Two pandigitals in the list can be multiplied
with two other pandigitals to arrive at a palindrome.
| 1728603594 * | k m | 1367049528 = 2363086727276803632 palindrome (444) x 4790813562 = 8281417541457141828 palindrome (828) |
| 2170654398 * | k m | 1280795364 = 2780164089804610872 palindrome (322) x 3908145276 = 8483232731372323848 palindrome (1117) |
Three palindromes can be expressed as products of
two pandigitals in two different but math-related ways.
| 2174017991997104712 = | k m | 1039267458 * 2091875364 N1 * N2 (42) x x 1045937682 * 2078534916 N2/2 * N1*2 (51) |
| 6495526015106255946 = | k m | 1308652947 * 4963520718 N1 * N2 (378) x x 2481760359 * 2617305894 N2/2 * N1*2 (1187) |
| 8272731805081372728 = | k m | 2015793846 * 4103957268 N1 * N2 (1007) x x 2051978634 * 4031587692 N2/2 * N1*2 (1041) |
Two palindromes are connected through a reversal
transformation of one of its pandigital factors
(Resp. index numbers 87 and 1226). This is a unique case !
1342649619169462431 =
1058963427 *
1267890453
K [ REVERSE DIGITS ! ] K
3540987621 * 2603581749 =
9219250743470529129 |
There are six cases whereby the sum of the pandigital factors is pandigital as well !
| 3220829395939280223 = | k m | 1068273549 * 3014985627 (118)
1068273549 + 3014985627 = 4083259176 |
| 2115763961693675112 = | k m | 1094658372 * 1932807546 (179)
1094658372 + 1932807546 = 3027465918 |
| 2893714315134173982 = | k m | 1672034859 * 1730654298 (789)
1672034859 + 1730654298 = 3402689157 |
| 7247133287823317427 = | k m | 1892463507 * 3829470561 (940)
1892463507 + 3829470561 = 5721934068 |
| 9758213125213128579 = | k m | 2064378159 * 4726950381 (1048)
2064378159 + 4726950381 = 6791328540 |
| 5595037421247305955 = | k m | 2150698473 * 2601497835 (1103)
2150698473 + 2601497835 = 4752196308 |
There are two cases whereby the difference of the pandigital factors is pandigital as well !
| 4031959181819591304 = | k m | 3805614972 * 1059476382 (91)
3805614972 1059476382 = 2746138590 |
| 5438225545455228345 = | k m | 2907438165 * 1870452693 (934)
2907438165 1870452693 = 1036985472 |
This is a really beautiful gem !
Only the third and fourth digit of the pandigital factors need be swapped to arrive at the palindrome.
| 4270936083806390724 = | k m | 2057643918 (1044)  2075643918 |
The palindrome is a concatenation of five 3-digit numbers separated by zeroes :
| 4270936083806390724 |  | A | B | C | D | E |
A + B + D = 2002
D - A = 212 = B - E
E - A = 297 = B - D
A lot of small palindromes pop up when the pandigitals are grouped into five sets of two digits.
| 20 | 57 | 64 | 39 | 18 |  | A | BN | C | D | E |
| 20 | 75 | 64 | 39 | 18 |  | A | BR | C | D | E |
Here they are sorted from smaller to larger :
A + BN = 77
A + D + E = 77
BN + C = 121
C + D + E = 121
A + BN + C = 141
A + C + D + E = 141
2A + (BN + BR) + 2C + 2D + 2E = 414
Some equalities can be detected :
BN = 57 = D + E
Playing around further (this may become politically incorrect...)
The largest factor of the palindrome is 115313551 which is a near-palindromic !
Break this down into three sets of 3-digit numbers
115 | 313 | 551
The middle part is palindromic prime 313 which when added
to the next palprime 353 gives 666, the numbers of the Beast !
Note that the sum of the left and right parts of the factor equals 666 as well !
115 + 551 = 666
The second pandigital gives another surprise when stripped of its last digit '8'.
207564391_[8]
Factorisation of this number gives us three primes
131 x 139 x 11399
Note the use of only three distinct digits 1, 3 and 9 as indicated
by the last three digits of the 'crippled' pandigital 207564391.
The second pandigital when stripped of its 'zero' digit is also in for a surprise !
2_[0]_75643918
Divide this ninedigital now into three sets of 3-digit numbers :
275 | 643 | 918
and you will immediately notice that
275 + 643 = 918