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Chukchi
Topic Author
Posts: 915
Joined: December 15th, 2001, 3:43 am

### 1913

Prove that A. 1910^1912 - 1912^1910 is divisible by 1913.B. (p-3)^(p-1) - (p-1)^(p-3) is divisible by p for all prime p>3.

cm27874
Posts: 69
Joined: July 2nd, 2007, 12:10 pm

### 1913

mod p:$(p-3)^{p-1} - (p-1)^{p-3} = (-3)^{p-1} - (-1)^{p-3} = 3^{p-1} - 1 = 3^p / 3 - 1 = 3 / 3 - 1 = 0$in step 2: p is odd and p > 3in step 3: p > 3in step 4: Fermat

yegulalp
Posts: 20
Joined: February 18th, 2008, 11:39 am

### 1913

It follows from Fermat's little theorem: a^(p-1) = 1 mod p for p prime, a not divisible by p.(p-3)^(p-1) = 1 mod p (by setting a = p-3 in Fermat)(p-1)^(p-1) = 1 mod p (by setting a = p-1 in Fermat)Now expand:(p-1)^(p-1) = ((p-1)^(p-3)) * (p-1)^2 = ((p-1)^(p-3)) * (p^2 - 2p + 1) = (p-1)^(p-3) mod p = 1 mod pCombining the above, we get (p-3)^(p-1) - (p-1)^(p-3) = 0 mod p.

yegulalp
Posts: 20
Joined: February 18th, 2008, 11:39 am

### 1913

Doh! Took too long to type it in.

wileysw
Posts: 593
Joined: December 9th, 2006, 6:13 pm

### 1913

how about proving it for p=1729 (the "dull" Hardy-Ramanujan #)? and how about p=286 or 24046?----- ----- ----- ----- -----spoiler/comments:(1) simply expand the powers using binomial theorem, one gets 3^(p-1)=1 (mod p). besides all primes (as indicated above by Fermat's little theorem), there are composite numbers that satisfy this relation: Fermat pseudo-primes to base 3;(2) one might consider the general form (p+m)^(p+n)-(p+n)^(p+m)=0 (mod p). if sufficiently large prime p satisfies the equation, seems one needs either m or n=-1 or m=-n

ppauper
Posts: 70239
Joined: November 15th, 2001, 1:29 pm

### 1913

QuoteOriginally posted by: ChukchiProve that A. 1910^1912 - 1912^1910 is divisible by 1913.1910^1912 - 1912^1910=2166692594352408499924447933736491626283407649922349056989156738398908954448711039960410358002285823305151178931503746864208580476846758354923604808394430569155692524918947114669043346511848537297677188078148554948907472683648739741779177646396994065873077526260647064444388112891001920512965071311112000755698253366467898905569972225903368365546910855972284135643434032818102005387759858495178332194844318798525183468100517353225532661054748679306413232319280011069610439178387382008859948234384386515067616962829254371748951330534133005498194101189358597654442590611106373147306195497379217851556027851537367635475677553113719998379692661079736246392107773639437988185671410286922031695298445913761848984240867805158168266139403195756737923251527062567268381621460914819471230977805684147535843805466366652053013049109276584615738981452133431376364059312225865762881978555160421199453077967873465499185392346196010332520055488778244024952593142305779167742450831765601736869167751557359031574231185194790402898230358662455324730545007228751103093913894144436271037350176417758225259032641129457584669091470165347113905870106271806034332513533865995688556862795959445747122324324054727194637917983145211570595179259640641946655484867170859718467455274138098462363523907564782486037787129513290156783884869438709131137058791290364446307641776439787122842938318731584593655685400490320277052726993098974294817218332278686779682808952667770195966683558708011457228446883943061813885302811022465807272553592922448835066531060269440298199658010580220662678783366603514373231266683313815116317928825626853262472904879351234690820574555993186986473684564752625167769618548288165720595358514635836294444744506239167399711328410695743078231466646519429764312544033773607927862305156733494832266649836526000008241434598050972575384881201575448534764487977181166057458105723679960970625077583033256961217384193832487434047049377912900073804272237023530584735786861578751146715180716162042963068201799881710101835895144465043518153772269208364280933532809148745321604228304709352501131642394633621408136747192102428992170990939402137748435580057432553501618402326777732906725083555693781144675518651720030705949631977255940362910924404002915611702367526862482798230682055585888644619617154133680143839906631830088558304618041264726473350856412476763298846610742124416322230526051502331502271667588085994296736712227428647295804786392265403805610848337267337091987341577731044628464368417615444504502353821911093580577983931089231203176628135167621935762673150118323217149265006191042696512181666186715128644420165867390877969290778859830222046935263914487366490228733585682168640060183923110816308733890702726030493539859379001042119692908780713709295709778039014047181862139260254524258171370648726936180166912120150399376558023746026840196709728209574157212417615466152968238235551751637352641881581129274565525418276357092859452151291297439348906742948806233824128535238154728201365026196426068035266040855745801009220312114299149863470928887739208035659857845279200404648531169842382730236241925282203377565143657130222197907553046598593352888249255961071342200605899342139166258403753287085690030155217347621579055546889585333395373073854240736006995653037047724399199640984196680160942057617180790772663048036501088423869520607523401679418242033893936335274595698430569413832826243814864629580527772348496567187400971069678869877064686387300337319708703545217283771313976923642938736459386645945382057241216492489795910280159419694085404711385457444750330355531795814573548450134987523435280754781256763413067598851903555037883457820742796258656779951668231595439018619529973902099009887656955209207221828079832275760624038587096288767832892704901195314208656877083663250249208737779179292864644218553529981770308497336738253931134341116736599003385999994938338120287378518609447838926950753418183593924551047499305810810652277901301337835913235477118605299814055133450398192958314433956461646110020726457513205895043444815061867592897262955354936210982039371517264654284019466001999984573814574041649276208915498942362513289416385583192209073522209035709436889966428821435288952174470741037115537948030738755640889237468574381965414168789645136532220793338602081511148044900545083849181982549381104014564403881347047502523960083459016568583782980475974461834800791705278671644826277263264619661631981381052835553893391516454222978858447964088614231376907519118293980203014415486189281006891790866654528212556488489628512751279062473515821670373041729209803022984996658943565450249780438240376399569879419277286931446785562566916709678164081953077257097743763508406760801429627346199225024024315997067296391296368352299571832195265177900623768289161472383442532040785499410750649064981112433283662982547454844245273131273436500554016402580838217341134666703181383989333742598416858439777712623387097223914235346787530183007464728261363788241544519289289975138185927098692030344932294619364111044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by 1913 and you get11326150519353938839124139747707745040686919236394924500727426755875112150803507788606431563002016849478051118303731034313688345409549181154854180911627969519893844876732603840402735737124142902758375264391785441447503777750385466501720740441176132074610964591012269024800774244072148042409644910146952434687392856071447458994092902383185407033700527213655431968862697505583387377876423724491261537871637840034109688803452782818742983068765021846870952599682592844064874224664858243642759792129557692185403120558438339632770263097407909072128562996285199151356208001103535667262447441178145414801652001314884305465110703361807213791843662629794752986890265413692827957060488292142822957110812576653224511156512638814208929775950879225074427199432969485453572303300893438679933251321514292459675085235056804244919043644063129036151275386576756044832012855787903114285844111631784742286738515252867043905830592504945166401045768367894636826725526096736953307592529178074238039044264252782849093435604731807581823827654776071381728858050220746215907443355432014826299202039604902029405431430429322831075112867068297684860982070602570862699072208749952930938614024024879486393739280313929572371343010889415638110795500573134563233954442588451958800143519467527958506866303750992067360364804649834240234102900519805066027898895929379845511278838350443215487940085304399292178022401466232724919251055897015025064386922803338665863475216689324465216762590479393682473750375765515221191245702096301441752600907438172759200556879562307581276527224310403662638153598361753864993367834204463225908614369187803728502210689385003840516573834584386758946543045294054496433714681381537719396734754389105260295058779436691936224314314744881040659861299772710035359710109951770036214648746245339753435790119375462624193630081537119563906873398858209349371481902651234532448813934781390282125588487104198938636787162539636630601395971656627810108752076513452851985027636495445641122400317387151296200042874019245740208237820932906657178012400779970539889877327805273106751721005252009161009574711942674445416292457878214474606225775950873694450787140917054644545835872068885169402640269501797078644409402256769632545476382526168010260903943565096265612662663496406078401624056746397745136537465520824255848614253172138466065366534455121413870103797242429499732601206179427163210823450347633724528341179237250362241174658794713263438603169849793015188765542798051945070503615147574723619782375418794796155265832838008853008771478991579880832078557946511353558613032671815594444063068511192068460596872274891349339489912001397756811130636585780607538143937605407670499164813034948678351490121191443339342962539940944725721346261679973868489909529057923229008341399780653497371572724198955852191626614587813522928281060701098538418068782141902910821728749232668971025679130242896774533393808425256860181086678416776134686489379385034463457669374235741972550479576313676078537002275160244266604103639291301976138348357844985891318454323650283062537746622518239456537688749844637743882114642811512716833435660874449573327575241281391647662872728941968600903754570078207377638267673284590377241288048893293704600262180958429804386071860374204565579576967350970914830087051944839123884995509588064193209315512896919972674663084351809139696129224294424472971344704829555338919365372182073023595571491081102271978988644915569767732291693523626131050063066839452693869940975972531193328652974265152341551157654922352040679860121361696235211240356295973309385283384137257944330007085895430290504696550627221763906329089290416954590003130433369341546698681760286441498467224002447629876837525454939748573439675314463958228997500408153854029067559009864591159267291650313127001119752518960186425772342962560267534202115233761388029178098842118853833005390142922722094087837877679011700669734905271921638514578813124691156213245815739924199193992737335483017780124599650995698476765811887376547048605295129714761926093288604394806364418222138230239493266427563498499505253195039111549249959564319310086074239414451509746858555406566695846831899438124879332909478265598920659744044482158337514003248485566921978414746665528164913908723239105537048973206387691657965445086965667646593510727033513580997911892754046952339515844767457210684168045203881463460969466892674248734869189620814609517991141028079690675782871221221440836285316533379790347054450884594766641128928847957299408821323148426451662385862447408682733360119532956237389213217405271891015572725946240573289979308568791442880214305587786541820437953153373635125822683843582263102178701320266161576989880198395594652889408068522774494518712113675897673936352932551318852019322364622580093215866519487196151428589448365830854846005151591274180078750554691235147218936473888365108589290504111346335684279042541648411576929969583815940061383159523567379914859505421313245761815717655672676463485249667348605549108830576528882060564177201735652613341843279477841319779163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