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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).

Sixth Page

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 458109 = 0,761239458

sin(0.124863975) = 0.124539768 544543...
tan(0.124785936) = 0.125437698 518344...

cos(0.761239458) = 0.723981564996210...
cos(0.761239485) = 0.723981546371089...
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 !

 26 = 39 = 58174

 24 = 36 = 79158

Arrange all the digits (1 to 9), each once
to form the following fractions :
Examples

 3 * 45 + 7 = 2 * 81 + 6 + 9 1

 1 * 86 + 9 = 2 * 43 + 5 + 7 815

 5 * 83 + 7 = 4 * 91 + 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 + 97 + 8 = 3 + 42 + 5 = 1

 5 + 84 + 9 = 3 + 62 + 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.

Indexabcdefghiab * cde * fghiABCDEFGHI
116927845316 * 927 * 8453125374896
218742935618 * 742 * 9356124958736
318762945318 * 762 * 9453129657348
419864752319 * 864 * 7523123497568
521657934821 * 657 * 9348128974356
621864975321 * 864 * 9753176958432
721954683721 * 954 * 6837136972458
823571948623 * 571 * 9486124579638
923648915723 * 648 * 9157136475928
1023756841923 * 756 * 8419146389572
1123981746523 * 981 * 7465168432795
1224537961824 * 537 * 9618123956784
1324738951624 * 738 * 9516168547392
1424756819324 * 756 * 8193148653792
1524918567324 * 918 * 5673124987536
1626531984726 * 531 * 9847135947682
1726783954126 * 783 * 9541194235678
1827516894327 * 516 * 8943124593876
1927538914627 * 538 * 9146132854796
2027561943827 * 561 * 9438142957386
2127635984127 * 635 * 9841168723945
2227819564327 * 819 * 5643124783659
2328936514728 * 936 * 5147134892576
2428936754128 * 936 * 7541197634528
2529871536429 * 871 * 5364135489276
2631725864931 * 725 * 8649194386275
2731927485631 * 927 * 4856139546872
2832459867132 * 459 * 8671127359648
2934621875934 * 621 * 8759184937526
3036517829436 * 517 * 8294154367928
3136871459236 * 871 * 4592143986752
3236892457136 * 892 * 4571146783952
3336912754836 * 912 * 7548247815936
3437521869437 * 521 * 8694167594238
3537594861237 * 594 * 8612189274536
3637621948537 * 621 * 9485217936845
3738619542738 * 619 * 5427127653894
3838972654138 * 972 * 6541241598376
3939576812439 * 576 * 8124182497536
4039768524139 * 768 * 5241156978432
4139867412539 * 867 * 4125139478625
4241783925641 * 783 * 9256297145368
4341927835641 * 927 * 8356317586492
4442531976842 * 531 * 9768217845936
4542687935142 * 687 * 9351269813754
4642819365742 * 819 * 3657125793486
4742819576342 * 819 * 5763198235674
4842876539142 * 876 * 5391198345672
4942891635742 * 891 * 6357237891654
5043981752643 * 981 * 7526317469258
5145693872145 * 693 * 8721271964385
5245871692345 * 871 * 6923271346985
5346359827146 * 359 * 8271136587294
5446513827946 * 513 * 8279195367842
5546513982746 * 513 * 9827231897546
5646819573246 * 819 * 5732215947368
5746917325846 * 917 * 3258137428956
5846981325746 * 981 * 3257146975382
5948519637248 * 519 * 6372158739264
6048621935748 * 621 * 9357278913456
6148651793248 * 651 * 7932247859136
6249365827149 * 365 * 8271147926835
6351369782451 * 369 * 7824147239856
6451429683751 * 429 * 6837149586723
6551693842751 * 693 * 8427297835461
6651762843951 * 762 * 8439327956418
6751834926751 * 834 * 9267394162578
6851972846351 * 972 * 8463419527836
6952369718452 * 369 * 7184137846592
7052416783952 * 416 * 7839169573248
7152641937852 * 641 * 9378312587496
7252819643752 * 819 * 6437274138956
7353721984653 * 721 * 9846376245198
7453891274653 * 891 * 2746129674358
7554238961754 * 238 * 9617123597684
7654279861354 * 279 * 8613129763458
7754326871954 * 326 * 8719153489276
7854367982154 * 367 * 9821194632578
7954613729854 * 613 * 7298241578396
8054617938254 * 617 * 9382312589476
8154673982154 * 673 * 9821356914782
8254761382954 * 761 * 3829157348926
8354783612954 * 783 * 6129259146378
8454826713954 * 826 * 7139318427956
8554867923154 * 867 * 9231432176958
8654871369254 * 871 * 3692173649528
8754873296154 * 873 * 2961139587462
8854917268354 * 917 * 2683132856794
8954921783654 * 921 * 7836389715624
9054926318754 * 926 * 3187159362748
9154931867254 * 931 * 8672435976128
9256718943256 * 718 * 9432379241856
9356819423756 * 819 * 4237194325768
9457348692157 * 348 * 6921137284956
9557642381957 * 642 * 3819139752486
9658342719658 * 342 * 7196142739856
9758729364158 * 729 * 3641153948762
9859274863159 * 274 * 8631139528746
9959342817659 * 342 * 8176164975328
10061243875961 * 243 * 8759129834657
10161389572461 * 389 * 5724135824796
10261578493261 * 578 * 4932173892456
10361749823561 * 749 * 8235376248915
10462531847962 * 531 * 8479279145638
10563458921763 * 458 * 9217265947318
10663478952163 * 478 * 9521286715394
10763487512963 * 487 * 5129157362849
10863495872163 * 495 * 8721271964385
10963528479163 * 528 * 4791159367824
11063592481763 * 592 * 4817179654832
11163917824563 * 917 * 8245476321895
11263918275463 * 918 * 2754159274836
11363918725463 * 918 * 7254419527836
11463927458163 * 927 * 4581267534981
11563941527863 * 941 * 5278312895674
11663951782463 * 951 * 7824468759312
11764728351964 * 728 * 3519163957248
11864728935164 * 728 * 9351435681792
11968751349268 * 751 * 3492178329456
12069321785469 * 321 * 7854173958246
12169573821469 * 573 * 8214324756918
12269813427569 * 813 * 4275239814675
12371892546371 * 892 * 5463345982716
12472354691872 * 354 * 6918176325984
12572413569872 * 413 * 5698169435728
12672631549872 * 631 * 5498249785136
12772648513972 * 648 * 5139239765184
12872834596172 * 834 * 5961357946128
12972891645372 * 891 * 6453413972856
13072894561372 * 894 * 5613361297584
13172958463172 * 958 * 4631319427856
13272958614372 * 958 * 6143423719568
13372963514872 * 963 * 5148356941728
13472968513472 * 968 * 5134357819264
13574916235874 * 916 * 2358159834672
13674952813674 * 952 * 8136573164928
13775963482175 * 963 * 4821348196725
13876342951876 * 342 * 9518247391856
13976819352476 * 819 * 3524219347856
14076923415876 * 923 * 4158291675384
14178459613278 * 459 * 6132219537864
14279623845179 * 623 * 8451415932867
14379641352879 * 641 * 3528178654392
14479684251379 * 684 * 2513135792468
14579846251379 * 846 * 2513167953842
14681293745681 * 293 * 7456176953248
14781342596781 * 342 * 5967165297834
14881347956281 * 347 * 9562268759134
14981354762981 * 354 * 7629218753946
15081394572681 * 394 * 5726182739564
15181394725681 * 394 * 7256231567984
15281526749381 * 526 * 7493319246758
15381527643981 * 527 * 6439274861593
15481527694381 * 527 * 6943296375841
15581623947581 * 623 * 9475478136925
15681654279381 * 654 * 2793147956382
15781679254381 * 679 * 2543139862457
15881724365981 * 724 * 3659214578396
15981749325681 * 749 * 3256197538264
16081753924681 * 753 * 9246563941278
16181792645381 * 792 * 6453413972856
16281973245681 * 973 * 2456193564728
16382719654382 * 719 * 6543385762194
16484216975384 * 216 * 9753176958432
16584276935184 * 276 * 9351216793584
16684357612984 * 357 * 6129183796452
16784357621984 * 357 * 6219186495372
16884513296784 * 513 * 2967127853964
16984591736284 * 591 * 7362365479128
17084612753984 * 612 * 7539387564912
17186423759186 * 423 * 7591276145398
17286549372186 * 549 * 3721175683294
17386751923486 * 751 * 9234596387124
17487291563487 * 291 * 5634142635978
17587294531687 * 294 * 5316135972648
17687321456987 * 321 * 4569127598463
17787321549687 * 321 * 5496153486792
17887342516987 * 342 * 5169153798426
17987432615987 * 432 * 6159231479856
18087453926187 * 453 * 9261364985271
18187654931287 * 654 * 9312529834176
18287924153687 * 924 * 1536123475968
18389316527489 * 316 * 5274148325976
18489361752489 * 361 * 7524241738596
18591278536491 * 278 * 5364135698472
18691567243891 * 567 * 2438125793486
18791567384291 * 567 * 3842198235674
18891846537291 * 846 * 5372413568792
18992378516492 * 378 * 5164179583264
19092738465192 * 738 * 4651315784296
19192754183692 * 754 * 1836127359648
19292871465392 * 871 * 4653372854196
19393768215493 * 768 * 2154153847296
19493852674193 * 852 * 6741534129876
19593867241593 * 867 * 2415194723865
19694513826794 * 513 * 8267398651274
19794617235894 * 617 * 2358136759284
19896183754296 * 183 * 7542132497856
19996348521796 * 348 * 5217174289536
20096753248196 * 753 * 2481179346528
20197468325197 * 468 * 3251147582396
20297684325197 * 684 * 3251215697348
20397864152397 * 864 * 1523127639584
20498271635498 * 271 * 6354168749532
20598631475298 * 631 * 4752293854176

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

724135698
+
72 * 413 * 5698 = 169435728
=
893571426

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!

Indexabcdefghi
(sorted)
abc * def * ghiABCDEFGHI
1163827945163 * 827 * 945127386945
2234561987234 * 561 * 987129567438
3237618954237 * 618 * 954139728564
4243691875243 * 691 * 875146923875
5248751963248 * 751 * 963179356824
6251738964251 * 738 * 964178569432
7256743891256 * 743 * 891169475328
8261538947261 * 538 * 947132975846
9261594837261 * 594 * 837129763458
10263871945263 * 871 * 945216473985
11281547936281 * 547 * 936143869752
12291534867291 * 534 * 867134726598
13312564897312 * 564 * 897157843296
14319572846319 * 572 * 846154367928
15324659871324 * 659 * 871185972436
16329576841329 * 576 * 841159372864
17342671958342 * 671 * 958219843756
18364581792364 * 581 * 792167495328
19381657942381 * 657 * 942235798614
20387641952387 * 641 * 952236159784
21413568927413 * 568 * 927217459368
22418756923418 * 756 * 923291675384
23423581796423 * 581 * 796195627348
24428657913428 * 657 * 913256731948
25432571869432 * 571 * 869214357968
26432597618432 * 597 * 618159384672
27438516729438 * 516 * 729164759832
28452871936452 * 871 * 936368495712
29463581927463 * 581 * 927249365781
30472518693472 * 518 * 693169435728
31495681723495 * 681 * 723243719685
32513872946513 * 872 * 946423179856
33531768924531 * 768 * 924376814592
34531876942531 * 876 * 942438176952
35536841927536 * 841 * 927417869352
36567843912567 * 843 * 912435918672
37579612843579 * 612 * 843298715364
38612743958612 * 743 * 958435617928
39639725841639 * 725 * 841389614275

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

" I have
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
"

1143205846618773803766935293126793854
21418520-128-1295041040253759254017127894536
34261800172817275043600257471254017129765384
4712193-95-151168216161444831360135926784
513117026-216895202889629008135926784
6131170168-2126642602845936397139847526
74331105-1073-1087264309403945669892145829376
8129225634921663004822446852145829376
9814260-132-24922460840559963865146738592
1093186508648864218610380013833634632154382976
111181851767757366302829613050159674328
12108164160-4736678962468744145183967524
13243585567541144304204954421412185492736
14991620-49162158762384328645189265734
151314365-27-313642920132264132512192576384
16233538520491138312044836319885193548726
1711102212202821848644805737305193765824
1811615785-13113267824748864080213497856
192019761760-75839231039645572305845269783514
202014596560538204262246577242100295184736
21626712-640-7013128686415543101065295814376
222717307286645493440920047012317965824
23623565-493-544276576302328978777327815964
24117170154-8272802402217660880376814592
251112265-23-44264111306828970137378594216
26133178160-10478939842086875780429713856
27126180108-139144889201307981745465193872
28181325323157361079008097628852483912576
29182328320-3117241393610863403345493527816
302396104484414148540177619171445514739862
3125231154115011039610847611510711425519362784
32181325323146361150508430932625537216948
33454338743870385217613946416997231300538471296
3418940524322732412960112496105232583179264
35134185153-108104965702256178937612395784
36174305136-2962732635205409261153637915824
37174305136107273176908157675370645837192
383999810-9792-979659478480274484352852652173984
391814520128-5185046718402076671332675912384
4016326524711896683705630125857697231584
41135194130-1371441035961886792025725398416
421974102662443121804010856497828745183296
4320240480683969696158592156960753629184
44151226-30-127224438443494759585759328416
45525153055304527810327586028574111285765214398
46195386-190-21233616984104052113380769152384
4750494901490048729927445628341710585785631924
48221058444042738415184158727147625792365184
495292785262326229365570881341876097824935176
50924657-495-50643214454175613186745834512976
5111920240-114198622162780862920847921536

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 !

1      137791837222965126573894
2      47804532848100127349856
3      271441020829000138542976
4      205021393624790142857936
5      251021173627710147298536
6      71808435671940156397824
7      86112366686190157843296
8      96393304834425159274836
9      156242083226040162739584
10      46417318873335169832754
11      237791537228315182765394
12      124412941231935182957346
13      169922265628320192485376
14      1465112652146535194273586
15      336961267236000213497856
16      250921705630340213984576
17      256231713630825219537864
18      449821142446410256937184
19      69069820869555283459176
20      71383795671825283961574
21      459361285247700295184736
22      136294672848675318427956
23      92781739293075342918576
24      339482073639780351972864
25      315182402439630378594216
26      299492656840035397842516
27      281223009641190423179856
28       1600566042160170483529176
29      1072269768107670523691784
30      128318290883895531896274
31      8883119756120085531896274
32      241574712452955569187234
33      503582354455590592814376
34      309244123251540637529184
35      731431857675465679352184
36      9999137532137895687591234
37      911681587692540723691584
38      205597048873425724581396
39      760411931278455734251896
40      762931937678715739126584
41      5529268128268185741239856
42      635312389267875758941326
43      558832804462525783591426
44      13617115344116145785319624
45      789392004881445791284536
46      1844048928184620823179456
47      1835969072183820832791456
48      678182462472150834975216
49      449583794458830852943176
50      6431042778643110893271456
51      864362195289180948721536
52      17874108832110290972631584

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.

2 4 5 Ninedigital123456789 x 246913578 x 493827156 x 617283945 x 864197523 x 987654312

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.

2 4 5 Pandigital1098765432 x 2197530864 x 4395061728 x 5493827160 x 7691358024 x 8790123456

[ March 21, 2021 ]

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.

123456789 x 1975308624 = 4938271562
123456789 x 3086419725 = 6172839452
123456879 x 3086421975 = 6172843952
- - -

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 !

123456789 [n] x 1975308624 [16n] = 4938271562 [4n or sqrt(16n2)]
123456789 [n] x 3086419725 [25n] = 6172839452 [5n or sqrt(25n2)]
123456879 [n] x 3086421975 [25n] = 6172843952 [5n or sqrt(25n2)]

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 :

The product of two ninedigitals is a square.
123456789 [n] x 493827156 [4n] = 2469135782 [2n or sqrt(4n2)]
The multiplier need not be a nine- or pandigital
123456789 [n] x 6049382661 [49n] = 8641975232 [7n or sqrt(49n2)]
123456789 [n] x 7901234496 [64n] = 9876543122 [8n or sqrt(64n2)]

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!).
"

[ 1 ] 123456789 [n] * 1975308624 [16n] = 4938271562 [4n or sqrt(16n2)]
[ 2 ] 129465573 [n] * 2071453968 [16n] = 5178634922 [4n or sqrt(16n2)]
[ 3 ] 158729463 [n] * 2539671408 [16n] = 6349178522 [4n or sqrt(16n2)]
[ 4 ] 158794623 [n] * 2540713968 [16n] = 6351784922 [4n or sqrt(16n2)]
[ 5 ] 184573629 [n] * 2953178064 [16n] = 7382945162 [4n or sqrt(16n2)]
[ 6 ] 237841956 [n] * 3805471296 [16n] = 9513678242 [4n or sqrt(16n2)]
[ 7 ] 237946581 [n] * 3807145296 [16n] = 9517863242 [4n or sqrt(16n2)]
[ 8 ] 246913578 [n] * 3950617248 [16n] = 9876543122 [4n or sqrt(16n2)]

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 ?

 Solution by Alexandru Dan Petrescu [ March 12, 2021 ] The pandigital equivalent of the two questions proposed by Daniel Hardisky. P1) What is the largest prime found after dividing one of the pandigital numbers by 9 ? A1) 9876541023 = 9 x 1097393447. (1097393447 being prime) P2) Which of pandigital number has the greatest number of divisors ? A2) 7691523840 has 1728 divisors. Factorization: 7691523840 = 2^8 x 3^2 x 5 x 7 x 11 x 13 x 23 x 29

In Pari/gp the number of divisors can be reproduced with the command
n=7691523840;
length(divisors(n));

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

493827160595061728394

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

4938271605 (9) 5061728394 = L (9) R

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.

 4938271605 (9) 5061728394

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

Divide the pandigital L = 4938271605 with our 9 and see what happens...
Indeed a new palindrome pops up nl. 548696845

And if we add the two pandigitals together we have
L + R = 99999999999, an eleven digit repdigit.

Have you noticed that L * 2 equals the largest pandigital number !
L * 2 =  9876543210

Now, if we divide R by 2 surprisingly we see another pandigital number popping up !
R / 2 = 5061728394 / 2 =  2530864197

Remember we started with 22222222222 and 22222222227 with a difference of 5.
Can we bring that number 5 into the game? Sure, we can. Here is how: divide L by 5
L / 5 = 4938271605 / 5 =  987654321
and we end up with the largest ninedigital number!

But wait, have you tried R – L ?
We get the very first ninedigital  123456789  !

So many topics with a wow factor come together in this story.
Yet, I feel not every chapter is written... can you add more ?

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

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

There are a lot of them but I will concentrate on those with the highest number of ninedigital substrings.
In the case of the power 2 this maximum is with 3 substrings.
162978354 2 =
26561943872549316
26561943872549316
26561943872549316

267453981 2 =
71531631952748361
71531631952748361
71531631952748361

294137658 2 =
86516961853724964
86516961853724964
86516961853724964

418739652 2 =
175342896157081104
175342896157081104
175342896157081104

981425736 2 =
963196475283141696
963196475283141696
963196475283141696v

Let us continue with minimal four ninedigital substrings. I found one with power 3.
It is a nice four in a row solution.
896134527 3 =
719647185932647507781421183
719647185932647507781421183
719647185932647507781421183
719647185932647507781421183

Now, looking for at least 5 ninedigital substrings we have to go to power 7 already.
Powers 4, 5 & 6 yield no records.
351724698 7 =
665916932344685257919368457329168452133457461919769912185472
665916932344685257919368457329168452133457461919769912185472
665916932344685257919368457329168452133457461919769912185472
665916932344685257919368457329168452133457461919769912185472
665916932344685257919368457329168452133457461919769912185472

614925783 7 =
33247494252534659817234650550975941577997092127916131038242727
33247494252534659817234650550975941577997092127916131038242727
33247494252534659817234650550975941577997092127916131038242727
33247494252534659817234650550975941577997092127916131038242727
33247494252534659817234650550975941577997092127916131038242727

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')

Let us find all smallest solutions from 2 to 9 ninedigital substrings.

{ 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

For reference goals and easy searching I list here all the nine- & pandigitals implicitly displayed in these topics.

Topic 6.10 → 761239458, 723981564, 761239485, 723981546

Topic 6.9 → 235869174, 237946158
345728169, 186924357, 583749126
697834251, 584936271

Contributions