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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

Index abcdefghi ab * cde * fghi ABCDEFGHI 1 169278453 16 * 927 * 8453 125374896 2 187429356 18 * 742 * 9356 124958736 3 187629453 18 * 762 * 9453 129657348 4 198647523 19 * 864 * 7523 123497568 5 216579348 21 * 657 * 9348 128974356 6 218649753 21 * 864 * 9753 176958432 7 219546837 21 * 954 * 6837 136972458 8 235719486 23 * 571 * 9486 124579638 9 236489157 23 * 648 * 9157 136475928 10 237568419 23 * 756 * 8419 146389572 11 239817465 23 * 981 * 7465 168432795 12 245379618 24 * 537 * 9618 123956784 13 247389516 24 * 738 * 9516 168547392 14 247568193 24 * 756 * 8193 148653792 15 249185673 24 * 918 * 5673 124987536 16 265319847 26 * 531 * 9847 135947682 17 267839541 26 * 783 * 9541 194235678 18 275168943 27 * 516 * 8943 124593876 19 275389146 27 * 538 * 9146 132854796 20 275619438 27 * 561 * 9438 142957386 21 276359841 27 * 635 * 9841 168723945 22 278195643 27 * 819 * 5643 124783659 23 289365147 28 * 936 * 5147 134892576 24 289367541 28 * 936 * 7541 197634528 25 298715364 29 * 871 * 5364 135489276 26 317258649 31 * 725 * 8649 194386275 27 319274856 31 * 927 * 4856 139546872 28 324598671 32 * 459 * 8671 127359648 29 346218759 34 * 621 * 8759 184937526 30 365178294 36 * 517 * 8294 154367928 31 368714592 36 * 871 * 4592 143986752 32 368924571 36 * 892 * 4571 146783952 33 369127548 36 * 912 * 7548 247815936 34 375218694 37 * 521 * 8694 167594238 35 375948612 37 * 594 * 8612 189274536 36 376219485 37 * 621 * 9485 217936845 37 386195427 38 * 619 * 5427 127653894 38 389726541 38 * 972 * 6541 241598376 39 395768124 39 * 576 * 8124 182497536 40 397685241 39 * 768 * 5241 156978432 41 398674125 39 * 867 * 4125 139478625 42 417839256 41 * 783 * 9256 297145368 43 419278356 41 * 927 * 8356 317586492 44 425319768 42 * 531 * 9768 217845936 45 426879351 42 * 687 * 9351 269813754 46 428193657 42 * 819 * 3657 125793486 47 428195763 42 * 819 * 5763 198235674 48 428765391 42 * 876 * 5391 198345672 49 428916357 42 * 891 * 6357 237891654 50 439817526 43 * 981 * 7526 317469258 51 456938721 45 * 693 * 8721 271964385 52 458716923 45 * 871 * 6923 271346985 53 463598271 46 * 359 * 8271 136587294 54 465138279 46 * 513 * 8279 195367842 55 465139827 46 * 513 * 9827 231897546 56 468195732 46 * 819 * 5732 215947368 57 469173258 46 * 917 * 3258 137428956 58 469813257 46 * 981 * 3257 146975382 59 485196372 48 * 519 * 6372 158739264 60 486219357 48 * 621 * 9357 278913456 61 486517932 48 * 651 * 7932 247859136 62 493658271 49 * 365 * 8271 147926835 63 513697824 51 * 369 * 7824 147239856 64 514296837 51 * 429 * 6837 149586723 65 516938427 51 * 693 * 8427 297835461 66 517628439 51 * 762 * 8439 327956418 67 518349267 51 * 834 * 9267 394162578 68 519728463 51 * 972 * 8463 419527836 69 523697184 52 * 369 * 7184 137846592 70 524167839 52 * 416 * 7839 169573248 71 526419378 52 * 641 * 9378 312587496 72 528196437 52 * 819 * 6437 274138956 73 537219846 53 * 721 * 9846 376245198 74 538912746 53 * 891 * 2746 129674358 75 542389617 54 * 238 * 9617 123597684 76 542798613 54 * 279 * 8613 129763458 77 543268719 54 * 326 * 8719 153489276 78 543679821 54 * 367 * 9821 194632578 79 546137298 54 * 613 * 7298 241578396 80 546179382 54 * 617 * 9382 312589476 81 546739821 54 * 673 * 9821 356914782 82 547613829 54 * 761 * 3829 157348926 83 547836129 54 * 783 * 6129 259146378 84 548267139 54 * 826 * 7139 318427956 85 548679231 54 * 867 * 9231 432176958 86 548713692 54 * 871 * 3692 173649528 87 548732961 54 * 873 * 2961 139587462 88 549172683 54 * 917 * 2683 132856794 89 549217836 54 * 921 * 7836 389715624 90 549263187 54 * 926 * 3187 159362748 91 549318672 54 * 931 * 8672 435976128 92 567189432 56 * 718 * 9432 379241856 93 568194237 56 * 819 * 4237 194325768 94 573486921 57 * 348 * 6921 137284956 95 576423819 57 * 642 * 3819 139752486 96 583427196 58 * 342 * 7196 142739856 97 587293641 58 * 729 * 3641 153948762 98 592748631 59 * 274 * 8631 139528746 99 593428176 59 * 342 * 8176 164975328 100 612438759 61 * 243 * 8759 129834657 101 613895724 61 * 389 * 5724 135824796 102 615784932 61 * 578 * 4932 173892456 103 617498235 61 * 749 * 8235 376248915 104 625318479 62 * 531 * 8479 279145638 105 634589217 63 * 458 * 9217 265947318 106 634789521 63 * 478 * 9521 286715394 107 634875129 63 * 487 * 5129 157362849 108 634958721 63 * 495 * 8721 271964385 109 635284791 63 * 528 * 4791 159367824 110 635924817 63 * 592 * 4817 179654832 111 639178245 63 * 917 * 8245 476321895 112 639182754 63 * 918 * 2754 159274836 113 639187254 63 * 918 * 7254 419527836 114 639274581 63 * 927 * 4581 267534981 115 639415278 63 * 941 * 5278 312895674 116 639517824 63 * 951 * 7824 468759312 117 647283519 64 * 728 * 3519 163957248 118 647289351 64 * 728 * 9351 435681792 119 687513492 68 * 751 * 3492 178329456 120 693217854 69 * 321 * 7854 173958246 121 695738214 69 * 573 * 8214 324756918 122 698134275 69 * 813 * 4275 239814675 123 718925463 71 * 892 * 5463 345982716 124 723546918 72 * 354 * 6918 176325984 125 724135698 72 * 413 * 5698 169435728 126 726315498 72 * 631 * 5498 249785136 127 726485139 72 * 648 * 5139 239765184 128 728345961 72 * 834 * 5961 357946128 129 728916453 72 * 891 * 6453 413972856 130 728945613 72 * 894 * 5613 361297584 131 729584631 72 * 958 * 4631 319427856 132 729586143 72 * 958 * 6143 423719568 133 729635148 72 * 963 * 5148 356941728 134 729685134 72 * 968 * 5134 357819264 135 749162358 74 * 916 * 2358 159834672 136 749528136 74 * 952 * 8136 573164928 137 759634821 75 * 963 * 4821 348196725 138 763429518 76 * 342 * 9518 247391856 139 768193524 76 * 819 * 3524 219347856 140 769234158 76 * 923 * 4158 291675384 141 784596132 78 * 459 * 6132 219537864 142 796238451 79 * 623 * 8451 415932867 143 796413528 79 * 641 * 3528 178654392 144 796842513 79 * 684 * 2513 135792468 145 798462513 79 * 846 * 2513 167953842 146 812937456 81 * 293 * 7456 176953248 147 813425967 81 * 342 * 5967 165297834 148 813479562 81 * 347 * 9562 268759134 149 813547629 81 * 354 * 7629 218753946 150 813945726 81 * 394 * 5726 182739564 151 813947256 81 * 394 * 7256 231567984 152 815267493 81 * 526 * 7493 319246758 153 815276439 81 * 527 * 6439 274861593 154 815276943 81 * 527 * 6943 296375841 155 816239475 81 * 623 * 9475 478136925 156 816542793 81 * 654 * 2793 147956382 157 816792543 81 * 679 * 2543 139862457 158 817243659 81 * 724 * 3659 214578396 159 817493256 81 * 749 * 3256 197538264 160 817539246 81 * 753 * 9246 563941278 161 817926453 81 * 792 * 6453 413972856 162 819732456 81 * 973 * 2456 193564728 163 827196543 82 * 719 * 6543 385762194 164 842169753 84 * 216 * 9753 176958432 165 842769351 84 * 276 * 9351 216793584 166 843576129 84 * 357 * 6129 183796452 167 843576219 84 * 357 * 6219 186495372 168 845132967 84 * 513 * 2967 127853964 169 845917362 84 * 591 * 7362 365479128 170 846127539 84 * 612 * 7539 387564912 171 864237591 86 * 423 * 7591 276145398 172 865493721 86 * 549 * 3721 175683294 173 867519234 86 * 751 * 9234 596387124 174 872915634 87 * 291 * 5634 142635978 175 872945316 87 * 294 * 5316 135972648 176 873214569 87 * 321 * 4569 127598463 177 873215496 87 * 321 * 5496 153486792 178 873425169 87 * 342 * 5169 153798426 179 874326159 87 * 432 * 6159 231479856 180 874539261 87 * 453 * 9261 364985271 181 876549312 87 * 654 * 9312 529834176 182 879241536 87 * 924 * 1536 123475968 183 893165274 89 * 316 * 5274 148325976 184 893617524 89 * 361 * 7524 241738596 185 912785364 91 * 278 * 5364 135698472 186 915672438 91 * 567 * 2438 125793486 187 915673842 91 * 567 * 3842 198235674 188 918465372 91 * 846 * 5372 413568792 189 923785164 92 * 378 * 5164 179583264 190 927384651 92 * 738 * 4651 315784296 191 927541836 92 * 754 * 1836 127359648 192 928714653 92 * 871 * 4653 372854196 193 937682154 93 * 768 * 2154 153847296 194 938526741 93 * 852 * 6741 534129876 195 938672415 93 * 867 * 2415 194723865 196 945138267 94 * 513 * 8267 398651274 197 946172358 94 * 617 * 2358 136759284 198 961837542 96 * 183 * 7542 132497856 199 963485217 96 * 348 * 5217 174289536 200 967532481 96 * 753 * 2481 179346528 201 974683251 97 * 468 * 3251 147582396 202 976843251 97 * 684 * 3251 215697348 203 978641523 97 * 864 * 1523 127639584 204 982716354 98 * 271 * 6354 168749532 205 986314752 98 * 631 * 4752 293854176

[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

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

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 !

Index  m   n   q   p   k   n2   a   b   c   Area (a*b)/2  1             13779 18372 22965 126573894 2             47804 5328 48100 127349856 3             27144 10208 29000 138542976 4             20502 13936 24790 142857936 5             25102 11736 27710 147298536 6             71808 4356 71940 156397824 7             86112 3666 86190 157843296 8             9639 33048 34425 159274836 9             15624 20832 26040 162739584 10             4641 73188 73335 169832754 11             23779 15372 28315 182765394 12             12441 29412 31935 182957346 13             16992 22656 28320 192485376 14             146511 2652 146535 194273586 15             33696 12672 36000 213497856 16             25092 17056 30340 213984576 17             25623 17136 30825 219537864 18             44982 11424 46410 256937184 19             69069 8208 69555 283459176 20             71383 7956 71825 283961574 21             45936 12852 47700 295184736 22             13629 46728 48675 318427956 23             92781 7392 93075 342918576 24             33948 20736 39780 351972864 25             31518 24024 39630 378594216 26             29949 26568 40035 397842516 27             28122 30096 41190 423179856 28             160056 6042 160170 483529176 29             107226 9768 107670 523691784 30             12831 82908 83895 531896274 31             8883 119756 120085 531896274 32             24157 47124 52955 569187234 33             50358 23544 55590 592814376 34             30924 41232 51540 637529184 35             73143 18576 75465 679352184 36             9999 137532 137895 687591234 37             91168 15876 92540 723691584 38             20559 70488 73425 724581396 39             76041 19312 78455 734251896 40             76293 19376 78715 739126584 41             5529 268128 268185 741239856 42             63531 23892 67875 758941326 43             55883 28044 62525 783591426 44             13617 115344 116145 785319624 45             78939 20048 81445 791284536 46             184404 8928 184620 823179456 47             183596 9072 183820 832791456 48             67818 24624 72150 834975216 49             44958 37944 58830 852943176 50             643104 2778 643110 893271456 51             86436 21952 89180 948721536 52             17874 10883 2110290 972631584

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

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)

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

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

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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( © All rights reserved ) - Last modified : October 4, 2023.

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

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