[ November 12, 2005 ]
Wonders of Numerals
written by B.S. Rangaswamy (email)

[ 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

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

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.

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 !!

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...

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

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 !

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.

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.

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.

4926 *

↗ → ↘

53187 = 261999162 palindromic    x 85713 = 4222222380 six_soldier    x 87153 = 429315678 ninedigital

---

4629 *

↗ → ↘

73851 = 341856279 ninedigital I    x 81537 = 377434773 palindromic    x 85173 = 394265817 ninedigital II

---

73851 *

↗ ↘

4629(0) = 341856279(0) nine- or pandigital    x 90264 = 6666086664 heptadic

---

3156 *

↗ ↘

94278 = 297541368 ninedigital    x 94782 = 299131992 palindromic

---

28407 *

↗ ↘

13569 = 385454583 palindromic    x 63591 = 1806429537 pandigital

---

62835 *

↗ ↘

4197(0) = 263718495(0) nine- or pandigital    x [0]9417 = 591717195 palindromic

---

90174 *

↗ ↘

63825 = 5755355550 heptadic    x 86253 = 7777778022 six_soldier

---

6945 *

↗ ↘

38721 = 268917345 ninedigital    x 78321 = 543939345 palindromic

---

82713 *

↗ ↘

49605 = 4102978365 pandigital    x 54096 = 4474442448 heptadic

---

78321 *

↗ ↘

[0]6945 = 543939345 palindromic    x 40965 = 3208419765 pandigital

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 !

Below is shown an exceptionally nice pair of equations.
The two 5-digit factors are each other's reversals.
A truly _unique_ pandigital phenomenon !