A large Ruth-Aaron pair with multiplicity was discovered by Jens Kruse Andersen on 12 May 2006. The (below) Mathematica inputs and outputs (indented) show this to be so and prints the smaller of the 3109-digit adjacent numbers. s1={ 123975575752249320138474866732136604982540531830832904021439414006353638364451539699377898693775903444503760495043748685 702140799775352775440335286758680850748997297473972389858042823714196343390530570667408069887860866950908061600838845088 129481657551421333620461070585937502025110426909631123657892171239053067396538921343617885076150747743088305240998076863 663123269845811683040891649644986590048234484013638265648943660276991864575242408894844996026942259061491686263654378098 695492007673427711907653660606215123671401170452880587993083309422282880244976199046520290117264539809756889548651484623 358430498406565541130669963866202474399263087375154535089517640932570811840515915302658681545000758847651251184217058105 176771726877477916962266319096697464097918820931772110601979451781968664919801796741314236181198037412797932271090737529 768831841299847270299789844447564062330101394615322520483970387513178347298653868274230780459207685871887701822509435967 73087017409241140184344105893149796903458738325663727250912105607982059880741, 153699433831017095809320421011276618543727099959990709676488148507548567589709530032424151255861860582761302203638665577 880067813315074822398075277277697386036125415911379348173661119160321212565091697834741312591349038248933283628398516582 667166277332466419713347309784055507055347850600286489318539591198578858797402834589340768473206247362306221688769681928 829213806915235027659863050957195507207760639909805998252133391877758746876226301372344779793299890222810421783829509243 805054424949895928678442151837472843783446981179243586273741453785029357483004686307311712083463182546476439911530839745 622423004005421746728610371474616548008784293332071750273361377200457419407858743811806909009673113372189629874623947972 083385837147885759105779153037121717234406387283000052452498212760730225718968272073468102017073986127034931450910981804 910918906223604953332452139594541002206645125016867694794505690393223174650548575149571242115890167009661617672663924268 842156802117206964097283461084935014651292047100689897498864333296554829605377995198118192733764501758976480757472046550 460723957234952586661211622873253738171491907269023090878647878830837047595393671031095353324569845590957534009868366090 969823666772456702255226482811950797088438181580152564095803463225087155288106057200603419892033750954164785114798636626 985669940974422795012927868102829529178144996109676106696472817347562470674054018784863519573908095570130006902964532392 667863116649972331861415540197309095375180519764180612891954523527168174774891862399172059746423186262200489692466459393 051660556916177290643114260160327964114472019692049241775885582174208410922236309398779082763304712395941314284049559014 414499231778408453954487498088075405985089421111981333443432946707913009559473937385290677229265191039219579185426420530 194879500430612222739148670938547649500076543547194340343637008674897825127668272274832664590660291331274707274782586627 488828868856578047925799735269555724554183599398118123785495512276586575475239857857153669072458032999681631718273146453 637588673480930653512928049410999}; s2={2,2,3,5,7,11,13,17,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107,109,113,127,131,137,139,149, 151,157,163,167,173,179,181,191,193,197,199,211,223,227,229,233,239,241,251,257,263,269,271,277,281,283,293,307,311,313, 317,331,337,347,349,353,359,367,373,379,383,389,397,401,409,419,421,431,433,439,443,449,457,461,463,467,479,487,491,499, 503,509,521,523,541,547,557,563,569,571,577,587,593,599,601,607,613,617,619,631,641,643,647,653,659,661,673,677,683,691, 701,709,719,727,733,739,743,751,757,761,769,773,787,797,809,811,821,823,827,829,839,853,857,859,863,877,881,883,887,907, 911,919,929,937,941,947,953,967,971,977,983,991,997,1009,1013,1019,1021,1031,1033,1039,1049,1051,1061,1063,1069,1087, 1091,1093,1097,1103,1109,1117,1123,1129,1151,1153,1163,1171,1181,1187,1193,1201,1213,1217,1223,1229,1231,1237,1249,1259, 1277,1279,1283,1289,1291,1297,1301,1303,1307,1319,1321,1327,1361,1367,1373,1381,1399,1409,1423,1427,1429,1433,1439,1447, 1451,1453,1459,1471,1481,1483,1487,1489,1493,1499,1511,1523,1531,1543,1549,1553,1559,1567,1571,1579,1583,1597,1601,1607, 1609,1613,1619,1621,1627,1637,1657,1663,1667,1669,1693,1697,1699,1709,1721,1723,1733,1741,1747,1753,1759,1777,1783,1787, 1789,1801,1811,1823,1831,1847,1861,1867,1871,1873,1877,1879,1889,1901,1907,1913,1931,1933,1949,1951,1973,1979,1987,1993, 1997,1999,2003,2011,2017,2027,2029,2039,2053,2063,2069,2081,2083,2087,2089,2099,2111,2113,2129,2131,2137,2141,2143,2153, 2161,2179,2203,2207,2213,2221,2237,2239,2243,2251,2267,2269,2273,2281,2287,2293,2297,2309,2311,2333,2339,2341,2347,2351, 2357,2371,2377,2381,2383,2389,2393,2399,2411,145336801, 153699433831017095809320421011276618543727099959990709676488148507548567589709530032424151255861860582761302203638665577 880067813315074822398075277277697386036125415911379348173661119160321212565091697834741312591349038248933283628398516582 667166277332466419713347309784055507055347850600286489318539591198578858797402834589340768473206247362306221688769681928 829213806915235027659863050957195507207760639909805998252133391877758746876226301372344779793299890222810421783829509243 805054424949895928678442151837472843783446981179243586273741453785029357483004686307311712083463182546476439911530839745 622423004005421746728610371474616548008784293332071750273361377200457419407858743811806909009673113372189629874623947972 083385837147885759105779153037121717234406387283000052452498212760730225718968272073468102017073986127034931450910981804 910918906223604953332452139594541002206645125016867694794505690393223174650548575149571242115890167009661617672663924268 842156802117206964097283461084935014651292047100689897498864333296554829605390392755693417665778349245649694417970300603 643807247637096528061846986709698892141429697138400681223098254880341422463963885111072888602113879119633402094943265820 717220905758260984626646117151003854155178988568938650790894269385171039796919005366358562025395797061223378865001147669 676633053340212012136833174842483421312506784617291181470781647871662278360420331111848100742212184735094505561969355841 069226943214866697889114726654833336264665019366874838798103692153533612584761411599939402517613951628261111204833599510 096948615715485621585342548184825584019124048703775695756861271129073559384572152448619739317417779392327934531489485323 152014685287360218047744579272126997515355289266481409328198071826334715369991614557978425020961417671129248931836212412 288056711490810167917345537430527829174207967165314144084916801902006898880645155458962649317690270315719463681015596766 950361120904975086677117569999421111381606677444038892372684282458837519072012944874562910212642377105574781515176605191 963252400731842759120909963556149}; {Length[s1], Length[s2]} {2, 362} {Union[PrimeQ[s1]], Union[PrimeQ[s2]]} {{True}, {True}} {Apply[Times,s1]+1==Apply[Times,s2], Apply[Plus,s1]==Apply[Plus,s2]} {True, True} Apply[Times,s1] 19054975801995091892841703401606177294545075192844473808940321254012992166052319594047464064914560760875374525006154 508644752945095748516948725589651476374581428478511699255043845971568391863427747037942188350799147872242010237043819103 578920506331446671267010256952749172672449040466315474124512177648926826658205646027331175895735049485362278213459231921 294807091610202618114492119102220443487474823794307259997436322217473523698959129185378060249830878764130156560512866303 625145443362506557422754903901236270336426028168460827736180476180572417128643396666416289233045197933584936379840570567 084315048567166664868527830645588233032143110313278439853299902509416617590580762348002839934035070762443755809596744321 761659877221771576109490250145402622414404562259774813298262754396378211139784677133475922772727787618773694866208950523 533230894662751867621547307603403189123541397908068975756961387034308212963084621035690171533984432674817862062108867677 248189661469468441733341191144487433197420177996522509720140979753709444020097028258927374280309487671007976256865740168 283068965101775855138076432471110284363877990385064403793644294595733636718976480384710552299590438658255722005009647337 109636402024202878961531603302762816182274872986175402554762800745593508922547625022108178944431677209358465025695605061 106676915994912087319190249524213638707617118035012391855385740693919833427199557170588828908963681381934941470045873311 518431486753675492289670860296443204478590651209629520895963744471162330854341328738476094967534729859033934337892316948 130994783610007486115808648265930110618476418387135567134712547259322583688642836836922006923712091065347705723072188710 429164135931834853953655063797343474600633673325362229040801599152680504884635916666778832738867735156194538242834770255 044544321237433940461250134609712394978008903717008981895088142458026051720760339117439620269040698061008958052801601054 968730013158277153905162808236342340287726311877039111529871303090509865823524354630477879545375977434466534593948949618 212598262648309622394734267833669790459575906748008273864755152588391502658327784374799064070328234254695164910217280013 797154318997193661545125635025472852125912558383160266664559497529222437300648427865975622445449788848957272232738918030 871988678662590796655553032858913315605720456625542685359667095936565892360640208306548521998441373210008572291852048026 901204138271520166907079134060324589045574103918399069762016004787287876488780959182600777634511518027371579743888186937 511870196985062693728993667914611154562567453548237593791259687391817725303771985704315679083479722112646152015630214681 580003440144118404559647502894546585869429255614819914890571832054841142679487341294610095570381163739613052839467798053 572752580529880811423387860522050110929577651005569710461687254056991965048766388054134374883254252902154022905265246101 634620142146031815900490794552543759035210282395392015118060754586572923985886310282898348243033088005552295208816531533 70337142941704610130531032478966096949380681751035597648875605646925429060966480912553804692548277672433233670259 Length[IntegerDigits[%]] 3109