[ November 12, 2005 ]
Wonders of Numerals
written by B.S. Rangaswamy (email)
ISBN: 81-7478-492-6
The first edition of this book appeared in February, 2004
by the publisher Sura Books (Pvt) Ltd., Chennai, India
[ article by yours truly PDG ]
A personal investigation in the products of two 5-digit factors
(or one 4-digit and one 5-digit) which taken together
form a pandigital (or a ninedigital) number.
Last month I received a copy by the author himself. He noticed
earlier that I was interested in recreational mathematics as well
regarding topics as palindromes, powers, primes, etc. In total
57 episodes are presented. A few of them I reprogrammed with
Ubasic and I will highlight some extra interesting offshoots.
**
The first topic 
Scintillating equations - Table (pp. 120)
The challenge here is to find all pandigital numbers
(digits 0 through 9 occurring only once) that can be expressed
as a product of two 5-digit factors so that these two factors
taken together also form a pandigital number.
These are also known as pandigital Vampire Numbers and all
vampire numbers are also Friedman Numbers - thx Bruno Curfs.
The computer rendered 1289 solutions... a 'prime' total !
The smallest factor from the output list happens with
1023598746 = 10482 * 97653
The largest factor from the output list happens with
6231574908 = 63102 * 98754
Note that the largest factor doesn't coincide
with the largest pandigital.
You will surely like this nice pandigital with a _unique_ solution
| 1420869375 = 20481 * 69375 |
The largest 5-digit factor is also the right substring of the pandigital itself !
Two factors occur each four times nl. 69135 and 98640 !
1968273450 = 28470 * 69135
2784619530 = 40278 * 69135
2819463570 = 40782 * 69135
2963817450 = 42870 * 69135 |
1356497280 = 13752 * 98640
1729356480 = 17532 * 98640
2479513680 = 25137 * 98640
5142793680 = 52137 * 98640 |
**
Nine pandigitals can be factorized into twin sets of 5-digit
factors of which three are included in episode 33 (pp. 20).
The remaining six twin sets are
1239756840 = 18495 * 67032
1239756840 = 23940 * 51786
1479635280 = 15864 * 93270
1479635280 = 18654 * 79320
1954283760 = 37620 * 51948
1954283760 = 37962 * 51480
2483619750 = 26475 * 93810
2483619750 = 39750 * 62481
4031985672 = 49137 * 82056
4031985672 = 58071 * 69432
4293156780 = 49260 * 87153
4293156780 = 54186 * 79230
**
I did a likewise investigation but with ninedigitals this time
expressed as a product of a 4-digit and a 5-digit factor.
The computer rendered 346 solutions !
The smallest such ninedigital is
123698574 = 2598 * 47613
The largest one can be found at episode 26 (p.15 & 101).
The smallest 4-digit factor happens with
128943576 = 1368 * 94257
The largest 4-digit factor happens with
539184276 = 9873 * 54612
The smallest 5-digit factor happens with
135649728 = 9864 * 13752
The largest 5-digit factor happens with
317592864 = 3216 * 98754
We notice that 5-digit number 98754 is the largest factor
for the ninedigitals as well as the pandigitals. Nice coincidence !
Here also two factors occur each four times
as with the pandigitals nl. 6351 and 9864 !
158692437 = 6351 * 24987
174582639 = 6351 * 27489
312964578 = 6351 * 49278
569837124 = 6351 * 89724 |
135649728 = 9864 * 13752
172935648 = 9864 * 17532
247951368 = 9864 * 25137
514279368 = 9864 * 52137 |
Now the reader is invited to multiply together
these two special 4-digit factors...
6351 * 9864 = 6264_6264
An eyecatching tautonymic number ! This topic is presented
in detail at wonplate 152. Follow the link to learn more.
Parallel to episode 33 I compiled four ninedigital numbers
that can be factorized into twin sets of a 4- & 5-digit factor.
147963528 = 7932 * 18654
147963528 = 9327 * 15864
195428376 = 3762 * 51948
195428376 = 5148 * 37962
248361975 = 3975 * 62481
248361975 = 9381 * 26475
429315678 = 4926 * 87153
429315678 = 7923 * 54186
Note that the first one is very beautiful since the 4-digit
and 5-digit factors are 'digital anagrams' amongst themselves.
7932 * 18654
x x
9327 * 15864
**
Episode 32 : Heptads (p. 19) is a spin-off of the above searching for
10-digit numbers that have at least 7 identical digits but still
must be expressible as a 'pandigital' product of two 5-digit factors.
The following lists the smallest and largest solutions per digit.
Overall we count 113 solutions... again a 'prime' total !
Note that no solutions exist with at least seven 9's !!
digit 0 ( # 16 )
1010600000 = 10432 * 96875
6012000000 = 64128 * 93750
digit 1 ( # 11 )
1111011128 = 12403 * 89576
6111141113 = 64273 * 95081
digit 2 ( # 24 )
2262126222 = 23046 * 98157
5222222258 = 56738 * 92041
digit 3 ( # 17 )
3334733336 = 34718 * 96052
3931533333 = 54093 * 72681
digit 4 ( # 21 )
1148444444 = 13657 * 84092
5444424440 = 69524 * 78310
digit 5 ( # 10 )
1555585551 = 26403 * 58917
7555551552 = 83472 * 90516
digit 6 ( # 7 )
4666636664 = 54017 * 86392
6766616636 = 79201 * 85436
digit 7 ( # 2 )
complete set !
2777379777 = 32091 * 86547
7797777872 = 81736 * 95402
digit 8 ( # 5 )
1888388388 = 20157 * 93684
7688288888 = 80612 * 95374
digit 9 ( # 0 )
nihil
A _twofold_ solution pops up if we keep the seven identical
digits in one uninterrupted cluster.
Note that it is a very beautiful construction since the two
5-digit factors are 'digital anagrams' amongst themselves.
5222222258 = 56738 * 92041
7055555558 = 76358 * 92401 |
56738 * 92041
x x
76358 * 92401
Two 10-digits contain more than seven identical digits
i.e. eight identicals occurring with digit 6 in both cases
(no higher sequences exists). Quite a Beastly affair !
Concatenate the remaining digits and you'll agree with me
that 2 0 0 5 was the best year to discover this in.
My destiny with the World!Of Numbers is on schedule !
6626666660 = 72308 * 91645
6666606665 = 79021 * 84365 |
Restricting the remaining digits to be identical leads us
finally to the next three nice solutions.
2222323232 = 29104 * 76358
5553355355 = 67391 * 82405
6000660000 = 63750 * 94128 |
The third solution again reveals to us the presence of
666 or the Number of the Beast !
The extended versions of the Beast are available as well !
For instance as differences between the two 5-digit factors.
4690873152 = 65238 * 71904
and
71904 – 65238 = 6666 |
1802967435 = 20649 * 87315
and
87315 – 20649 = 66666 |
To conclude this expansion on the subject two equations
yielding palindromes, the first one is of a repdigit kind,
our 5-digit Beast turned topsyturvy !
2495671308 = 47931 * 52068
and
47931 + 52068 = 99999 |
2846031795 = 30645 * 92871
and
92871 – 30645 = 62226 |
The palindrome 62226 will come back in another format,
so keep it in mind...
**
Inspired by B.S. Rangaswamy's book I set out to look for
ninedigital numbers that are the product of two 5-digit factors
and that taken together form pandigitals.
141 solutions came up which is a palindromic total !
The smallest one is
315867942 = 15486 * 20397
The largest one is
987561234 = 28179 * 35046
The uniqueness of the following result is that
the addition of the two factors is palindromic.
And we came across that one before, didn't we...
857264193 = 20589 * 41637
and
20589 + 41637 = 62226 |
The Beast took refuge in one of the 141 solutions !
The result is composed of the Number of the Beast and
the digitsum of 666 i.e., 6 + 6 + 6 = 18 !
846173952 = 17082 * 49536
and
17082 + 49536 = 66618 |
**
Episode 55 : Six Soldiers (p. 77)
This chapter prompts you explore 10 digit numbers having six
identical numerals positioned in a continuous line.
While recomputing all the possible solutions (total of 44),
including those with two 5-digit factors not evenly
divisible by 3, I stumbled over the following curios ¬
Particularly beautiful is this item because it
uses only two distinct digits namely '3' & '0'.
3333330000 = 34125 * 97680
Some not continuous solutions with only two distinct digits
are the following three items (from a total of 1581 - 44 or 1537).
1111551155 = 23705 * 46891
4442244242 = 45317 * 98026
5566656665 = 62473 * 89105
Also special is this item because the digits of the
tendigit number are in 'ascending order' and consecutive !
6777777888 = 72561 * 93408
**
Awesome x anagrammatical x equations
emerged while comparing various factors from
nine- and pandigital output lists
The first construct is with multiplicand 27489 and his
anagram mate 49278. The same couple of multipliers
can be applied to arrive at nine- and pandigital numbers.
Note that 27489 + 49278 = 76767 and palindromic !
| 27489 * | ↗
↘ | 5361(0) = 147368529(0)
x x
6351(0) = 174582639(0) |
| x |
| 49278 * | ↗
↘ | 5361(0) = 264179358(0)
x x
6351(0) = 312964578(0) |
A second construct produces a ninedigital and a pandigital with
this couple of multiplicand and multiplier anagrams.
ps. the second 49278 was also used in the above setup !
The third multiplication with the palindromic outcome
and the fourth equation with a heptadic result
finish this illustration of our four interrelated concepts
in a wonderful and astonishing manner [ Dec 4, 2005 ].
Finally three _still interesting_ leftovers from my search.
24561, 45618 & 61329 are the resp. multiplicands.
Note that in the third case the largest 4-digit multiplier
is the reversal of the smallest 4-digit multiplier.
| 24561 * | ↗
↘ | 8739(0) = 214638579(0)
x x
8793(0) = 215964873(0) |
| 45618 * | ↗
↘ | 3792(0) = 172983456(0)
x x
9372(0) = 427531896(0) |
| 61329 * | ↗
↘ | 7458(0) = 457391682(0)
x x
8547(0) = 524178963(0) |
**
Palindromes as products of two 5-digit factors.
There are only ten palindromes consisting of nine digits
and three palindromes consisting of ten digits that are
the products of two 5-digit factors which taken together
form a pandigital. Prime factors are highlighted.
all odd digits! 393555393
385454583
690555096
431292134
707595707
629979926
919222919
966737669
all even digits! 804464408
883000388
2936556392
4461771644
4878998784 |
= 10857 * 36249
= 13569 * 28407
= 15708 * 43962
= 16978 * 25403
= 17563 * 40289
= 19658 * 32047
= 25471 * 36089
= 25801 * 37469
= 25897 * 31064
= 28517 * 30964
= 32564 * 90178
= 51029 * 87436
= 53724 * 90816 |
One anagrammatical combination shows up here.
10857 * 36249 = 393555393
x x
15708 * 43962 = 690555096
|
Palindromes as products of a 4-digit and a 5-digit factor.
There are three palindromes consisting of eight digits and
thirtythree palindromes consisting of nine digits that are
the products of a 4-digit and a 5-digit factors which taken
together form a ninedigital.
Five double anagrammatical combinations also show up here.
1453 * 29678 = 43122134
x x
4351 * 68792 = 299313992
|
9256 * 87143 = 806595608
x x
9526 * 87413 = 832696238
|
4197 * 28563 = 119878911
x x
9417 * 62835 = 591717195
|
4659 * 82137 = 382676283
x x
6945 * 78321 = 543939345
|
5824 * 79136 = 460888064
x x
8425 * 61793 = 520606025
|
One triple anagrammatical combination exists as well !
2964 * 71358 = 211505112
x x
4629 * 81537 = 377434773
x x
4926 * 53187 = 261999162
|
**
Mixed anagram equations starting from a common factor.
The first and the last two constructions are remarkable in the
sense that all their factors are also anagrams among each other !
Many examples I found are displayed as well but without
the pretention of having made an 'exhaustive' list.
**
Palindromic lookalikes.
Behold the next two palindromes and get enchanted.
The first equation is an 8-digit palindrome expressed
as a product of a 4- and a 5-digit factor which taken
together form a ninedigital.
The second equation is a 9-digital palindromic anagram
from the previous one except for the extra zero digit 0
expressed as a product of two 5-digit factors which
taken together form a pandigital.
A zero is also what is needed to make the crossing
from ninedigital to pandigital numbers !
8803_3088 = 4576 * 19238
8830_0_0388 = 28517 * 30964
Compare the next two palindromes and get enchanted.
The first equation is an 8-digit palindrome expressed
as a product of a 4- and a 5-digit factor which taken
together form a ninedigital.
The second equation is a 9-digit palindrome identical
to the previous one except for the extra middle digit 9
expressed as a product of two 5-digit factors which
taken together form a pandigital.
4312_2134 = 1453 * 29678
4312_9_2134 = 16978 * 25403
There exist imho no better constructions that can
synthesize this wonplate in such a beautiful way !
**
Scintillating equations with two 5-digit factors and their reversals.
Below is shown an exceptionally nice pair of equations.
The two 5-digit factors are each other's reversals.
A truly _unique_ pandigital phenomenon !
4905361782
=
52137 * 94086
↔↕ ↕↔
73125 * 68049
=
4976083125 |
For reference goals and easy searching all the nine- & pandigitals
implicitly displayed in these topics are listed here.
Topic → 1048297653, 6231574908, 6310298754, 1420869375, 2048169375, 2847069135, 4027869135,
4078269135, 4287069135, 1375298640, 1753298640, 2513798640, 5213798640, 1849567032, 2394051786,
1586493270, 1865479320, 3762051948, 3796251480, 2647593810, 3975062481, 4913782056, 5807169432,
4926087153, 5418679230, 259847613, 136894257, 987354612, 986413752, 321698754, 635124987,
635127489, 635149278, 635189724, 986413752, 986417532, 986425137, 986452137, 793218654,
932715864, 376251948, 514837962, 397562481, 938126475, 492687153, 792354186, 1043296875,
6412893750, 1240389576, 6427395081, 2304698157, 5673892041, 3471896052, 5409372681, 1365784092,
6952478310, 2640358917, 8347290516, 5401786392, 7920185436, 3209186547, 8173695402, 2015793684,
8061295374, 5673892041, 7635892401, 7230891645, 7902184365, 2910476358, 6739182405, 6375094128,
6523871904, 7190465238, 2064987315, 8731520649, 4793152068, 4793152068, 3064592871, 9287130645,
1548620397, 2817935046, 2058941637, 2058941637, 1708249536, 1708249536, 3412597680, 2370546891,
4531798026, 6247389105, 7256193408, 274895361, 147368529, 274896351, 174582639, 492785361,
264179358, 492786351, 312964578, 2748953610, 1473685290, 2748963510, 1745826390, 4927853610,
2641793580, 4927863510, 3129645780, 2784913506, 4927865031, 947823156, 9478203156, 7294836015,
245618739, 214638579, 2456187930, 2146385790, 245618793, 215964873, 2456187930, 2159648730,
456183792, 172983456, 456189372, 427531896, 4561837920, 1729834560, 4561893720, 4275318960,
613297458, 457391682, 613298547, 524178963, 6132974580, 4573916820, 6132985470, 5241789630,
1085736249, 1356928407, 1570843962, 1697825403, 1756340289, 1965832047, 2547136089, 2580137469,
2589731064, 2851730964, 3256490178, 5102987436, 5372490816, 1085736249, 1570843962, 145329678,
457619238, 536812947, 296471358, 315694782, 341872569, 387265941, 419728563, 435168792,
437861529, 461295837, 462981537, 465982137, 492653187, 498162573, 547362891, 579243681,
582479136, 631945278, 651834729, 694578321, 698237541, 716359248, 752831469, 781426593,
786429513, 794231856, 823175946, 823964175, 829731546, 842561793, 847561923, 925687143,
934162785, 941762835, 952687413, 145329678, 435168792, 925687143, 952687413, 419728563,
941762835, 465982137, 694578321, 582479136, 842561793, 296471358, 462981537, 492653187,
492653187, 492685713, 492687153, 738514629, 462981537, 462985173, 738514629, 7385146290,
7385190264, 315694278, 315694782, 2840713569, 2840763591, 628354197, 6283541970, 628359417,
6283509417, 9017463825, 9017486253, 694538721, 694578321, 8271349605, 8271354096, 783216945,
7832106945, 7832140965, 457619238, 2851730964, 145329678, 1697825403, 5213794086, 7312568049
A000167 
ninedigital