Problems & Puzzles: Puzzles

Puzzle 341. Multiplicative persistence, Erdos style.

The original 'Multiplicative Persistence' concept is:

“Multiply all the digits of a number N  by each other, repeating with the product until a single digit is obtained. The number of steps required is known as the multiplicative persistence...

... The smallest numbers having multiplicative persistence of 1, 2, ... are 10, 25, 39, 77, 679, 6788, 68889, 2677889, 26888999, 3778888999, 277777788888899, ... (Sloane's ; Wells 1986, p. 78)....

There is no number N<10^50 with multiplicative persistence >11 (Wells 1986, p. 78)….”

(Excerpt from Eric W. Weisstein. "Multiplicative Persistence." From --A Wolfram Web Resource. http://mathworld.wolfram.com/MultiplicativePersistence.html )

This is the sequence of numbers arising from 277777788888899 (*):

277777788888899,  4996238671872, 438939648, 4478976, 338688, 27648, 2688, 768, 336, 54, 20, 0

2) Multiplicative Persistence, Erdos style, goes this way:

“Multiply all the non-zero digits of a number N  by each other, repeating with the product until a single digit is obtained...."

Walter Schnider says that "Using the modification from Erdos the persistence seems not to be bounded (proof?)"

I have obtained the following n values for persistence greater than 11:

 persistence n (smallest?) value 12 (5)16 (7)13 13 (7)42(8)2(9)14 14 (2)1(6)1(7)130(9)8

This is the sequences for persistence 12:

55555555555555557777777777777, 14784089722747802734375, 49962386718720, 438939648, 4478976, 338688, 27648, 2688, 768, 336, 54, 20, 2

Questions:

1. The n values that I have obtained for persistence 12, 13 & 14 are the smallest values?

2. Can you obtain the smallest n values for persistence from 15 to 20?

3. Is the persistence (Erdos style) unbounded, as Schnider thinks?

_____
(*) BTW,
277777788888899 is the minimized expression of the number really computed originally, that is to say 22222222222222222223333777777. I have computed only one more distinct number composed only of digits 2, 3, & 7 having persistence 11: 22223333333333333333333377777, whose minimized expression is 27777789999999999. I bet that these two numbers (and all its permutations) are the only solutions - for numbers composed only of digits 2, 3 & 7 - for persistence 11.

Contributions came from Wilfred Whiteside & Phil Carmody:

***

WW wrote:

Puzzle 341 - minimal solutions to persistence 12 through 16

(Question1: case 12,13,14 solutions below are smaller than those posted by CR)

case of persist=12: (7)9 (8)12 (9)5 = 26 digits = minimum
77777777788888888888899999
163747527548676077518848
4996238671872000
438939648
4478976
338688
27648
2688
768
336
54
20
2

case of persist=13: (3)1 (7)27 (8)6 (9)19 = 53 digits = minimum
37777777777777777777777777778888889999999999999999999
69809726478979191696277299862872985495399563264
1404492781386126176471740317696000
24975483372306432
4389396480
4478976
338688
27648
2688
768
336
54
20
2

case of persist=14: (2)1 (6)1 (7)1 (8)99 (9)10 = 112 digits = minimum
2678888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888889999999999
74578405298868358061499745611046025877675420408378594099674468879366865988719527897725397266121883648
797466391371269339802853647433701338561899365731842129920000000000
1404492781386126176471740317696000
24975483372306432
4389396480
4478976
338688
27648
2688
768
336
54
20
2

Question2: In Progress

case of persist=15: (6)1 (7)157 (8)46 (9)25 = 229 digits = minimum
6777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777788888888888888888888888888888888888888888888889999999999999999999999999
7190277805656896237266279928179772059796081093791892500479162183359964757353956754361577464590374547304773629978879224056047498957910439999134424123798313948633871716972951634608296401150495846039552
6761649844889427899714485661836135982253298692044782055127319048849899874922975246243864709837052968960000000000000000000
478171229990877345192708930286667593551490390339639585013760000000
76808198982053775275798298624000000
24975483372306432000
4389396480
4478976
338688
27648
2688
768
336
54
20
2

case of persist=16: (3)1 (7)54 (8)82 (9)353 = 490 digits = minimum
3777777777777777777777777777777777777777777777777777777888888888888888888888888888888888888888888888888888888888888888888888888888888888899999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999
10313360079197038826077089364230457379432574159540187898047551374783782810031383346839773167884842406424617596932068662374149873895560771196842434034106623844332439337889427880983704625242093937990922999271273650634388894340185325262986279807209440122274073367109798977182804436148033972294151277222034341895648455313751129874315672779020852685493964028556348278697469898081571840854299775963529025551901462407830348077712377075714821070019280962881566277632
1976933552197413612159628902627430978010851694042125816551551749164417593934812396670920758208266066117717375273336106396150759715023578737998568950408664664249561214198331979010971731603753185698045337798713099904267181475705651200000000000000000000000000000000
6761649844889427899714485661836135982253298692044782055127319048849899874922975246243864709837052968960000000000000000000000000
478171229990877345192708930286667593551490390339639585013760000000
76808198982053775275798298624000000
24975483372306432000
4389396480
4478976
338688
27648
2688
768
336
54
20
2

case of persist=17: (3)1 (7)27 (8)622 (9)399 = 1049 digits = minimum
37777777777777777777777777778888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999
574779817804390904890286981578964902781742714480320102678593558479177180582606355263905759684722351577904879247007053974584208744001683336359195814522257396582927450017759678139720092317593733121971548881593674829563639377365298522834542493377570399998139939997206867358132738209223116041773016336986393741009927170577275675516663119149156228199482628326795805605312859951078492366250880322089839944278485593967559483726298470729579638656671875978230497021973201423578244054142219544517517323730416132555518599958484932098350355976621826733775403353089175239106643656216106760961306951699339732730932329296161823783063407785293611090217664324462726977118668779311130064512708621524267150762503281355753486157669279028423101647557295097183503597630687186194628637661562365592379177125730665243432740056266434025929842174673602217195744165721547122876094729663369401715352412452489940215312319347759504339884598957640155052631453987231074795542859487259944471949737984
828368498364837666825200579884512979791181474317446829688303086334406913900460872056195260292948438666866245620375776835468872736609243559801402574861042255424847483347504475870445862265647578973527779685846893028443330223238867730328672262060722901509882728834544281291989982007388436910606023096673175842971489593137979851445568782398280193277780127646514774127076336897004902895175816443798802252380437662173047599682950648247296300971464012595200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
1047621388678982390523975179652349599477748169802451387943572170708460577790233282360940599731586471226145975582852646268709258217365333296402680064553776311480647684769193600633649249762058762715383618293106618568275937262672281600000000000000000000000000000000000
1027981992925234360360281305873470377951167677089379366226989667956442830719965534999107611234813238663498319265792000000000000000000000
69040999053800788558896662715231492793697575191617305376194560000000
7680819898205377527579829862400000000
24975483372306432000
4389396480
4478976
338688
27648
2688
768
336
54
20
2

Since number of digits approx. doubles for each increase of persistence ... will the next one be about 2200 digits?

...

I just found that Sloane lists up to N=13 for puzzle 341.
http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A014120

***

Phil wrote:

The Mathworld article you quote is out of date. See
http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0107&L=NMBRTHRY&P=R1036&I=-3

***

case of persist=18: (3)1 (7)140 (8)258 (9)1946 = 2345 digits = minimum
3777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777778888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888889999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999
9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999
5546089446864993529211309277721902623921032942997752811155459225454653157758875848216708167539884748935256311209720493184916224374011538798648276693622901916484264375777973990118071990405792714953864728927670012650548636232907369331755715064164141721771477290463405073729397681549209707582251480573313202793384449720114125873722456567562884760557716994630379279118028796491275961180600131863361999229306561907902495863924270980338162667096065910767199498570847066832858969748012938702706242895453715720942053399871597306066885767178386889651213451677381892954610218189664453784722038903379574718002867001824430885557853947001735683243880129729029326901329441072385714478176524063707618271395123427178392244597065078280601777333294867808651451993623108209418834504133607836522285959938918017616536647743660794464010778591673043817853791825738359957977731732339111065870691499407301026683534386928847776717097640892983356648443269088620338566007602443823504685441911392453914455515814960267177013710458463913662218117868875394356288738109936930025612349179764792739182147743766499893101451159488613401271011428659470767788899767225322028665874820423656707171148149345649701223424038606655403674471053129707799443096383054475131893524143060453492257965492908867560190324635167174136171969783733686211350810179264527500393801257803155996718586021917228213197087852008170483234719413171993921675905733555859244024039834888419510755002530218965368674915121610588239589313431121133377748320919368045315707926392390573792935382942764152735873140708200903102871357107928179207372569882789469200172668691784641049939859933638688576573068442782610447792967163415984420992534012210694727327402017165902469331478723572913714337604928465150355702762005047183733799690728706783422518348196433756560679980996138369586377726077245285392106166028866336175476921565145323228832943092270880811262867881292120461534401133519772059778555272625658599275254701443447484193914491970829305644077317506854180148207593585397487833743271651902684680051069928953143296132134878
004081045658853357675509678872299543196936893471271548167163098933196555956991970543615801477763277391640108271269167995416226338481922874767886525174355362578432
467338002977719346258635144194285880902335387146283511718071933474044456888783142691112725858124329244990441918382247336664254516542386750620984965587975475569799829106885984759652281988913364369724740961798915637708611954432931159979158114951095682019026006507502163256855675727249077525851367064446411174638256732956862893695340701395932509242548101702336832171179906929978279089043353918445104687806701933961692810499705360223461538019631997602791280161641012393929504868786066660127991769859891806683048666885286312636697448189392773404726062086934371836289043776441956148930392693334789998938242867177309932777596083665170729974555087375022873915131740301901340080187317590584808459367963873486550406189099383135020761554472032741278749942096060900983773251658639447264177829960064934488350906777019936284789806104263475489585429876692932887656975405614281052659569663873471683378070950996019224571716010153807508395660676699836493866257770779183016238093468466275300297683143931779399818379871564965579206760474245623899389300143337714483200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
5282528022771627962616697821876118575425232316103621107312801043126357684026493696376850636823916301534209864376562322666511263786106379015698812556929987122537856402332615807203303209820558200812088431298970866764261440361930929766254975271956297170432736665516954279982826687403285241255658656754484389619293415109709835696321181437683470340536773600363074739226159462530122576218292298490399124152530248701689677889813140153335392203459989622290032378832683687838134781221439849041952310952292407276430483523892997521408000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
67185664822176592087211503596296259026338640541054584459171760071743803304205757654780034195294194593728172227168327493651763035191142648597386774481837004832375464349798584529195748969608198380851909577961747885909503301738862534678267559936000000000000000000000000000000000000000000000
10279819929252343603602813058734703779511676770893793662269896679564428307199655349991076112348132386634983192657920000000000000000000000000000
69040999053800788558896662715231492793697575191617305376194560000000
7680819898205377527579829862400000000
24975483372306432000
4389396480
4478976
338688
27648
2688
768
336
54
20
2

***

Marc Lapierre wrote on August 12, 2013:

To complete your list, here are some results with numbers having greater persistence than 18 :
p( (5)1(8)6457 ) = 19
p( (8)14829 ) = 20
p( (8)32935 ) = 21
p( (8)68764 ) = 22
Please note that these solutions are not minimal.
They can be tested here :
or better with your own tools to be sure there is no mistake :)
It really seems to be unbounded when you look at the shape of the curve in above url...
I heard about this problem thanks to the last paper of JP Delahaye author of mathematics pages in Pour la Science (french version of Scientific American).

***

On Set 23, 2017, Edouard Debonneuil wrote:

I found your page http://www.primepuzzles.net/puzzles/puzz_341.htm as a person at work told me about the persistence-limit
conjecture and I found the answer; I am going to make an article, but your page indicates to email you to add a solution; could you add this? :

a) I looked for persistence greater than 11 up to 10^1000 and found none. It took a few few hours, so I am planning to parallelize
it to go to 10^10000.Method: I cycled through 2^i*3^j*7*k and 3^i*5^j*7^k up to that limit (limit: eg 237788999 is written
in less than 1000 digits). The known persistance 11 numbers are found quasi immediately and then no other persistance 11 number appears.

b) I found why persistance is limited: the number of possible "links" (to go down from one number to a single digit, when computing
the persistence of a number) that don't have a "0" is limited (so it is impossible to build a longer chain of links, ie a greater
persistence, than that limit):

-- these links are written as 2^i*3^j*7*k and 3^i*5^j*7^k (otherwise they have a zero at the end, when written)
-- there are asymptotically O(n^2) numbers "2^i*3^j*7*k and 3^i*5^j*7^k" of size n [proof (*)]; what counts is that it is polynomial
-- an asymptotic proportion A*exp(-B n) of them don't have zeros (if "2^i*3^j*7*k and 3^i*5^j*7^k" have as many "0" as other
numbers of the same size without a 0 at the end, the proportion is (9/10)^(n-1)
-- summing over all lengths n (to sum over all integers), the integral of a polynomial times an exponential is defined (not infinite)

c) The reasoning is valid of other fixed numbering bases > 3

d) For persistance a la Erdos, the same type of reasoning suggests(**) that there is no limit:
-- There are O(n^3) possible links of size n [2^i*3^j*5^k*7^l]
-- (**) Assuming that the product of their digits behaves like other numbers of the same size, there are O(n^a) links-of-links of size n, etc.
-- So whatever the desired persistence p there is an infinite number of chains that get to that persistence, a la Erdos.

(*) Considering 2^i*3^j*7*k numbers,
10^n <= 2^i*3^j*7*k < 10^(n+1) can be written n <= i*log(2)+j*log(3)+k*log(7) < n+1
i*log(2)+j*log(3)+k*log(7) < K can be visualized as the volume of half a cube whose connected tips are (0,0,0), (K/log(2),0,0),
(0,K/log(3),0), (0,0,K/log(7)) and the cube is cut in half by the plane that goes through  these 3 latter tips.
So asymptotically the volume is K^3/(log2*log3*log7).

We want the tranche from K-1 to K (n to n+1) so we derive by K: the volume of the tranche is 3*K^2/(log2*log3*log7)
So card({2^i*3^j*7*k or 3^i*5^j*7*k | size n) ~ 3*n^2*(1/log2+1/log5)/(log3*log7)

***

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