE3I.E3>M: }I¤8;0GE3:-JM>9I¢8;4.:¤ÃÄ0VED>9:-I¢BG:-4v /8503IÂU+2;dM:S85W!N903:-d2;J!E3>M2;0uJMICED:T2;0§E3>M: KzI¢8;85IåU+25JM\QUL:-858npYJ9IåU+JKx@¤<SE-v ÍkÚ Ö I ©BK ¢¤£Ü¦¥¦Fªa ©F¦=¢Q£a¤ ¥o¦c§A¨Y¥.©F¦F¤¨§ z ¢¤§10 ;¡Zª«A¦Z«Y q«A¦?¡Fª£4 V¦F¤ ÎqÐ K I 4i% 0¤ 9X 0¤ § yªz«A¦F§yª«A¦ «Y q«A¦?¡Fªà£4 V¦F¤ 4i% 0¤ 9X 0¤ § ;¡ ¬ § Ô ' (¡ /' "*, J'ED>M250u03:-<FED25I¢JUL:T03>MIXU E3>M: KzI¢8;85IXU+2;JM\ Æ Í OqY z ·e ;¡Â¦ 9¤¦F§ª«A¦F§ { Í ¬®­ 47IYdÅÍ ²³ ­!± ~ ¯ Í ¬®­ à«9¢¤ª# ;¡#ªz«A¦ 784q§9¤¦F¤D¡X¦P4i! (4;: ¡Fªm¦F§9«4;: ¥o¦6 ¡"à«A¦84q¤D¦F¥| ;¡gªÊ¤¨A¦# 4q¤T¦ 9¤¦F§£4X¡F ªx 9¤¦g §Aªm¦ 2 Y¦F¤D¡¨§ }:TPa:-\¢25J'POW'J9I¤ED25JM\ #$£&%¤ e'à«A¦c©- § 4q¥Â x¢":£784X¦)( 7F ^¦F§Aª { Í ¬æ­ ~ Í ¬®­ Í ;¡ 4$0 07 c¢¤§10 4q§:+*.
Load more

Recommended publications
  • Arxiv:1301.0252V1 [Math.NT]
    VARIATIONS OF LUCAS’ THEOREM MODULO PRIME POWERS ROMEO MESTROVIˇ C´ Abstract. Let p be a prime, and let k,n,m,n0 and m0 be nonnegative integers such that k ≥ 1, and 0 and m0 are both less than p. K. Davis and W. Webb established that for a prime p ≥ 5 the following variation of Lucas’ Theorem modulo prime powers holds k ⌊(k−1)/3⌋ np + n0 np n0 k k ≡ ⌊(k−1)/3⌋ (mod p ). mp + m0 mp m0 In the proof the authors used their earlier result that present a general- ized version of Lucas’ Theorem. In this paper we present a a simple inductive proof of the above con- gruence. Our proof is based on a classical congruence due to Jacobsthal, and we additionally use only some well known identities for binomial coefficients. Moreover, we prove that the assertion is also true for p =2 and p = 3 if in the above congruence one replace ⌊(k − 1)/3⌋ by ⌊k/2⌋, and by ⌊(k − 1)/2⌋, respectively. As an application, in terms of Lucas’ type congruences, we obtain a new characterization of Wolstenholme primes. 1. INTRODUCTION AND MAIN RESULTS In 1878, E.´ Lucas proved a remarkable result which provides a simple a way to compute the binomial coefficient b modulo a prime p in terms of the binomial coefficients of the base-p digits of nonnegative integers a and b with b ≤ a. Namely, if p is a prime, and n, m, n0 and m0 are nonnegative integers with n0, m0 ≤ p − 1, then a beautiful theorem of Lucas ([11]; also see [6]) states that for every prime p, arXiv:1301.0252v1 [math.NT] 29 Dec 2012 np + n n n 0 ≡ 0 (mod p) (1) mp + m0 mm0 1991 Mathematics Subject Classification.
    [Show full text]
  • On a Congruence Modulo N Involving Two Consecutive Sums of Powers
    1 2 Journal of Integer Sequences, Vol. 17 (2014), 3 Article 14.8.4 47 6 23 11 On a Congruence Modulo n3 Involving Two Consecutive Sums of Powers Romeo Meˇstrovi´c Maritime Faculty University of Montenegro 85330 Kotor Montenegro [email protected] Abstract For various positive integers k, the sums of kth powers of the first n positive integers, k k k Sk(n):=1 +2 + +n , are some of the most popular sums in all of mathematics. In ··· 3 this note we prove a congruence modulo n involving two consecutive sums S2k(n) and S2k+1(n). This congruence allows us to establish an equivalent formulation of Giuga’s conjecture. Moreover, if k is even and n 5 is a prime such that n 1 ∤ 2k 2, then ≥ − − this congruence is satisfied modulo n4. This suggests a conjecture about when a prime can be a Wolstenholme prime. We also propose several Giuga-Agoh-like conjectures. Further, we establish two congruences modulo n3 for two binomial-type sums involving sums of powers S2i(n) with i = 0, 1,...,k. Finally, we obtain an extension of a result of Carlitz-von Staudt for odd power sums. 1 Introduction and basic results n k The sum of powers of integers i=1 i is a well-studied problem in mathematics (see, e.g., [9, 40]). Finding formulas for these sums has interested mathematicians for more than 300 years since the time of Jakob BernoulliP (1654–1705). Several methods were used to find the sum Sk(n) (see, for example, Vakil [49]).
    [Show full text]
  • Wolstenholme and Morley, Primes and Pseudoprimes 3
    WOLSTENHOLME AND MORLEY, PRIMES AND PSEUDOPRIMES ATFERTHOUGHTS FOLLOWING MORLEY’S OTHER MIRACLE & CATALAN NUMBERS, PRIMES AND TWIN PRIMES CHRISTIAN AEBI Z 3 From the two congruences here bellow valid in p it is quite natural to wonder if there Z 4 exists primes p for which they take place in p . 2p−1 3 Wolstenholmes’s congruence. [11, 1862] If p is prime then p−1 ≡ 1 (mod p ) Morley’s congruence. [8, 1895] If p> 3 is prime then (p−1)/2 p−1 p−1 3 (−1) · p−1 ≡ 4 (mod p ). 2 Z Z 2 Z 3 In the same vein, replacing p by an odd integer n, and working in n, n or n could these congruences characterize the fact for n of being prime ? These two questions motivate the following. Definition. A prime p is a called a Wolstenholme prime [respectively a Morley prime] if 2p−1 4 (p−1)/2 p−1 p−1 4 p−1 ≡ 1 (mod p ) [respect. (−1) · (p−1)/2 ≡ 4 (mod p )] and a composite odd number n is defined as a Wolstenholme pseudoprime of order k=1, 2 or 3 [respect. Morley 2n−1 k (n−1)/2 n−1 n−1 pseudoprime of order k] if n−1 ≡ 1 (mod n ) [repect. (−1) · (n−1)/2 ≡ 4 k (mod n )]. Presently the only Wolstenholme primes known are 16843 and 2124679 [7]. In this note, we prove p2 is a Morley pseudoprime (of order 2) iff p2 is a Wolstenholme pseudoprime (of order 2) iff p is a Wolstenholme prime iff p is a Morley prime.
    [Show full text]
  • On the Converse of Wolstenholme's Theorem
    ACTA ARITHMETICA LXXI.4 (1995) On the converse of Wolstenholme’s Theorem by Richard J. McIntosh (Regina, Sask.) 1. Introduction. Gauss ([Disquisitiones Arithmeticae, 1801, art. 329]) wrote: The problem of distinguishing prime numbers from composite numbers (...) is known to be one of the most important and useful in arithmetic. (...) The dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated. Wilson’s Theorem states that if p is prime then (p − 1)! ≡ −1 (mod p). It is easy to see that the converse of Wilson’s Theorem also holds. Thus Wilson’s Theorem can be used to identify the primes. Another congruence identifying the primes is (p + 1)(2p + 1)(3p + 1) ... ((p − 1)p + 1) ≡ 0 (mod (p − 1)!). 2p−1 (For a proof see [21].) It is not difficult to show that p−1 ≡ 1 (mod p) for all primes p. In 1819 Babbage [5, p. 271] observed that the stronger con- gruence 2p−1 ≡ 1 (mod p2) holds for all primes p ≥ 3, and Wolstenholme p−1 [5, p. 271], in 1862, proved that 2p−1 ≡ 1 (mod p3) for all primes p ≥ 5. p−1 2n−1 3 9 The congruence n−1 ≡ 1 (mod n ) has no composite solutions n < 10 . J. P. Jones ([9, problem B31, p. 47], [23, p. 21] and [12]) has conjectured that there are no composite solutions. Unlike that of Wilson’s Theorem the converse of Wolstenholme’s Theorem is a very difficult problem. A set S of positive integers is a Diophantine set if there exists a polyno- mial P (n, x1, .
    [Show full text]
  • Arxiv:1111.3057V2 [Math.NT] 25 Dec 2011 Ffermat
    WOLSTENHOLME’S THEOREM: ITS GENERALIZATIONS AND EXTENSIONS IN THE LAST HUNDRED AND FIFTY YEARS (1862–2012) ROMEO MESTROVIˇ C´ ABSTRACT. In 1862, 150 years ago, J. Wolstenholme proved that for any prime p ≥ 5 the numerator of the fraction 1 1 1 1+ + + ··· + 2 3 p − 1 written in reduced form is divisible by p2 and that the numerator of the fraction 1 1 1 1+ + + ··· + 22 32 (p − 1)2 written in reduced form is divisible by p. The first of the above congruences, the so-called Wolstenholme’s the- orem, is a fundamental congruence in Combinatorial Number Theory. In this article, consisting of 11 sections, we provide a historical survey of Wolstenholme’s type congruences, related problems and conjectures. Namely, we present and compare several generalizations and extensions of Wolstenholme’s theorem obtained in the last hundred and fifty years. In particular, we present about 80 variations and generalizations of this theorem including congruences for Wolstenholme primes. These con- gruences are discussed here by 33 remarks. 1. INTRODUCTION Congruences modulo primes have been widely investigated since the time of Fermat. Let p be a prime. Then by Fermat little theorem, for each integer arXiv:1111.3057v2 [math.NT] 25 Dec 2011 a not divisible by p ap−1 ≡ 1 (mod p). Furthermore, by Wilson theorem, for any prime p (p − 1)!+1 ≡ 0 (mod p). 2010 Mathematics Subject Classification. Primary 11B75; Secondary 11A07, 11B65, 11B68, 05A10. Keywords and phrases: congruence modulo a prime (prime power), Wolstenholme’s theorem, Bernoulli numbers, generalization of Wolstenholme’s theorem, Ljunggren’s congruence, Jacobsthal(-Kazandzidis) congruence, Wolstenholme prime, Leudesdorf’s theorem, converse of Wolstenholme’s theorem, q-analogues of Wolstenholme’s type congruences.
    [Show full text]
  • Subject Index
    Subject Index Many of these terms are defined in the glossary, others are defined in the Prime Curios! themselves. The boldfaced entries should indicate the key entries. γ 97 Arecibo Message 55 φ 79, 184, see golden ratio arithmetic progression 34, 81, π 8, 12, 90, 102, 106, 129, 136, 104, 112, 137, 158, 205, 210, 154, 164, 172, 173, 177, 181, 214, 219, 223, 226, 227, 236 187, 218, 230, 232, 235 Armstrong number 215 5TP39 209 Ars Magna 20 ASCII 66, 158, 212, 230 absolute prime 65, 146, 251 atomic number 44, 51, 64, 65 abundant number 103, 156 Australopithecus afarensis 46 aibohphobia 19 autism 85 aliquot sequence 13, 98 autobiographical prime 192 almost-all-even-digits prime 251 averaging sets 186 almost-equipandigital prime 251 alphabet code 50, 52, 61, 65, 73, Babbage 18, 146 81, 83 Babbage (portrait) 147 alphaprime code 83, 92, 110 balanced prime 12, 48, 113, 251 alternate-digit prime 251 Balog 104, 159 Amdahl Six 38 Balog cube 104 American Mathematical Society baseball 38, 97, 101, 116, 127, 70, 102, 196, 270 129 Antikythera mechanism 44 beast number 109, 129, 202, 204 apocalyptic number 72 beastly prime 142, 155, 229, 251 Apollonius 101 bemirp 113, 191, 210, 251 Archimedean solid 19 Bernoulli number 84, 94, 102 Archimedes 25, 33, 101, 167 Bernoulli triangle 214 { Page 287 { Bertrand prime Subject Index Bertrand prime 211 composite-digit prime 59, 136, Bertrand's postulate 111, 211, 252 252 computer mouse 187 Bible 23, 45, 49, 50, 59, 72, 83, congruence 252 85, 109, 158, 194, 216, 235, congruent prime 29, 196, 203, 236 213, 222, 227,
    [Show full text]
  • Arxiv:2101.11157V1 [Math.NT] 27 Jan 2021 Congruence, the Constant 2 May Be Replaced by Any Integer H Not Divisible by P
    WOLSTENHOLME AND VANDIVER PRIMES SHEHZAD HATHI, MICHAEL J. MOSSINGHOFF, AND TIMOTHY S. TRUDGIAN 2p 4 Abstract. A prime p is a Wolstenholme prime if p ≡ 2 mod p , or, equiv- alently, if p divides the numerator of the Bernoulli number Bp−3; a Vandiver prime p is one that divides the Euler number Ep−3. Only two Wolstenholme primes and eight Vandiver primes are known. We increase the search range in the first case by a factor of 6, and show that no additional Wolstenholme primes exist up to 60 billion, and in the second case by a factor of 8, proving that no additional Vandiver primes occur up to 40 billion. To facilitate this, we develop a number of new congruences for Bernoulli and Euler numbers mod p that are favorable for computation, and we implement some highly parallel searches using GPUs. 1. Introduction In 1862, Wolstenholme [34] established that the following three congruences hold for every prime p ≥ 5: X X 2p − 1 k−2 ≡ 0 mod p; k−1 ≡ 0 mod p2; ≡ 1 mod p3: p − 1 0<k<p 0<k<p These results are closely related: if the stronger congruence obtained by replacing the modulus pm by pm+1 holds in one of the expressions above for a particular prime p, then a similar strengthening for that same prime p holds in all of them. See [9] for a proof, [1] for remarks on higher-order congruences, and [21] for a survey on other generalizations and extensions of Wolstenholme's theorem. A Wolstenholme prime (first defined by McIntosh [18]) is an odd prime p for which any of the following congruences holds: X X 2p − 1 k−2 ≡ 0 mod p2; k−1 ≡ 0 mod p3; ≡ 1 mod p4: p − 1 0<k<p 0<k<p (1.1) Many conditions equivalent to these are also well known.
    [Show full text]
  • Paulo Ribenboim: « the Little Book of Bigger Primes
    2 How to Recognize Whether a Natural Number is a Prime In the article 329 of Disquisitiones Arithmeticae, Gauss (1801) wrote: The problem of distinguishing prime numbers from com- posite numbers and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic... The dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated. The first observation concerning the problem of primality and fac- torization is clear: there is an algorithm for both problems. By this, I mean a procedure involving finitely many steps, which is applicable to every number N and which will indicate whether N is a prime, or, if N is composite, which are its prime factors. Namely, given the natural√ number N, try in succession every number√ n =2, 3,... up to [ N] (the largest integer not greater than N) to see whether it di- vides N. If none does, then N is a prime. If, say, N0 divides N, write N = N0N1,soN1 <N, and then repeat the same procedure with N0 and with N1. Eventually this gives the complete factorization into prime factors. What I have just said is so evident as to be irrelevant. It should, however, be noted that for large numbers N, it may take a long time with this algorithm to decide whether N is prime or composite. 16 2. How to Recognize Whether a Natural Number is a Prime This touches the most important practical aspect, the need to find an efficient algorithm—one which involves as few operations as possible, and therefore requires less time and costs less money to be performed.
    [Show full text]
  • Peter Lombaers Dep. De Mathemática, Centro De
    #A42 INTEGERS 20 (2020) GENERALIZATIONS OF WOLSTENHOLME’S THEOREM VIA THE P -ADIC LOGARITHM Peter Lombaers 1 Dep. de Mathem´atica, Centro de Mathem´atica, Universidade do Porto, Porto, Portugal [email protected] Received:4/8/19, Revised: 2/17/20, Accepted: 5/10/20, Published: 5/26/20 Abstract 2p 1 We use the p-adic logarithm to express the binomial coefficient p −1 in terms of harmonic sums, where p is an odd prime. We use the same logarithmic− method on norms of cyclotomic integers to obtain several congruences. For example, we show: p 1 − 2 2 k 1 1 2k (1 Lp)(Lp 3) 2 ( 1) − − − mod p , − k k ⌘ 2p kX=1 ✓ ◆ where Lp is a the p-th Lucas number. 1. Introduction In 1862, Joseph Wolstenholme [10] proved the following theorem: Theorem 1. For any prime p 5 we have: ≥ 2p 1 − 1 mod p3, (1) p 1 ⌘ ✓ − ◆ p 1 − 1 0 mod p2, (2) k ⌘ kX=1 p 1 − 1 0 mod p. (3) k2 ⌘ kX=1 (i) n 1 (1) Let us set Hn = k=1 ki and Hn = Hn . So Wolstenholme’s theorem says 2 (2) Hp 1 0 mod p and Hp 1 0 mod p. Over the years many generalizations of − ⌘ P − ⌘ 1The author was partially supported by CMUP (UID/MAT/00144/2013), which is funded by FCT (Portugal) with national (MEC) and European structural funds (FEDER), under the part- nership agreement PT2020 and by FCT (Portugal) through the PhD grant PD/BD/128063/2016. INTEGERS: 20 (2020) 2 this theorem have been investigated.
    [Show full text]
  • A Search for Primes $ P $ Such That Euler Number $ E {P-3} $ Is
    A search for primes p such that Euler number Ep−3 is divisible by p Romeo Meˇstrovi´c Department of Mathematics, Maritime Faculty Kotor, University of Montenegro Dobrota 36, 85330 Kotor, Montenegro e-mail: [email protected] Abstract Let p > 3 be a prime. Euler numbers Ep−3 first appeared in H. S. Vandiver’s work (1940) in connection with the first case of Fermat Last Theorem. Vandiver proved that p p p x + y = z has no solution for integers x,y,z with gcd(xyz,p)=1 if Ep−3 ≡ 0 (mod p). Numerous combinatorial congruences recently obtained by Z.-W. Sun and by Z.-H. Sun involve the Euler numbers Ep−3. This gives a new significance to the primes p for which Ep−3 ≡ 0 (mod p). For the computation of residues of Euler numbers Ep−3 modulo a prime p, we use the congruence which runs significantly faster than other known congruences involv- ing Ep−3. Applying this congruence, a computation via Mathematica 8 shows that 7 only three primes less than 10 satisfy the condition Ep−3 ≡ 0 (mod p) (such primes are 149, 241 and 2946901, and they are given as a Sloane’s sequence A198245). By using related computational results and statistical considerations similar to those on search for Wieferich and Fibonacci-Wieferich and Wolstenholme primes, we conjec- ture that there are infinitely many primes p such that Ep−3 ≡ 0 (mod p). Moreover, we propose a conjecture on the asymptotic estimate of number of primes p in an inter- val [x, y] such that Ep−3 ≡ A (mod p) for some integer A with |A| ∈ [K,L].
    [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]
  • On Divisibility Properties of Some Differences of Motzkin Numbers
    Annales Mathematicae et Informaticae 41 (2013) pp. 121–136 Proceedings of the 15th International Conference on Fibonacci Numbers and Their Applications Institute of Mathematics and Informatics, Eszterházy Károly College Eger, Hungary, June 25–30, 2012 On divisibility properties of some differences of Motzkin numbers Tamás Lengyel Occidental College, Los Angeles, California, USA [email protected] Abstract We discuss divisibility properties of some differences of Motzkin numbers Mn. The main tool is the application of various congruences of high prime power moduli for binomial coefficients and Catalan numbers combined with some recurrence relevant to these combinatorial quantities and the use of infinite disjoint covering systems. We find proofs of the fact that, for different settings of a and b, more and more p-ary digits of Mapn+1+b and Mapn+b agree as n grows. Keywords: Catalan number, Motzkin number, harmonic number, divisibility MSC: 11B83, 11B50, 11A07, 05A19 1. Introduction The differences of certain combinatorial quantities, e.g., Motzkin numbers, exhibit interesting divisibility properties. Motzkin numbers are defined as the number of certain random walks or equivalently (cf. [2]) as n n M = C , n 0, (1.1) n 2k k ≥ kX=0 where Ck is the kth Catalan number 1 2k C = , k 0. k k + 1 k ≥ We need some basic notation. Let n and k be positive integers, p be a prime, dp(k) and νp(k) denote the sum of digits in the base p representation of k and the 121 122 T. Lengyel highest power of p dividing k, respectively. The latter one is often referred to as the p-adic order of k.
    [Show full text]