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 The Nine Digits Page 5with 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 5.13 [ July 15, 2015 ]
Finding a ninedigital as a substring in the decimal expansion
of that same ninedigital raised to a power p
Here I am looking for ninedigitals raised to a power p so that the same
ninedigital pops up as a substring in the decimal expansion of that number.
For that purpose I use UBASIC. One limitation here is that for the largest
ninedigital an overflow occurs when p is greater than 119. Someone who
is equipped with better tools can raise that exponent to higher values and,
no doubt, will certainly find much more solutions. Good hunting! P@rick.
See WONplate 195 for the pandigital version of this topic.
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365781249 51 =
365781249 101 =
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Proof By Alexandru Petrescu (PhD in applied mathematics) [ April 17, 2022 ] Let a = 365781249. Condition is: 109 | (a51–a) or 109 | a(a50–1). But gcd(a,109) = 1 so 109 | (a50–1). Factorization: a50–1 = (a–1)(a+1)(a4–a3+a2–a+1)(a4+a3+a2+a+1)(a20–a15+a10–a5+1)(a20+a15+a10+a5+1) a–1 = 365781248 = 28 x 7 x 17 x 12007 (1) a+1 = 365781250 = 2 x 57 x 2341 (2) For any number b having the units digit equal to 9 we have: b2k = 1 (mod 10) and b2k+1 = 9 (mod 10) So: a4–a3+a2–a+1 = 1–9+1–9+1 = 5 (mod 10) (3) a20–a15+a10–a5+1 = 1–9+1–9+1 = 5 (mod 10) (4) From (1)-(4) we have 29 x 59 = 109 | (a50–1). Generally 109 | (a50p+1–a) because a50p–1 = (a50)p–1 = (a50–1)(....) |
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635781249 101 =
13616930906420442043259405929999866138890181953431470395921699407085117760783 635781249 201 =
29164245973260567384593998725899801391036529801483811790123121455125852321698 In both cases the expansion ends with our ninedigital !! |
Proof By Alexandru Petrescu (PhD in applied mathematics) [ April 18, 2022 ] Let a = 635781249. Condition is: 109 | (a101–a) or 109 | a(a100–1). But gcd(a,109) = 1 so 109 | (a100–1). Factorization: a100–1 = (a–1)(a+1)(a2+1)(a4–a3+a2–a+1)(a4+a3+a2+a+1)(a8–a6+a4–a2+1)(a20–a15+a10–a5+1)(a20+a15+a10+a5+1)(a40–a30+a20–a10+1) a–1 = 635781248 = 27 x 4967041 (1) a+1 = 635781250 = 2 x 57 x 13 x 313 (2) For any number b having the units digit equal to 9 we have: b2k = 1 (mod 10) and b2k+1 = 9 (mod 10) So: a4–a3+a2–a+1 = 1–9+1–9+1 = 5 (mod 10) (3) a20–a15+a10–a5+1 = 1–9+1–9+1 = 5 (mod 10) (4) 2 | (a2+1) (5) From (1)-(5) we have 29 x 59 = 109 | (a100–1). Generally 109 | (a100p+1–a) because a100p–1 = (a100)p–1 = (a100–1)(....) |
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759186432 101 =
82260666872235555762250266625924353034740692041723072613225221908129972517410 759186432 201 =
89132484842206877224072333243882506481115047313501101593088591184654074856787 In both cases the expansion ends with our ninedigital !! Here also I detected a pattern as for all exponents +100 (starting with 1) it occurs. |
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With 873264591 I found the largest ninedigital with p < 119 !
Topic 5.12 [ March 1, 2015 ]
Nine- & pandigitals equal to the sum of two squares
Some statistics and curios
This topic is a continuation from wonplate193
Statistics and curios for the ninedigital variation
In total there are 65795 ninedigitals that can be expressed as sums of two squares in 1 or more ways.
This is around 18% of the 9! possible ninedigitals. Here is the distribution list :
(*_1)  28763(*_2) 25968(*_3) 1464(*_4) 7513(*_5) 55(*_6) 848(*_7) 0(*_8) 901(*_9) 27(*_10) 34(*_11) 0(*_12) 161(*_13) 0(*_14) 0(*_15) 2 469738125 & 879463125(*_16) 44 |
(*_17) 0 (*_18) 4 346978125, 647193825, 783961425 & 968417325(*_19) 0(*_20) 2 763498125 & 796843125(*_21) 0(*_22) 0(*_23) 0(*_24) 8(*_25) 0(*_26) 0(*_27) 0(*_28) 0(*_29) 0(*_30) 0(*_31) 0(*_32) 1 439817625 |
In total there are exactly 100 ninedigitals expressible as a sum of two squares whereby
the concatenation of its basenumbers forms a ninedigital (77 in total) or a pandigital (23 in total).
As a coincidence there are also 23 ninedigitals whereby the basenumbers A and B are the same.
They show up on the right side of the table.
One ninedigital stands out from the rest namely 317928645 because it is the
only one that has more than one solution. Beware this is a unique case!
This couple are the first two from an eightfold (*_8) solution for this ninedigital.
Note: this curio was first observed by Peter Kogel already in 2005.
| 317928645 = | ↗ ↘ |
25382 + 176492 29432 + 175862 |
Here is the complete list of all the hundred ninedigital solutions.
What is under construction is the list regarding the pandigitals. B.S. Rangaswamy sent me already one example
to wet your appetite. See at the bottom of the table.
| 1 | 175236849 = 34952 + 127682 | 612859437 = 137492 + 205892 | 143752968 = 2 * 84782 | ||
| 2 | 179684325 = 36542 + 128972 | 618542937 = 136592 + 207842 | 145897362 = 2 * 85412 | ||
| 3 | 189237465 = 45362 + 129872 | 689537412 = 109742 + 238562 | 162973458 = 2 * 90272 | ||
| 4 | 197485632 = 53762 + 129842 | 756928314 = 149672 + 230852 | 164275938 = 2 * 90632 | ||
| 5 | 218367945 = 54692 + 137282 | 783691245 = 185732 + 209462 | 178945362 = 2 * 94592 | ||
| 6 | 231649785 = 65282 + 137492 | 786425193 = 159482 + 230672 | 183974562 = 2 * 95912 | ||
| 7 | 234169785 = 76592 + 132482 | 814593672 = 158942 + 237062 | 219367458 = 2 * 104732 | ||
| 8 | 234718965 = 87932 + 125462 | 817439625 = 137492 + 250682 | 346581792 = 2 * 131642 | ||
| 9 | 234971685 = 56792 + 142382 | 824691537 = 103592 + 267842 | 423579618 = 2 * 145532 | ||
| 10 | 237916845 = 89732 + 125462 | 841769325 = 170582 + 234692 | 453968712 = 2 * 150662 | ||
| 11 | 238176549 = 69452 + 137822 | 867154293 = 169532 + 240782 | 461593728 = 2 * 151922 | ||
| 12 | 238971465 = 76592 + 134282 | 874932516 = 174962 + 238502 | 497638152 = 2 * 157742 | ||
| 13 | 243956817 = 27842 + 153692 | 891746325 = 137852 + 264902 | 537198642 = 2 * 163892 | ||
| 14 | 249813657 = 82592 + 134762 | 912645873 = 164972 + 253082 | 571963842 = 2 * 169112 | ||
| 15 | 251649873 = 53672 + 149282 | 916782345 = 185072 + 239642 | 618534792 = 2 * 175862 | ||
| 16 | 256847193 = 49682 + 152372 | 921847653 = 160982 + 257432 | 637459218 = 2 * 178532 | ||
| 17 | 268371954 = 69752 + 148232 | 925781634 = 143972 + 268052 | 639174258 = 2 * 178772 | ||
| 18 | 283974165 = 43592 + 162782 | 927814653 = 157982 + 260432 | 654279138 = 2 * 180872 | ||
| 19 | 286753194 = 82952 + 147632 | 934687125 = 143702 + 269852 | 654713298 = 2 * 180932 | ||
| 20 | 312897645 = 89342 + 152672 | 943725186 = 187052 + 243692 | 765421938 = 2 * 195632 | ||
| 21 | 317928645 = 25382 + 176492 = 29432 + 175862 | 948571236 = 136802 + 275942 | 913524768 = 2 * 213722 | ||
| 22 | 326785149 = 49652 + 173822 | 972354861 = 174692 + 258302 | 943256178 = 2 * 217172 | ||
| 23 | 328459617 = 54362 + 172892 | 973416285 = 174062 + 258932 | 958431762 = 2 * 218912 | ||
| 24 | 341978265 = 64592 + 173282 | ||||
| 25 | 345172689 = 87452 + 163922 | ||||
| 26 | 346297185 = 79532 + 168242 | ||||
| 27 | 362794185 = 54962 + 182372 | ||||
| 28 | 365984721 = 24362 + 189752 | ||||
| 29 | 368529417 = 53762 + 184292 | ||||
| 30 | 369471825 = 59642 + 182732 | ||||
| 31 | 374921865 = 83522 + 174692 | ||||
| 32 | 378129645 = 83492 + 175622 | ||||
| 33 | 385267914 = 23672 + 194852 | ||||
| 34 | 413629578 = 64532 + 192872 | ||||
| 35 | 435869712 = 78242 + 193562 | ||||
| 36 | 436521978 = 96272 + 185432 | ||||
| 37 | 478192653 = 46982 + 213572 | ||||
| 38 | 481379265 = 48572 + 213962 | ||||
| 39 | 489731265 = 56972 + 213842 | ||||
| 40 | 497163825 = 48962 + 217532 | ||||
| 41 | 497231865 = 59762 + 214832 | ||||
| 42 | 514926873 = 57632 + 219482 | ||||
| 43 | 523687194 = 63452 + 219872 | ||||
| 44 | 523861794 = 78632 + 214952 | ||||
| 45 | 532978641 = 87962 + 213452 | ||||
| 46 | 592876413 = 61982 + 235472 | ||||
| 47 | 598314762 = 64712 + 235892 | ||||
| 48 | 613259874 = 18752 + 246932 | ||||
| 49 | 613478925 = 74582 + 236192 | ||||
| 50 | 614829357 = 78692 + 235142 | ||||
| 51 | 619423785 = 15962 + 248372 | ||||
| 52 | 631548297 = 87962 + 235412 | ||||
| 53 | 635127849 = 91682 + 234752 | ||||
| 54 | 639218457 = 73562 + 241892 | ||||
| 55 | 648912537 = 38642 + 251792 | ||||
| 56 | 651429378 = 68132 + 245972 | ||||
| 57 | 674921853 = 41972 + 256382 | ||||
| 58 | 682143597 = 61892 + 253742 | ||||
| 59 | 694725138 = 96872 + 245132 | ||||
| 60 | 712839546 = 97352 + 248612 | ||||
| 61 | 726389145 = 89762 + 254132 | ||||
| 62 | 726439185 = 71642 + 259832 | ||||
| 63 | 729563481 = 39842 + 267152 | ||||
| 64 | 758394621 = 43952 + 271862 | ||||
| 65 | 789631245 = 97412 + 263582 | ||||
| 66 | 812596473 = 56132 + 279482 | ||||
| 67 | 817453962 = 65492 + 278312 | ||||
| 68 | 823415697 = 35162 + 284792 | ||||
| 69 | 825697314 = 39152 + 284672 | ||||
| 70 | 839147625 = 91562 + 274832 | ||||
| 71 | 846352197 = 73592 + 281462 | ||||
| 72 | 865972341 = 38462 + 291752 | ||||
| 73 | 872156394 = 16352 + 294872 | ||||
| 74 | 914367285 = 64712 + 295382 | ||||
| 75 | 931427586 = 68312 + 297452 | ||||
| 76 | 934816725 = 74312 + 296582 | ||||
| 77 | 951246738 = 85172 + 296432 |
Another ninedigital that stands out from the rest is 934167285 because it is the only
one that can be written as a sum of two squares in two different ways such that both
their basenumbers forms a ninedigital when multiplied together.
A nice unique case!
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In total there are 219 ninedigitals expressible in this way.
The smallest is 248635917 = 106142 + 116612 and 10614 * 11661 = 123769854
The largest is 987431562 = 215192 + 228992 and 21519 * 22899 = 492763581
Statistics and curios for the pandigital variation
In total there are 568801 pandigitals that can be expressed as sums of two squares in 1 or more ways.
This is around 17,416 % of the 9*9! possible pandigitals. Here is the distribution list :
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(*_1) (23112)1 + (23193)2 + (23072)3 + (22132)4 + (24904)5 + (21085)6 + (22492)7 + (21296)8 + (20600)9 ( 201886 )tot (*_2) (26329)1 + (26858)2 + (26881)3 + (25208)4 + (25865)5 + (24452)6 + (26628)7 + (24705)8 + (24742)9 ( 231668 )tot(*_3) (1840)1 + (919)2 + (1592)3 + (1372)4 + (511)5 + (1434)6 + (1526)7 + (1324)8 + (1471)9 ( 11989 )tot(*_4) (10254)1 + (10559)2 + (11129)3 + (10170)4 + (8929)5 + (9983)6 + (11204)7 + (10259)8 + (10737)9 ( 93224 )tot(*_5) (60)1 + (0)2 + (110)3 + (116)4 + (5)5 + (57)6 + (105)7 + (96)8 + (107)9 ( 656 )tot(*_6) (1408)1 + (740)2 + (1164)3 + (1127)4 + (377)5 + (1185)6 + (1257)7 + (1081)8 + (1156)9 ( 9495 )tot(*_7) (5)1 + (0)2 + (7)3 + (7)4 + (0)5 + (1)6 + (5)7 + (5)8 + (4)9 ( 34 )tot(*_8) (1601)1 + (1606)2 + (1864)3 + (1748)4 + (1197)5 + (1739)6 + (2088)7 + (1915)8 + (1992)9 ( 15750 )tot(*_9) (32)1 + (19)2 + (25)3 + (17)4 + (5)5 + (23)6 + (30)7 + (29)8 + (25)9 ( 205 )tot(*_10) (30)1 + (2)2 + (42)3 + (90)4 + (0)5 + (40)6 + (82)7 + (53)8 + (72)9 ( 411 )tot(*_11) (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot(*_12) (251)1 + (133)2 + (281)3 + (266)4 + (73)5 + (270)6 + (297)7 + (290)8 + (294)9 ( 2155 )tot(*_13) (1)1 + (0)2 + (0)3 + (1)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 2 )tot(*_14) (0)1 + (0)2 + (1)3 + (4)4 + (0)5 + (4)6 + (3)7 + (1)8 + (7)9 ( 20 )tot(*_15) (2)1 + (0)2 + (4)3 + (6)4 + (0)5 + (0)6 + (5)7 + (1)8 + (2)9 ( 20 )tot(*_16) (88)1 + (71)2 + (111)3 + (114)4 + (49)5 + (128)6 + (142)7 + (156)8 + (153)9 ( 1012 )tot(*_17) (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot(*_18) (9)1 + (3)2 + (11)3 + (14)4 + (2)5 + (7)6 + (9)7 + (14)8 + (10)9 ( 79 )tot(*_19) (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot(*_20) (1)1 + (0)2 + (4)3 + (6)4 + (0)5 + (7)6 + (17)7 + (3)8 + (8)9 ( 46 )tot(*_21) (1)1 + (0)2 + (1)3 + (0)4 + (0)5 + (0)6 + (1)7 + (0)8 + (0)9 ( 3 )tot(*_22) (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot(*_23) (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot(*_24) (11)1 + (9)2 + (14)3 + (12)4 + (3)5 + (12)6 + (23)7 + (8)8 + (22)9 ( 114 )tot(*_25) (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot(*_26) (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot(*_27) (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot(*_28) (0)1 + (0)2 + (0)3 + (1)4 + (0)5 + (0)6 + (0)7 + (1)8 + (1)9 ( 3 )tot(*_29) (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot(*_30) (0)1 + (0)2 + (2)3 + (2)4 + (0)5 + (0)6 + (1)7 + (1)8 + (0)9 ( 6 )tot(*_31) (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot(*_32) (2)1 + (0)2 + (3)3 + (1)4 + (0)5 + (3)6 + (2)7 + (2)8 + (3)9 ( 16 )tot(*_33) (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot(*_34) (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot(*_35) (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot(*_36) (0)1 + (0)2 + (1)3 + (0)4 + (0)5 + (1)6 + (1)7 + (0)8 + (2)9 ( 5 )tot 3049186725, 6137904825, 7261938450, 9132768450 & 9702436185(*_37) (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot(*_38) (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot(*_39) (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot(*_40) (0)1 + (0)2 + (0)3 + (1)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 1 )tot 4398176250(*_41) (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot(*_42) (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot(*_43) (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot(*_44) (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot(*_45) (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot(*_46) (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot(*_47) (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (0)7 + (0)8 + (0)9 ( 0 )tot(*_48) (0)1 + (0)2 + (0)3 + (0)4 + (0)5 + (0)6 + (1)7 + (0)8 + (0)9 ( 1 )tot 7316984025(*_subt) (65037)1 + (64112)2 + (66319)3 + (62415)4 + (61920)5 + (60431)6 + (65919)7 + (61240)8 + (61408)9 ( 568801 )total |
| 1 | 1023458976 = 194762 + 253802 | 1 | 1024397685 = 57692 + 314822 | |
| 2 | 1024537896 = 193862 + 254702 | 2 | 1063789245 = 78422 + 316592 | |
| 3 | 1024736985 = 165392 + 274082 | 3 | 1073982645 = 89462 + 315272 | |
| 4 | 1032768549 = 146702 + 285932 | 4 | 1074893265 = 15692 + 327482 | |
| 5 | 1038249657 = 185042 + 263792 | 5 | 1236785940 = 51962 + 347822 | |
| 6 | 1038259476 = 198602 + 253742 | 6 | 1237694085 = 56822 + 347192 | |
| 7 | 1058267349 = 184502 + 267932 | 7 | 1258047369 = 67952 + 348122 | |
| 8 | 1063275498 = 159032 + 284672 | 8 | 1270365489 = 81752 + 346922 | |
| 9 | 1073982645 = 198542 + 260732 | 9 | 1294037685 = 92582 + 347612 | |
| 10 | 1075348296 = 145862 + 293702 | 10 | 1324958760 = 71942 + 356822 | |
| 11 | ° = ° + ° | 11 | ° = ° + ° | |
| 12 | ° = ° + ° | 12 | ° = ° + ° |
One lovely pandigital already popped up on my screen 1073982645
It combines in an elegant way the two numberformats i.e. ninedigital and pandigital !
This couple are the middle two from a fourfold (*_4) solution for this pandigital.
| 1073982645 = | ↗ ↘ |
89462 + 315272 198542 + 260732 |
Then three more pandigitals with double solutions popped up later on !
They are fully pandigital solutions throughout. Enjoy them!
| 7095281346 = | ↗ ↘ |
207452 + 816392 398612 + 742052 |
| 7125094386 = | ↗ ↘ |
312692 + 784052 497312 + 682052 |
| 7465102389 = | ↗ ↘ |
369422 + 781052 469832 + 725102 |
Two pandigitals that stand out from the rest are 5921803476 and 8097452136 because
they are the only ones that can be written as a sum of two squares such that both their
basenumbers form a pandigital when concatenated as well as when multiplied together.
A nice couplet !
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While tracing for long(er) chains of pandigitals I stumbled across the following unique and
unexpectedly double expression. A thing of beauty !
Note that A and B are in descending order here. I didn't find a case whereby A and B are ascending.
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Can you find longer chains of pandigital expressions using perhaps other operations than concatenation ?
More subcategories
All the pandigitals (41) equal to 2 x A2 are A =
22887, 23124, 24957, 25941, 26409, 26733, 27276, 29685, 31389, 35367,
39036, 39147, 39432, 39702, 40293, 41997, 42843, 43059, 44922, 45258,
45624, 46464, 49059, 50889, 53568, 54354, 57321, 59268, 59727, 60984,
61098, 61611, 61866, 62634, 65436, 68823, 68982, 69087, 69696, 69732,
69798.
Note that two of them (see underlined) are palindromic !
All the pandigitals (46) equal to (A)2 + (AReversed)2 are A (smallest) =
10716, 12804, 14496, 14967, 16053, 18126, 18528, 19317, 20493, 21423, 21792,
22839, 23205, 23286, 23544, 24267, 26058, 27324, 28557, 28563, 30597, 32325,
32838, 33105, 37824, 41676, 41718, 41736, 42378, 43497, 43725, 45018, 46464,
47217, 49245, 49305, 51327, 52866, 53436, 54456, 55296, 56247, 60927, 63666,
67137, 69696.
Note that the same two palindromes appear again !
The most beautiful one however seems to me the next one
The reversals of above paragraph are evidently included in this total.
Topic 5.11 [ April 27, 2014 ]
Order out of chaos using the ninedigits
Numberphile has released a video file that relates to the ninedigits.
Near the end of the video the zero comes in as well!
https://www.youtube.com/watch?v=CwIAfkuXc5A&feature=em-uploademail
And here is a video explaining the mysterious math from above
which is in fact the Erdös-Szekeres theorem.
https://www.youtube.com/watch?v=LBPj8E1JKaQ&feature=em-uploademail
Topic 5.10 [ December 15, 2013 ]
Nearing the end of the year 2013
Ain't that nice!
Who likes to do this exercise for the coming years 2014, 2015, ...
And what about the pandigital version of this topic ?
Topic 5.9 [ January 1, 2013 ]
The ninedigital primes version of WONplate 181
The following ninedigital numbers are all (probable (3-PRP!)) prime.
Note that the 5-digit displacements are also prime !
|
1325! + 87649 1462! + 59387 1475! + 96823 1547! + 68239 1685! + 29437 2351! + 74869 2354! + 86197 2468! + 75193 2486! + 15973 2615! + 84793 2876! + 15493 3218! + 45697 3845! + 76129 4618! + 32579 4895! + 76213 4981! + 62753 8153! + 72649 8573! + 41269 9625! + 34781 9824! + 61357 |
1438! - 57269 1478! - 52369 1486! - 59273 1576! - 28493 1849! - 67523 2471! - 63589 2536! - 84179 3469! - 28751 4285! - 71963 4562! - 93871 4691! - 52837 5239! - 16487 5417! - 86923 5462! - 98317 5743! - 68219 5749! - 26183 7948! - 23561 7982! - 45361 8123! - 47569 9836! - 25147 |
Topic 5.8 [ December 30, 2010 ]
From a posting to [SeqFan]
by Eric Angelini
"As an afterthought, here's one I like (because of
the symmetry in the operations) that was appropriate
for the countdown on Friday night { can you find out the exact year? }:
Happy New Year! "
Topic 5.7 [ April 2008 ]
Fractions using the same digits as their decimal representation
A webpage by Christian Boyer
An example using all the nine digits from
http://www.christianboyer.com/fractiondigits/ is
The above link came from a reply in the SeqFan mailing list where
Alexander R. Povolotsky's topic was about approximating Pi
using just nine- and pandigitals. Here is his *best* combination !
Alexander R. Povolotsky [ October 8, 2022 ] writes Since then the better approximations were found per Using each number (1-9 EXACTLY ONCE) can you make 2 distinct 9 digits numbers, so the quotient of the two numbers is as close to Pi as possible? 429751836 / 136794258 = 3.14159265369164852899... (pi + 1.01855e-10) 467895213 / 148935672 = 3.14159265350479621759... (pi - 8.49969e-11) Pi = 3.1415926535897932384626433832795028842 |
Topic 5.6 [ March 31, 2008 ]
Blending palindromes with nine- & pandigitals using multiplication by 9
by B.S. Rangaswamy
This topic is a continuation from wonplate 173.
Pandigitals (Kmil = 1000 million) total = 559
Palindrome Pandigital | Palindrome Pandigital | Palindrome Pandigital | Palindrome Pandigital |
|---|---|---|---|
1 Kmil | 3 Kmil | 5 Kmil | 7 Kmil |
Topic 5.5 [ January 22, 2006 ]
Pandigital... throughout
A unique construction : start with multiplying these two
5-digit factors which taken together form a pandigital number :
| 54981 * 62037 |
equals
| 3410856297 |
as you can see the result of the multiplication is pandigital as well.
And now let us take the square of this pandigital 34108562972
| 11633940678784552209 |
... an order_2 pandigital emerges since all the digits from 0 to 9
occur exactly two times !
Discover more of these gems at wonplate 167 !
Topic 5.4 [ October 26, 2005 ]
From Palindromic Squares to Pandigitals
264 is a very interesting number since it is the 12th basenumber of a palindromic square.
The square itself is this nice palindrome 69696.
Note the presence of the number of the Beast ! 69696 .
Did you know that when we power up 264 with two exponents and add them up
that we arrive at a pandigital number... in two different ways !
| 2643 + 2644 = 4875932160 2644 + 2644 = 9715064832 |
|---|
69696 squared and then doubled
- B.S. Rangaswamy [ August 11, 2006 ] was enthused by this presentation
- and found the following solutions :
232322 * 8 = 4317806592 = 464642 * 2
232322 * 18 = 9715064832 = 696962 * 2
423242 * 4 = 7165283904
I found the next equation of some interest.
| 20162 + 20163 = 8197604352 |
|---|
Or the next one with two consecutive integers
| 233 * 244 = 4036718592 |
|---|
Topic 5.3 [ October 2005 ]
Figure this out : 4-1-4
It is a four digit number multiplied by a one digit number to equal another four digit number
and only the nine digits from 1 to 9 can be used once ?
Two solutions to this 9-digit problem can be found.
| 1738 * 4 = 6952 1963 * 4 = 7852 |
|---|
Topic 5.2 [ October 2006 ]
Figure this out : 4-1-5
and only the ten digits from 0 to 9 can be used once ?
Thirteen solutions to this pandigital problem can be found.
|
3094 * 7 = 21658 3907 * 4 = 15628 4093 * 7 = 28651 5694 * 3 = 17082 5817 * 6 = 34902 6819 * 3 = 20457 6918 * 3 = 20754 7039 * 4 = 28156 8169 * 3 = 24507 9127 * 4 = 36508 9168 * 3 = 27504 9304 * 7 = 65128 9403 * 7 = 65821 |
|---|
Topic 5.1 [ September 4, 2005 ]
Generating Pandigitals from Palindromes through Fibonacci iteration
by B.S. Rangaswamy
" I got inspired by your presentation of the derivation of 68 ninedigit numbers (with all numerals from 1 to 9)
from palindromes through Fibonacci iteration. I have developed it further by arriving at a dozen 10 digit numbers
(with all numerals from 0 to 9) from 2 to 10 digit palindromes. Some of the 10 digit numbers arrived at
together with their mother palindromes are :
75257
799997
8055508
42944924It is interesting to note that 6300036 leads to pandigital 1467908532, which matches identically
with each stage of iteration of 630036 to 146798532. I came across another curio as well :
630036
6300036
9530359
95300359Following closely resembling palindromes lead to closely matching pandigitals :
592070295 2960351478
952070259This task began at my son's residence in Florida US and was completed at Bangalore India.
I was thrilled at the discovery of each of these pandigitals.I am grateful to you for your encouragement and guidance in this venture.
B.S.Rangaswamy "
In total there are 117 palindromes that yield pandigital numbers
The smallest one is 75257 and the largest one is 4376006734 ¬
| 1 75257 75258 150515 225773 376288 602061 978349 1580410 2558759 4139169 6697928 10837097 17535025 28372122 45907147 74279269 120186416 194465685 314652101 509117786 823769887 1332887673 2156657560 3489545233 5646202793 9135748026 |
1 799997 799998 1599995 2399993 3999988 6399981 10399969 16799950 27199919 43999869 71199788 115199657 186399445 301599102 487998547 789597649 1277596196 2067193845 |
1 6431346 6431347 12862693 19294040 32156733 51450773 83607506 135058279 218665785 353724064 572389849 926113913 1498503762 |
1 408676804 408676805 817353609 1226030414 2043384023 3269414437 5312798460 |
1 561262165 561262166 1122524331 1683786497 2806310828 4490097325 7296408153 |
1 760474067 760474068 1520948135 2281422203 3802370338 6083792541 |
|---|---|---|---|---|---|
| 1 93811839 93811840 187623679 281435519 469059198 750494717 1219553915 1970048632 3189602547 |
1 1206776021 1206776022 2413552043 3620328065 6033880108 9654208173 |
1 3067447603 3067447604 6134895207 |
1 4068228604 4068228605 8136457209 |
||
| 117 in total |
| 75257 799997 2877782 4364634 4689864 5068605 6300036 6431346 6881886 8055508 15844851 42944924 54777745 93811839 95300359 |
146353641 177858771 185121581 185222581 187333781 207313702 238242832 245363542 295747592 310717013 314878413 324151423 348616843 350878053 354010453 |
358070853 370464073 370797073 390474093 394070493 395313593 395727593 407525704 408676804 460363064 473303374 475686574 506939605 527121725 527454725 |
530878035 536454635 547252745 561262165 590272095 592070295 629171926 642676246 655191556 695525596 740838047 760474067 782484287 841303148 914272419 |
950272059 952070259 1089229801 1097337901 1098228901 1206776021 1289009821 1295335921 1298008921 1395225931 1397007931 2068448602 2069339602 2158448512 2159339512 |
2359119532 2365445632 2369009632 2456336542 2458118542 2465335642 2468008642 3067447603 3069229603 3076446703 3079119703 3157447513 3159229513 3175445713 3179009713 |
3256446523 3259119523 3265445623 3269009623 3456226543 3457117543 3465225643 3467007643 3475115743 3476006743 4067337604 4068228604 4076336704 4078118704 4086226804 |
4087117804 4158228514 4175335714 4178008714 4185225814 4187007814 4258118524 4268008624 4286006824 4365225634 4367007634 4376006734 |
|---|
| Palindrome | Ninedigital | Palindrome | Pandigital | |||
|---|---|---|---|---|---|---|
| QUEEN | KING | |||||
| 630036 9530359 128909821 129808921 139707931 236909632 246808642 317909713 326909623 346707643 347606743 417808714 418707814 426808624 428606824 436707634 437606734 |
146798532 123894675 257819643 259617843 279415863 473819265 493617285 635819427 653819247 693415287 695213487 835617429 837415629 853617249 857213649 873415269 875213469 |
6300036 95300359 1289009821 1298008921 1397007931 2369009632 2468008642 3179009713 3269009623 3467007643 3476006743 4178008714 4187007814 4268008624 4286006824 4367007634 4376006734 |
1467908532 1238904675 2578019643 2596017843 2794015863 4738019265 4936017285 6358019427 6538019247 6934015287 6952013487 8356017429 8374015629 8536017249 8572013649 8734015269 8752013469 |
|||
|
For reference goals and easy searching I list here all the nine- & pandigitals implicitly displayed in these topics.
Topic 5.12 → 349512768, 365412897, 453612987, 537612984, 546913728, 652813749, 765913248, 879312546, 567914238, 897312546, 694513782, 765913428, 278415369, 825913476, 536714928, 496815237, 697514823, 435916278, 829514763, 893415267, 253817649, 294317586, 496517382, 543617289, 645917328, 874516392, 795316824, 549618237, 243618975, 537618429, 596418273, 835217469, 834917562, 236719485, 645319287, 782419356, 962718543, 469821357, 485721396, 569721384, 489621753, 597621483, 576321948, 634521987, 786321495, 879621345, 619823547, 647123589, 187524693, 745823619, 786923514, 159624837, 879623541, 916823475, 735624189, 386425179, 681324597, 419725638, 618925374, 968724513, 973524861, 897625413, 716425983, 398426715, 439527186, 974126358, 561327948, 654927831, 351628479, 391528467, 915627483, 735928146, 384629175, 163529487, 647129538, 683129745, 743129658, 851729643
1374920589, 1365920784, 1097423856, 1496723085, 1857320946, 1594823067, 1589423706, 1374925068, 1035926784, 1705823469, 1695324078, 1749623850, 1378526490, 1649725308, 1850723964, 1609825743, 1439726805, 1579826043, 1437026985, 1870524369, 1368027594, 1746925830, 1740625893
1947625380, 1938625470, 1653927408, 1467028593, 1850426379, 1986025374, 1845026793, 1590328467, 1985426073, 1458629370
576931482, 784231659, 894631527, 156932748, 519634782, 568234719, 679534812, 817534692, 925834761, 719435682
2074581639, 3986174205, 3126978405, 4973168205, 3694278105, 4698372510
3579068124, 3579068124, 2517086394, 2517086394
7512948360, 7530642918
1047629538, 1069438752, 1245703698, 1345870962, 1394870562, 1429306578, 1487960352, 1762398450, 1970538642, 2501649378, 3047618592, 3064975218, 3109765248, 3152497608, 3247051698, 3527496018, 3671045298, 3708154962, 4035972168, 4096573128, 4163098752, 4317806592, 4813570962, 5179380642, 5739061248, 5908714632, 6571394082, 7025391648, 7134629058, 7438096512, 7465931208, 7591830642, 7654803912, 7846035912, 8563740192, 9473210658, 9517032648, 9546027138, 9715064832, 9725103648, 9743521608
3921846057, 1830296457, 5032186497, 6143928570, 1486972530, 4195028637, 7162908345, 5469821370, 1972480653, 1509482673, 1357694208, 9324187605, 3061725849, 5197843620, 2537418960, 6401729853, 7914563208, 2541987360, 6528140973, 2154087693, 7256903418, 3782601954, 8104629573, 3609258714, 3268749105, 6308541972, 8417569320, 5801367492, 9421375860, 8201749365, 4692750381, 8596371240, 4317806292, 7309428165, 5372908461, 4970538261, 7863920154, 7260394581, 6879405321, 7248503961, 7853902641, 8679015234, 9027384165, 8493716052, 9862103745, 9715064832
1023849765, 1027935648, 3921846057, 7914563208, 9864025713, 9872350146, 9876135240
5678904132, 5678904132, 2690457813, 2690457813, 5490321768, 5490321768, 6029417538, 6029417538, 8576492130, 8576492130, 5047168932, 8140532769
Topic 5.9 → 132587649, 146259387, 147596823, 154768239, 168529437, 235174869, 235486197, 246875193, 248615973, 261584793, 287615493, 321845697, 384576129, 461832579, 489576213, 498162753, 815372649, 857341269, 962534781, 982461357
143857269, 147852369, 148659273, 157628493, 184967523, 247163589, 253684179, 346928751, 428571963, 456293871, 469152837, 523916487, 541786923, 546298317, 574368219, 574926183, 794823561, 798245361, 812347569, 983625147
Topic 5.7 → 124983576, 216984375
Topic 5.6 → 240545042, 2164905378, 240656042, 2165904378, 315989513, 2843905617, 360212063, 3241908567, 360989063, 3248901567, 361323163, 3251908467, 364989463, 3284905167, 420545024, 3784905216, 420656024, 3785904216, 480212084, 4321908756, 480989084, 4328901756, 485767584, 4371908256, 486989684, 4382907156, 513545315, 4621907835, 531545135, 4783906215, 531878135, 4786903215, 536989635, 4832906715, 624656426, 5621907834, 630212036, 5671908324, 630989036, 5678901324, 642656246, 5783906214, 642989246, 5786903214, 713545317, 6421907853, 724656427, 6521907843, 753878357, 6784905213, 753989357, 6785904213, 840212048, 7561908432, 840989048, 7568901432, 915989519, 8243905671, 936989639, 8432906751, 951434159, 8562907431, 951989159, 8567902431, 963878369, 8674905321, 963989369, 8675904321
Topic 5.5 → 5498162037
Topic 5.3 → 173846952, 196347852
Topic 5.2 → 3094721658, 3907415628, 4093728651, 5694317082, 5817634902, 6819320457, 6918320754, 7039428156, 8169324507, 9127436508, 9168327504, 9304765128, 9403765821B.S. Rangaswamy (email) - go to topic 1
B.S. Rangaswamy (email) - go to topic 2
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Patrick De Geest - Belgium
- Short Bio - Some PicturesE-mail address : pdg@worldofnumbers.com