Twin concatenations
Hello Math-fun,
L = 1, 11, 2, 6, 3, 21, 4, 536340852, 5, 105, 6, 176, 7, …
What comes next?
Answer:
Next comes k.
What is the value of k?
Answer:
I don’t know.
But look, there is a way to compute k as we have:
1 x 11 = 11
2 x 6 = 12
3 x 21 = 63
4 x 536340852 = 2145363408
5 x 105 = 525
6 x 176 = 1056
7 x k = 1767abcd…n
…
Reading vertically the numbers one by one of the left part (left of the = sign) gives L:
L = 1, 11, 2, 6, 3, 21, 4, 536340852, 5, 105, 6, 176, 7, k, …
Reading vertically the numbers of the right part gives R:
R = 11, 12, 63, 2145363408, 525, 1056, 1767abcd…n
And (concatenation L) = (concatenation R), if you consider only the digits.
What about extending L?
Best,
É.
____________________
November 6th 2022 update
Maximilian H. was quick to notice that a lexicographically earliest start than mine was possible:
1 * 1 = 1
2 * 6 = 12
3 * 2 = 6
4 * 8 = 32
5 * 97 = 485
...
Then Gilles E.-F. suggested to start with 0:
0 x 0 = 0
1 x 0 = 0
2 x 5 = 10
...
Brilliant idea, Gilles!
Hans H. was quick to compute... the first 1000 lines of the table!
0 * 0 = 0
1 * 0 = 0
2 * 5 = 10
3 * 846 = 2538
4 * 1 = 4
5 * 1283 = 6415
6 * 2 = 12
7 * 1194673 = 8362711
8 * 118342264792783 = 946738118342264
9 * 88 = 792
10 * 7839881 = 78398810
11 * 7127164651933786 = 78398811171271646
12 * 432781551 = 5193378612
13 * 332908885487176222196 = 4327815511333290888548
14 * 512587299724651848 = 7176222196145125872
15 * 664831 = 9972465
16 * 11550979 = 184815664
17 * 488918326528186758187250898933387168971117 = 8311611550979174889183265281867581872508989
18 * 18548427206176212141579289232 = 333871689711171818548427206176
19 * 11165346278380641637712348833582324409 = 212141579289232191116534627838064163771
20 * 11744167911622 = 234883358232440
21 * 4381511627 = 92011744167
22 * 414373733809778 = 9116222143815116
23 * 118365842336252599053529724279276631564805882488817076796332524296734268294259275 = 2722414373733809778231183658423362525990535297242792766315648058824888170767963325
24 * 1 = 24
25 * 1186937073177 = 29673426829425
26 * 35674 = 927524
27 * 4634 = 125118
28 * 247752613473698836695122437419736195489244167682944193562 = 6937073177263567427463428247752613473698836695122437419736
29 * 6741 = 195489
30 * 813892276480645207655804360463075882688173588526812 = 24416768294419356229674130813892276480645207655804360
31 * 149379316996185 = 4630758826881735
32 * 276646288473418103656130791336451965147943157392540872292641239108732236685169 = 8852681231149379316996185322766462884734181036561307913364519651479431573925408
33 * 219068609815481 = 7229264123910873
34 * 65784857921535 = 2236685169332190
35 * 196 = 6860
36 * 27265225962718 = 981548134657848
37 * 15654469 = 579215353
38 * 13674638754 = 519636272652
39 * 6657107274760371 = 259627183715654469
40 * 95341865968859916427681869 = 3813674638754396657107274760...See the full 1000-term table here.This is extraordinary! Those huge series of digits blow my mind! Look for instance at this multiplication, towards the end of the table;Hans:«The first 3152 digits of the multiplicand (71001719...35811456) are the same as the final 3152 digits of the product. If you highlight these digits on both sides of the equation, folk will get a sense of the correspondence that is the essence of the exercise. In the product, the seven digits preceding the highlight are 9910992. Those correspond of course to line 991 * 0 followed by line 992»:992 * 710017197676976979096579441930397183962282066467918270255261972707688190794592371625325076490100485342218963548708058937381361135253314664260948630677390118685868159944784315059472829205673589078210405751821588377747732390633776922252732133675906186348547598581047653853372335608363556519125858172936434974402224028895947361348464466332594686231709240460867623437084700229599691988556406562356927101556117705749369878928256745333488175838553639258621874296857347869112127124306322553872922149425703565492261456093634695651439056059008870196039766951353613709492541177091193166637211770777740291602607607667403856021773031225407076739220746958454668227332314024366484549822986442468896904055079300340652188600986680505698911793302670147359058862894253297380765894856550312155185678467458060713917990717567566054634345533083749454876171907769807469806990143040680313312392881185598311826075026305305511833681319139935478304870469704922298959031157077661870143305280014892378088082351033541545733458284914168104695670126396509308025036648842653816348078269654677643551299994693684321014751269625628196496014200365265835151232822549595620605500237593457818157026789250995859703300940875824177273552344619140554118684260999206363337330902608157474740494311037526356461051884642282162909174281269762073281386452994908522296077741246636815702699368981008327379805181840324775507179980552090647252717586156071303660687940169813490220407506945252424907187534339938367034947631622874172711962716467647062791759404977899787787958700822095330490671175475193345838464472384539752573899571517628646361670343494940220780548455676745200895280806281434080143780776765305814649106848334506539361865415554330547495673151398199929967743436731662105075709026365497324423882476623457224811128680373792846436375267521809282083634544117293477822045734306553906377986454455291936915359401797898309760337660848851360911047727602636797055368043937470090820652804856579995196030155297794222203729758705981585776472596519631098216838460318191638294822810701906166762224774402548526860180938359102427191657927098357126952518246807195544877201645642692256046426622041053701369572184607223859671388086037320605595458190768639837453058461528934974559639402638378222048031012808749169294856643123768947449679646706702688626559999329217587144393602797638497940677180307092910316884570586772232707225478346864443629665467145175784352496502602283176093582107514048575142697009127771741245100624992940296717412465938011664216367489649639504281917687220243057326650583956293989442290504967951979288721501358651415900635741321207458311941057654930983810665110716709711457746860016220871746734136773896710488300148194639178710166993951128632072106632482076988886966328788888757318197543457481536135112720931279720041401401831326837174838966585368969852657167583386255839425237613604084500686209228314234036859142364520809135286396542476771389577341114886228637705375061047929253101677402969058312654126509532221713975642965932261190807333585614400871282637998734846738579304706514774193480767901929878387378794203421439231885795656281944009131731071845486439349482302785248534994473365358114566794584997590078841483271025720029940436267312625514175906546400878771417937299562925037045327584627447773960405266437230509131447207055387327671871769352750879183604702763699660861020283009910477929199085784658333176483539285429208352824215813684146586368336361944385095584137013982879737293460547587913975048596470653928057337589592471592494300149348400717008158395133068689657442674076328108766792879936477082375764454259661232382823952024647246593672209096173070478060939597695089354490169029373949328741419982725575138615024992868256217480138009910549811140073812672779802718339945684557190885646612991056717540195045034130978297027699011439593538500995805179313931796516051601390691157454033178601922565069197515068780522706505543797718282068 = 704337060095561163263806806394954006490583809936174924093219876926026685268235632652322475878179681459481211840318394465882310246171288146946861041631970997736381214665226040538997046572028200365584722505807015670725750531508706706874710276606498936857759217792399272622545356923496648066972851307552943494607006236664779782457676750601933928741855566537180682449588022627762894452647955309858071684743668764103374919896830691370820270431845210144552899302482489086159230107311871973441938772230297936968323364444885618086227543610536799234471448815742784799816600847674463621304114076611518369269786746806064625173598846975603820125306980982787030881513655512171552673424402550929145728822638665937926971092178787061653320498956248786180186391991099271001719767697697909657944193039718396228206646791827025526197270768819079459237162532507649010048534221896354870805893738136113525331466426094863067739011868586815994478431505947282920567358907821040575182158837774773239063377692225273213367590618634854759858104765385337233560836355651912585817293643497440222402889594736134846446633259468623170924046086762343708470022959969198855640656235692710155611770574936987892825674533348817583855363925862187429685734786911212712430632255387292214942570356549226145609363469565143905605900887019603976695135361370949254117709119316663721177077774029160260760766740385602177303122540707673922074695845466822733231402436648454982298644246889690405507930034065218860098668050569891179330267014735905886289425329738076589485655031215518567846745806071391799071756756605463434553308374945487617190776980746980699014304068031331239288118559831182607502630530551183368131913993547830487046970492229895903115707766187014330528001489237808808235103354154573345828491416810469567012639650930802503664884265381634807826965467764355129999469368432101475126962562819649601420036526583515123282254959562060550023759345781815702678925099585970330094087582417727355234461914055411868426099920636333733090260815747474049431103752635646105188464228216290917428126976207328138645299490852229607774124663681570269936898100832737980518184032477550717998055209064725271758615607130366068794016981349022040750694525242490718753433993836703494763162287417271196271646764706279175940497789978778795870082209533049067117547519334583846447238453975257389957151762864636167034349494022078054845567674520089528080628143408014378077676530581464910684833450653936186541555433054749567315139819992996774343673166210507570902636549732442388247662345722481112868037379284643637526752180928208363454411729347782204573430655390637798645445529193691535940179789830976033766084885136091104772760263679705536804393747009082065280485657999519603015529779422220372975870598158577647259651963109821683846031819163829482281070190616676222477440254852686018093835910242719165792709835712695251824680719554487720164564269225604642662204105370136957218460722385967138808603732060559545819076863983745305846152893497455963940263837822204803101280874916929485664312376894744967964670670268862655999932921758714439360279763849794067718030709291031688457058677223270722547834686444362966546714517578435249650260228317609358210751404857514269700912777174124510062499294029671741246593801166421636748964963950428191768722024305732665058395629398944229050496795197928872150135865141590063574132120745831194105765493098381066511071670971145774686001622087174673413677389671048830014819463917871016699395112863207210663248207698888696632878888875731819754345748153613511272093127972004140140183132683717483896658536896985265716758338625583942523761360408450068620922831423403685914236452080913528639654247677138957734111488622863770537506104792925310167740296905831265412650953222171397564296593226119080733358561440087128263799873484673857930470651477419348076790192987838737879420342143923188579565628194400913173107184548643934948230278524853499447336535811456
... Merci and bravo Hans! (and Maximilian and Gilles).
I found a smaller (lex-earlier) solution :
RépondreSupprimer>> 1 x 1 = 1
>> 2 x 6 = 12
>> 3 x 2 = 6
>> 4 x 8 = 32
>> 5 x 97 = 485
>> 6 x 162 693 782 297 = 976 162 693 782
>>7 x k(7) = 2977...
To find the next term, consider
7 * 0.abc... = 2.977abc...
<=> (7 - 0.001) * 0.abc... = 2.977
<=> 0.abc... = 2.977 / (7 - 0.001) = 0.425346478068...
Now actually we need integers, 7 * 0.abc...wxyz = 2.977abc...w
(3 digits less because the prefix 2977 "shifts to the right" abc...)
so we have to truncate k to, say, m digits, and the r.h.s. to 4+m-3 digits the same length.
That's a simple loop, e.g.,
for(n=1,99, k=x\10^-n; 7*k==eval(Str(2977,k))\10^3 && return(k))
It will return k(7) = 425346478
and indeed, 7 * 425 346 478 = 2 977 425 346
is a match! Then it goes on 8 * k(8) = 4788 ... etc.
I already found k(6) = 162 693 782 297 above with the same method.
RépondreSupprimerIt also gives k(7)=abc...n for your non-minimal solution in the above post:
7 * .abc... = 1.767abc… <=> .abc... = 1.767 / (7 - 0.001) = 0.2524646378054...
We find that truncation to 10 digit works :
k(7) = abc...n = 2524646378 and 7 * 2 524 646 378 = 17 672 524 646.
Next term must satisfy 8 * k(8) = 3788..., we consider
0.abc... = 3788 / (8000 - 1.) = 0.47355919489936... and indeed k(8) = 47355919489936242 works, 8 * 47 355 919 489 936 242 = 378 847 355 919 489 936, etc.