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The Nine Digits Page 6 with some Ten Digits (pandigital) exceptions
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When I use the term ninedigital in these articles I always refer to a strictly zeroless pandigital (digits from 1 to 9 each appearing just once).

  Topic 6.10   [ February 17, 2022 ]
Pandigitals in Trigonometric Functions.
by Daniel Hardisky

Find two Pan-digit numbers x, y (☆as shown) such that F(x) = y and x ≈ y (as close as possible) F(x) is Sin, Cos or Tan. ☆Fraction form   Decimal form 761 239 458 109  =  0,761239458 E.g. (in radians) sin(0.124863975) = 0.124539768 544543... tan(0.124785936) = 0.125437698 518344... cos(0.761239458) = 0.723981564 996210... cos(0.761239485) = 0.723981546 371089... Note the two swaps between 58 & 85 and 64 & 46 Last two digits reversed!

  Topic 6.9   [ October 15, 2021 ]
Daniel Hardisky has some new ninedigital fractions to share.
(Many others can be found f.i. in Page 3 of this ninedigits section)

The equivalent fractions given here are quite interesting. Each one of them uses all the digits from 1 to 9 one ! 2 6  =  3 9  =  58 174 2 4  =  3 6  =  79 158

Arrange all the digits (1 to 9), each once to form the following fractions : Examples 3 * 4 5 + 7 = 2 * 8 1 + 6 + 9 1 1 * 8 6 + 9 = 2 * 4 3 + 5 + 7 8 15 5 * 8 3 + 7 = 4 * 9 1 + 2 + 6 4 What integers are possible ?                   1, 2, 4, 6 What fractions are possible ?  1/2, 2/7, 8/15, 3/4, 4/5, 8/7, 8/5

And finally a challenge for our readers! Arrange the digits from 1 to 9, each once to form fractions as shown: 6 + 9 7 + 8  =  3 + 4 2 + 5  =  1 5 + 8 4 + 9  =  3 + 6 2 + 7  =  1 Can you find other combinations with the last ratio > 1 ?

  Topic 6.8   [ April 4, 2021 ]
Separate ninedigitals in three groups of 2, 3 and 4 digits
and multiply them together
Presented by Alexandru Dan Petrescu

Separate ninedigitals in three groups of 2, 3 and 4 digits and multiply them together.
When the product is also ninedigital it should be one of the 205 entries of the table below.

[by PDG] A thing of beauty pops up with entry 125. And it is unique !
Enjoy this trio of ninedigitals.

  Topic 6.7   [ March 25, 2021 ]
Separate ninedigitals in three threedigit groups and multiply them together
Presented by Alexandru Dan Petrescu

Separate ninedigitals in three threedigit groups and multiply them together
When the product is also ninedigital it should be one of the 39 entries of the table below.

The ninedigital in line 34 is the largest product and was already noted by Janean Wilson.
Reference : Wonplate 95 - A ninedigital divertimento (second case)
At least we now have all 39 solutions!

  Topic 6.6   [ March 21, 2021 ]
Integral Triangles with Ninedigital Areas
Daniel Hardisky

a few of these integral triangles with ninedigital areas. These are oblique.

integral right triangles with ninedigital areas.

integral oblique triangles with two sides which are ninedigital numbers AND integer area.

So far I cannot find ninedigital numbers on all three sides of any triangle AND integer area.
We are working on this on my math page.

Checked with Wolfram Alpha for accuracy.

Regards, Daniel Hardisky "

Find integral oblique triangles with the Areas = 9 digit integers, using all digits 1 to 9 (re-arranged). Example: 413829576 (not the answer) /\ / \ a / \ b / Area \ /____________\ c Daniel Hardisky

Triangles with rational cosines: a = 2q(p–k), Cosϑ = p/q b = q2 – k2 c = q2 + k2 – 2pk –q < k < p Let q = m2 + n2, p = m2 – n2 n2 = 2mn or p = 2mn and n2 = m2 – n2 Then Sinϑ = (n2)/q Area = 1/2abSinϑ = 1/2ab(n2)/q

[ March 23, 2021 ]
" Hi Patrick,
I send you in addition the table for Integral Triangles (oblique) with ninedigital areas by Daniel Hardisky.
Best regards, Alexandru Dan Petrescu "

And here the solutions for the Integral Right Triangles with ninedigital areas
as proposed by Daniel Hardisky [ March 26, 2021 ].

Note that triangles 30 and 31 produce the same ninedigital area ( highlighted ). Just click on the table to see !

  Topic 6.5   [ March 18, 2021 ]
Nine- and pandigitals with 5 multiples
Alexandru Dan Petrescu

We checked for ninedigitals and pandigitals having the greatest number of multiples
that are also ninedigitals and pandigitals.

For ninedigital numbers there is only one solution with 5 multiples.

For pandigital numbers there are two solutions with 5 multiples.
One of them being the trivial case 1234567890 which is the above ninedigital x 10.
The other one being 1098765432.

[ March 21, 2021 ]
Addendum by Patrick De Geest

The above found multiples return in an investigation I started some years ago.
The goal was to find all  Products of ninedigitals with pandigitals that result in squares .
The first few solutions are identical with the multiples from Petrescu's table.

And here I stopped the search at the time in favor of other projects.
Of course there is a relation that I will highlight here.
The relation shows up when variables are put in place !

Some logical questions that arise are for instance
1. Can you extend and/or complete the list ?
2. Are solutions abundant or rare ?
3. Exist there non nine- or pandigital squares ?
   Must the nine- and pandigital numbers share their factors?

Variations on the theme could be :

Ps. note that there are some sporadic solutions given by Peter Kogel. Study this page first if you're interested.
See my webpage The Nine Digits Page 2 under 'digital diversions'

Soon after [ March 22, 2021 ] Alexandru Dan Petrescu wrote

" Relating to products of ninedigitals with pandigitals resulting in squares I extended/completed your list.
There are 8 solution for x4 and 512 solution for x5.
Interesting, powers of 2, and for x5 exponent is 9 (again 9!)."

  Topic 6.4   [ March 11, 2021 ]
A ninedigit problem Re-arrange the digits 1 to 9 to make a 9 digit number
Daniel Hardisky

Re-arrange the digits 1 to 9 to make a 9 digit number. These numbers are always divisible by 9 since the sum of the digits = 45 and is also divisible by 9. Example: 619428753 / 9 = 68825417 (not the answer) 1. What is the largest prime found after dividing one of these 9 digit numbers by 9 ? 2.Which of these 9 digit numbers has the greatest number of divisors ? Daniel Hardisky 1. 987654231 / 9 = 109739359 2. 769152384 768 divisors

Of course more questions can be posed around this ninedigital and prime topic.

3. Can you find the pandigital equivalent for the above problems 1 and 2 ?

4. What is the smallest prime that cannot divide any ninedigital or pandigital number ?

Solution by Alexandru Dan Petrescu [ March 13, 2021 ]

The smallest prime that cannot divide any ninedigital number is 44449 The smallest prime that cannot divide any pandigital number is 111119

In hindsight, now that we know Alexandru's solutions, we see that this problem
was already discussed in the past. Here is the source
Puzzle 926. pandigital and prime numbers
Nevertheless, a second opinion can't do any harm :)
Note also that both solutions belong to the same OEIS sequence A090148. What a coincidence!

Both numbers were also already registered in the Prime Curios! database.
Prime Curios! 44449
Prime Curios! 111119

  Topic 6.3   [ October 9, 2020 ]
From my collection of palindromic quasipronic numbers of the form n*(n+5)
may I present a remarkable repdigital number (see Index Nr 24)

Multiply this eleven digit repdigit 22222222222 with 22222222227
and we get the following palindrome with 21 digits

But what makes this equation beautiful is that left and right of the central 9
we unveil two curious pandigitals

Moreover there is a surprising order in the arrangement of the ten digits
From left to right we start with 4 downto 0 intertwined with, also from left to right, the sequence from 9 downto 5.

The story is not at its end. What about that middle 9 ?

  Topic 6.2   [ December 26, 2016 ]
An astonishing e_quation using just our familiar nine digits

Incredible Formula - Numberphile

When we put those nine digits in a row we get the number 194673285.
Anyone there who can turn this ninedigital into another curio ?

  Topic 6.1   [ July 23, 2015 ]
Finding one or more ninedigitals as a substring in the decimal expansion
of some ninedigital raised to a power p

What can we find in ninedigitals raised to the power 2

Two nice stepladder_5 solutions !

Let me thicken the plot at this point and leave behind us these rather trivial overlapping solutions.
Instead let me try to hunt for strictly NON_OVERLAPPING [ further on referred as NOV ] ninedigitals substrings.
(Written in Ubasic - program name 'ssninx.ub')

{ 2 NOV_substrings with power 2 → none found }

2 NOV_substrings with power 3 → 2 solutions

297146853 ³ (26 digits) =

26236953487129018576392477

368571429 ³ (26 digits) =

50068548279613994372186589

{ 3 NOV_substrings with power 4, 5, 6, 7, 8 & 9 → none found }

3 NOV_substrings with power 10 → 2 solutions.

It is only at this power 10 that three separated ninedigital substrings appear !

271593864 ¹⁰ (85 digits) =

218373300939673819524716945508072362497652813274728161018618967907688452196377
6638976

479635182 ¹⁰ (87 digits) =

644332915723468747863172845933720713156733737801139368957793052380596182495377
326490624

{ 4 NOV_substrings with power 11, 12, 13, 14, 15, 16 & 17 → none found }

4 NOV_substrings with power 18 → 1 unique solution.

126593784 ¹⁸ (146 digits) =

697305140930159178423626352855159661623909798533865336983126873785948503478454
37769987132465705319852613947799401827653080762942141372458699063296

{ 5 NOV_substrings with power 19 up to 53 → none found }

5 NOV_substrings with power 54 → 1 unique solution.

457621839 ⁵⁴ (468 digits) =

464904842381567901688169376170799425136784613762714883780984997750911233911508
936246640416070598231699589961455572370670134004976420087287007782271802716619
828591635274809169528631549679091382639154700948359979571366641842161148335787
380149741364100335489890532665464884193736506826807596856661519884956418475084
016370282412303054004020873233524647278558086135180436315240989878197644865743
219148203100219738559091235979416283395521378773569037854028732788952004958241

{ 6 NOV_substrings with power 55 up to 85 → none found }

6 NOV_substrings with power 86 → 1 unique solution.

483259761 ⁸⁶ (747 digits) =

691089971277284621583973839908321146944631601201133003854843772671615448200671
262685915616836442117193245682799504223144629651743867521615002321668923138270
732687951714743751378782961329254342951134325410090163829860183409176492183553
310746630772058428151699835838580691150692415388449686925020318868688964370912
798936026644796788897118723452353835177251420939765727091581345769228796322211
232349371258688904374257677622623182596912041405433457744567399420686374957586
747787843855865628218707234366023809207264616321153936468531652523622910413452
658685093044935968826596483821451367475207986095872295275022536365973380099994
536865981060519036886188996252316309481443530912678465755435071179141665451617
176067797792325538546657974609472790542707361

{ 7 NOV_substrings with power 87 up to 119 → none found }

From power 120 and above OVERFLOW occurred and marked the end of the game for me.

Not so for Alexandru Petrescu who went further where I left off!
After seven years he submitted a solution with seven NOV's [ October 16, 2022 ].

{ From power 120 up to 147 none found. }

7 NOV_substrings with power 148 → first solution.

468273951 ¹⁴⁸ (1284 digits) =

171395710512787569160389869261985042124386255961340690113730062303413294844100
198396075912174953868100690644559963205332251785133798752941967288501567135341
196132847513353879481373749678439251430315232600764017026736274230256374579137
203275356112168107818766177750054417547060219260789158264410121289832491310645
985505971150828170863898592408482964975061079388277839322891849505702927232162
055851896525942326913276186900071393011996709326514089199509459170753116910460
168885226620436759637454958956004535789023688963905385044104036972428261729696
391900248261953470828358552682947017285995515658797391324733317129328833147698
181343169213457683365730141401111637820404660109255309558965037169776156182668
817056458492312974290644916933121992571047044310289509075969603615575018522063
602604345853194689086474745961943932239905260118353254963446008370535379245386
791975239205803417560011492633608762104222430055785718804454810192768766257189
672174635024670236272051022965400738946773827345506604183549915096906845395432
114531541606169388534796211346224568330115404525038205901929481301684824555576
216153629793617603141295322472423182184870457666208506170404195410930512382935
168907991258693030444827759273792824505665725922515483917257278574443850230426
642017409630127380144126703089289601

And on [ November 9, 2022 ] Alexandru Petrescu sent in the first solution with eight ninedigital substrings.

8 NOV_substrings with power 222 → first solution.

417269835 ²²² (1914 digits) =

540221425800102179059832028643688548078371644141172370977432587561779829831783
784117331656607024211426731029534086377298554997081649820238206606369126091188
880259481341213243428015703398987587542132079200603217319182239144882817464541
802843114056411548427481423608448881332148674292711362946057011642355888776832
472820195562990636492418675613457390668742957732088220530486152512926574832882
513788101036573467251348917603397464532248651656268909984156656166628191673740
182550219168946745229143496597631146157713679371485620157914302932628585382173
623123302093880654428645476236592674476654904231944211009238971604536861262856
172264073356106447047137076638666379259622875648432862155138490781039741755608
547902267304388233193050112740977802421654455954279395793089360675030229547257
625598984402327982708096753798067901884780220614678317406223795623148354709741
108538562078782275909200072139536489218677915868019652748531090194276583828506
757626769202804167847008257722539522624340257439248571936816671863080186751329
467317930383922811548725949412039294933645361364726296169305656582673077634112
535307956395979288567318429450788503409234458511084077962961358854664593128937
516928207578292380467082697259074293631533261770612984646016018685385123221089
219332455825152176575236562318154434477173846366734426364717320655185150317209
203484204071333969892520021694823043702527651842460509166220976353923115370922
117958700199052891952219275989157702337013570892161835001003780799825994237796
707195539461079689427036592649599264449156980007734708548150299975291630316256
084950596354213035756704505171808857047022787644327651015047937470196230512162
928220346934084622924948185339738464061101943458661976906174308542560268829639
504261538094198778374669567447220017503166594472035884958438556796794821351896
656891091651199864873076999250868754761214040442623669546174276777834072330746
785495225736895008594729006290435791015625

A day later on [ November 10, 2022 ] Alexandru Petrescu sent in 'a' solution with nine ninedigital substrings.

“I have a solution for nine ninedigitals, but I don't know if

that power 300 is minimal. I didn't check all powers from 223 to 300! ”

On [ November 13, 2022 ] Alexandru Petrescu mailed me with the message that no solutions were
found with power 223 up to 271. But with power 272 he hit the final 9 NOV_substrings solution.
As a bonus notice that the last two solutions have palindromic powers (222 & 272).

{ From power 223 up to 271 none found. }

9 NOV_substrings with power 272 → first solution.

673298145 ²⁷² (2402 digits) =

187248212977000073829157559540364148052748376536267353109603075320600826572977
319538192740216006694047497673481658138772401727147088250465778799207618240083
184887707567881525142496611903837310709044888703457168521756741871344347342848
931036507593047129231379465111163894237964543413160004699501462210223727845479
621842844651463256754358979561738422007675393511764292408733362110009810951629
917470630233437014829995247017018448785159690363449570673024381321656243667656
166701019600528934568554744220152694933747943030434829227634437978622274255486
344347662168826238484586371283680892461868752420567491363777498113360659703046
569190809465353346156068888574052023689040088323564591511992780051542693815753
643477740679893912953494331591623784530724698346071882339376352952612394795322
182458719740674377634815661985847867071460747090371502611625685979284575623241
181488591745448661351071337697877771028866029172187773992684928167435633568347
610332702778253059539300526195137064412194282462226694392840185571485524410375
235451491437724974161223045922070039968391912233202252066215584926624007744811
640041886181724164839596619592333900267736534680032114051492148173382524364765
261264913760134929632686738782263156779811048892522117991061732317076164779826
735420723242636889879684263842803126458591467329809411542557463237924001756170
217914276835730789243118202509023791363354073403179684666890068394252413313643
178902212604456367058081173248887465053737431520420948430925404771681233346383
234419885578316814571321184301461355480117703942644600568452673552898487901325
635043703007745185834209536929737953260923835620696362300704393006028955164954
279868438114286131819562707170527372595684792537101835164292330858940657560778
692843759805980755509500958742813724675329128636363453081813929316503222939137
510931959531916915193274863593999016329847106271689317049916010315953943974939
817394134008470325410482364403307595895709020839428253463473525258647955513390
188027384249498183232378669297972492795854968576887614680513934032424468297153
356885451932440732091488496007153789019402165754008862080634416237417027422241
801966592271263200829317071500460453414974250068714562776801357550137784139545
637908083456558354963242031596444263387781851965206309366591633510497262311112
530964134634085189047035724760258671349081656645870140899579156466930237772141
24489659385697088432067014540649552145623601973056793212890625

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Patrick De Geest - Belgium - Short Bio - Some Pictures
E-mail address : pdg@worldofnumbers.com