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Details of Palindromic Triangulars
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Legend

F_ba Decomposition of the basenumber in its prime factors. F_pt Decomposition of the palindromic triangular. Note : a palindromic triangular will always be divisible by its basenumber when this last one is 'odd'. With 'even' basenumbers you'll notice that the triangulars are always divisible by half their basenumbers. To reconstruct the triangular you must take into account the green together with the purple prime factors but not the black 2. Date Date of discovery by the author. blue Hypertext links to related topics in number theory Comm Comments about the palindromic triangular number.


Factorizations

back [147]
F_ba  128184152897963669861552 =
      2  x  2  x  2  x  2  x  11  x  17  x  127  x  1873  x  3581  x  50295245131
F_pt  8215588527084263951180440811593624807258855128 =
      3  x  3243912979  x  13171762387769
Date  January 5, 2008 (Feng Yuan)

back [146] F_ba 115945380367510626966648 = 2 x 2 x 2 x 3 x 193 x 8017 x 3122288398366517 F_pt 6721665614283359365218338125639533824165661276 = 7 x 11 x 131 x 1543 x 169633 x 43915229833 Date January 5, 2008 (Feng Yuan)
back [145] F_ba 114860741505066911342296 = 2 x 2 x 2 x 7 x 7 x 7 x 11 x 31 x 468151 x 262208746399 F_pt 6596494969546900318903773098130096459694946956 = 101 x 479 x 15420611 x 153961859113 Date January 5, 2008 (Feng Yuan)
back [144] F_ba 26539182748774333465497 = 3 x 3 x 3 x 677 x 19661 x 141101 x 523358663 F_pt 352164110486420593101030101395024684011461253 = 11 x 509 x 123553 x 19181995591667 Date January 5, 2008 (Feng Yuan)
back [143] F_ba 5805651147054009150041 = 1429 x 153395789 x 26485322761 F_pt 16852792620644766088388388066744602629725861 = 3 x 3 x 3 x 11 x 11 x 13 x 17 x 23 x 281 x 153469 x 4053449 Date January 5, 2008 (Feng Yuan)
back [142] F_ba 3630238456595636157728 = 2 x 2 x 2 x 2 x 2 x 113444951768613629929 F_pt 6589315625872933253747473523392785265139856 = 3 x 23 x 29 x 17234557 x 105265956197 Date January 2, 2008 (Feng Yuan)
back [141] F_ba 3261059106801402665754 = 2 x 3 x 3 x 3 x 3 x 3 x 23 x 23 x 149 x 4477841 x 19011299 F_pt 5317253249026181079052509701816209423527135 = 5 x 23 x 23 x 149 x 652211821360280533151 Date January 2, 2008 (Feng Yuan)
back [140] F_ba 775781766082836455602 = 2 x 11 x 337817 x 104384348772323 F_pt 300918674293302389819918983203392476819003 = 167 x 4645399796903212309 Date May 31, 2001
Note the large palindromic substring in the largest prime factor
4645399796903212309
back [139] F_ba 585863634063453017106 = 2 x 3 x 19 x 83 x 61917526322495563 F_pt 171618098859017793192291397710958890816171 = 11 x 53260330369404819737 Date April 18, 2001
back [138] F_ba 333741662509416495210 = 2 x 3 x 3 x 5 x 97 x 127 x 227 x 4523 x 293183831 F_pt 55691748647274629891519892647274684719655 = 17 x 37 x 530590878393348959 Date April 18, 2000
back [137] F_ba 333470345690102634085 = 5 x 23 x 21661 x 303959 x 440418821 F_pt 55601235727338276212021267283372753210655 = 100045471 x 1666593911533 Date April 18, 2000
The first palindromic triangular discovered in this new millennium. Add the triplets of the palindrome from right to left : 55 + 601 + 235 + 727 + 338 + 276 + 212 + 021 + 267 + 283 + 372 + 753 + 210 + 655 = 5005 ... a nice palindromic number !
back [136] F_ba 191600462227318472121 = 2 x 3 x 17 x 47 x 197 x 503 x 8693 x 92795011 F_pt 18355368562861046305450364016826586355381 = 2477 x 38675910825054193 Date April 22, 2000
back [135] F_ba 35369824079822102851 = 53 x 667355171317398167 F_pt 625512227718781732353237187817722215526 = 2 x 30253 x 8817409 x 33148469 Date October 28, 1999 (after a two-year period of nonactivity...)
back [134] F_ba 32637734114649927570 = 2 x 3 x 3 x 5 x 37 x 113 x 86735587219033 F_pt 532610844069291845737548192960448016235 = 7 x 11 x 47 x 193 x 241027 x 193869019 Date December 30, 1997
Add the triplets of the basenumber from right to left : 32 + 637 + 734 + 114 + 649 + 927 + 570 = 3663 A nice palindromic number ! We perceive two palindromic substring of 4 digits in the basenumber. 326(3773)(4114)649927570 The difference between these numbers is 341. This difference is a substring of the concatenation of these two palindromes : 37734114
back [133] F_ba 19123354745855372721 = 3 x 17209 x 1199089 x 308912707 F_pt 182851348367914603505306419763843158281 = 9561677372927686361 Date August 2, 1997

The following basenumbers start and end with the same digits : [121] 19 35755375665009 721 [133] 19 123354745855372 721 A lot of palindromic subgroups can be extracted from this basenumber

19123354745855372721 19123354745855372721 19123354745855372721 19123354745855372721 19123354745855372721 19123354745855372721 19123354745855372721 19123354745855372721
Confert [130] the 'universal' number 42 is present : Add the digits of the above basenumber together and divide by 2 !
back [132] F_ba 16175904024612952346 = 2 x 3187 x 5347 x 474620309557 F_pt 130829935506744754616457447605539928031 = 3 x 7 x 13 x 746939 x 79326954001 Date April 11, 1997
The Number of the Beast present as always in the base. 16175904024612952346 Extract the odd digits from the base and concatenate them to form this number 117591953. Now add the three triplets together. 117 + 591 + 953 = 1661 Yep, it's palindromic ! The digit 8 is the only one not showing up in the basenumber. A triplet of palindromic duo's (77, 88 and 99) bu17 Sum up all the digits of the basenumber to arrive at a palindrome namely 77 . Note that 77 x 88 = 6776 ! bu17 The summation of the digits at either side of the middle number 1 of the palindromic triangular number is the palindrome 88 . bu17 Add together the digits at the even places in the triangular number : ( 3 + 8 + 9 + 3 + 5 + 6 + 4 + 7 + 4 ) x 2 + 1 = 99 Let me play a little with the triplets of the triangular number. The sum of the first two equals the sum of the last two triplets : 130 + 829 = 928 + 031 = the palindrome 959 Anagrams of the same numbers to be found in various places : The sum of the first and the last three triplets of the triangular is 130 + 829 + 935 = 1894 and 1498 = 539 + 928 + 031 The fourth and fifth from left and right gives 506 + 744 = 1250 and 1052 = 447 + 605 and 1520 = 616 + 457 ! The third and fourth from left and right gives 935 + 506 = 1441 and 1144 = 605 + 539 !
back [131] F_ba 11665833272979576316 = 2 x 2 x 9091 x 17761 x 18062451829 F_pt 68045832976478686977968687467923854086 = 11 x 844511 x 1255792165577 Date January 29, 1997
The number 131 can be written as the sum of three consecutive primes
131 = 41 + 43 + 47
back [130] F_ba 3654345456545434563 = PALINDROMIC 3 x 11 x 11 x 1429 x 9091 x 774923959 F_pt 6677120357887130286820317887530217766 = 2 x 17 x 101 x 149 x 849217 x 4205081 Date May 17, 1996
Proof : this palindromic triangular is worth twice its basenumber.

Method : Divide into groups of three beginning from the right to the left :

Base = 003 + 654 + 345 + 456 + 545 + 434 + 563 = 3000 Tria = 006 + 677 + 120 + 357 + 887 + 130 + 286 + 820 + 317 + 887 + 530 + 217 + 766 = 6000 Odd and even are in perfect balance. Method : split the odd and even numbers and add them together : 3 + 5 + 3 + 5 + 5 + 5 + 5 + 3 + 5 + 3 = 42 6 + 4 + 4 + 4 + 6 + 4 + 4 + 4 + 6 = 42 Now, let's apply this method with the palindromic triangular : 6 + 7 + 1 + 0 + 5 + 8 + 7 + 3 + 2 + 6 + 2 + 3 + 7 + 8 + 5 + 0 + 1 + 7 + 6 = 84 = 2 x 42 6 + 7 + 2 + 3 + 7 + 8 + 1 + 0 + 8 + 8 + 0 + 1 + 8 + 7 + 3 + 2 + 7 + 6 = 84 = 2 x 42 Again, the number 42 pops up ! The Number of the Beast present as always. 3654345456545434563 Note that the above sum 42 equals 6 * 6 + 6 and 6 equals 4 + 2 Each digit in the basenumber (except the border ones) is surrounded by two numbers which sums up to either 8 or 10. 3 6 --> 3 + 5 = 8 5 --> 6 + 4 = 10 4 --> 5 + 3 = 8 3 --> 4 + 4 = 8 4 --> 3 + 5 = 8 5 --> 4 + 4 = 8 4 --> 5 + 5 = 10 5 --> 4 + 6 = 10 6 --> 5 + 5 = 10 5 --> 6 + 4 = 10 etc... Clustered trios composed of odd digits emerge in this triangular number : 66 771 20 357 88 713 0286820 317 88 753 02 177 66 Assemble two numbers with those odd digits. The first is 771357713 and the second is 317753177.

Performing adding and subtracting with these numbers yields interesting results :

771357713 - 317753177 ---------------   = 453604536 note that 4536 occurs twice.

Remember that the digits 3, 4, 5 and 6 were the only 4 digits from our basenumber ! 771357713 + 317753177 ---------------  = 1089110890 and 1089 also showing up twice.

Note that 4536 + its reversal 6354 equals 10890 which is also 1089 + its reversal 9801 !

Finally divide the whole addition number by two and we see 544555445 ... a palindrome !

This palindromic 'mean' value is a property we find with many numbers and their reversals. It occurs when the sum of the number and its palindrome is even. 12345 + 54321 = 66666/2 = 33333 1089 + 9801 = 10890/2 = 5445 103 + 301 = 404/2 = 202


back [129] F_ba 3652242206567626036 = 2 x 173 x 2957 x 1784851760869 F_pt 6669436567716980985890896177656349666 = 43 x 285451 x 297549720509 Date May 16, 1996
This palindromic triangular is encapsulated by 'The Number of the Beast' itself
666 9436567716980985890896177656349 666
The sum of the digits of the middle part of the above triangular
equals the palindrome 181 ! A palindromic subgroup of 5 digits exists in the basenumber
3652242206567626036
back [128] F_ba 3637582781740258588 = 2 x 2 x 19 x 37 x 3930491 x 329117339 F_pt 6616004247006598875788956007424006166 = 7 x 277 x 320741 x 5848986211 Date May 16, 1996 back [127] F_ba 3405603489389985674 = 2 x 7 x 43 x 467 x 29027 x 417328993 F_pt 5799067563472623134313262743657609975 = 3 x 5 x 5 x 11 x 4128004229563619 Date April 30, 1996
Divide the basenumber in blocks of three from right to left and add them together :
003 + 405 + 603 + 489 + 389 + 985 + 674 = 3548
This is the basenumber of palindromic triangular [21]!
back [126] F_ba 3333229904907177885 = 3 x 3 x 5 x 29531 x 2508271838563 F_pt 5555210799483757064607573849970125555 = 13 x 109 x 761389 x 1544752211 Date April 25, 1996 back [125] F_ba 3271889143987456170 = 2 x 3 x 5 x 79 x 83 x 421 x 24109 x 1638743 F_pt 5352629285271484348434841725829262535 = 239 x 2023027 x 6767044007 Date April 22, 1996 back [124] F_ba 3237532871306389774 = 2 x 293 x 1483 x 1097909 x 3393197 F_pt 5240809546394698286828964936459080425 = 5 x 5 x 7 x 107 x 172898951738659 Date April 20, 1996 back [123] F_ba 2672899778869085022 = 2 x 3 x 83 x 241 x 499 x 727 x 61390523 F_pt 3572196613939201806081029393166912753 = 359 x 5941253 x 1253170549 Date April 12, 1996 back [122] F_ba 2470152923949718922 = 2 x 311 x 26293 x 151040465207 F_pt 3050827733848672937392768483377280503 = 3 x 73 x 571 x 19753480027427 Date April 6, 1996 back [121] F_ba 1935755375665009721 = 233 x 4721 x 1759788812897 F_pt 1873574437207991455541997027344753781 = 3 x 103 x 3132290251885129 Date March 25, 1996
blue A conjecture about the number 121 by Mike Keith. The number 121 can be written as the sum of three consecutive primes
121 = 37 + 41 + 43
back [120] F_ba 775527779120670322 = 2 x 1153 x 336308663972537 F_pt 300721668093919607706919390866127003 = 11 x 70502525374606393 Date November 17, 1995 back [119] F_ba 352465914177746748 = 2 x 2 x 3 x 3 x 3 x 271967 x 11999887043 F_pt 62116110328577368186377582301161126 = 107 x 413461 x 7967072987 Date December 22, 1995 back [118] F_ba 352449681516778876 = 2 x 2 x 19 x 30338993 x 152855957 F_pt 62110389000639430803493600098301126 = 43 x 8196504221320439 Date December 22, 1995 back [117] F_ba 247184344095525797 = 913687 x 270535034531 F_pt 30550049982967649594676928994005503 = 3 x 13 x 78367 x 40438322923 Date March 29, 1992 back [116] F_ba 112540786741633183 = 17 x 67 x 173 x 22277 x 25637957 F_pt 6332714340212879669782120434172336 = 2 x 2 x 2 x 2 x 11 x 81239 x 3935525353 Date October 11, 1995 back [115] F_ba 109113947229482909 = 7 x 41 x 47 x 843559 x 9589259 F_pt 5952926739999190550919999376292595 = 3 x 3 x 5 x 11 x 13 x 17 x 498715422229 Date September 30, 1995 back [114] F_ba 108100203429839929 = 7 x 11 x 17 x 43 x 367 x 6343 x 825007 F_pt 5842826990786388228836870996282485 = 5 x 10810020342983993 Date September 27, 1995 back [113] F_ba 78358935395500837 = 7 x 11 x 719 x 29501 x 47976899 F_pt 3070061378158136996318518731600703 = 13 x 17 x 89 x 769 x 4679 x 553601 Date October 2, 1995 back [112] F_ba 26555908318527497 = 7 x 13 x 3931 x 74236369457 F_pt 352608133311018969810113331806253 = 3 x 73 x 60629927667871 Date April 24, 1992 back [111] F_ba 10686132500837474 = 2 x 5351 x 165569 x 6030823 F_pt 57096713912727488472721931769075 = 3 x 5 x 5 x 11 x 181 x 14591 x 4904593 Date October 13, 1993 blue Test for divisibility by 111 back [110] F_ba 4500990532341401 = 7 x 12163 x 219311 x 241051 F_pt 10129457886113466431168875492101 = 3 x 11 x 29 x 223 x 10545357391 Date July 5, 1991 back [109] F_ba 4062262265478592 = 2 x 2 x 2 x 2 x 2 x 2 x 7 x 365201 x 24828929 F_pt 8250987356765633365676537890528 = 11 x 13513 x 19853 x 1376567 Date September 20, 1990
Arrange the numbers of this basenumber 004.062.262.265.478.592 in the form of a rectangle of 3 by 6
0 0 4 0 6 2 In the second row appears the Number of the beast 666 ! 2 6 2 2 6 5 4 7 8 5 9 2 Another way of displaying also reveals the number of the beast in an unexpecting way. Divide the basenumber in blocks of four : 4062.2622.6547.8592 and arrange these blocks in the form of a rhombus from the right downwards to the left. 4 2 0 6 6 6 8 5 2 2 5 4 2 9 7 2 Count the first three blocks of four together 4062 + 2622 + 6547 and a palindromic result appears namely : 13231 A large subsection of the basenumber is palindromic 40_6226226_5478592 The number 109 is a prime. The next following prime is 113. When added together we find a palindrome 222 which is a repdigit number

back [108] F_ba 3729035890325943 = 3 x 3 x 43 x 811 x 26951 x 440849 F_pt 6952854335669501059665334582596 = 2 x 2 x 11 x 4106521 x 10319053 Date March 27, 1991
This is a posting I came across in sci.math by Robert E Sawyer answering the following question of Ray Van Raamsdonk :
Does anyone know why the number 108 was so popular in China and India ?

In Buddhism, as in the Indian religion(s) that preceded it, 108 is a number that signifies "completion", "perfection", etc... It is surely no coincidence that
108 = (1)*(2*2)*(3*3*3)
although I've never seen this fact mentioned anywhere. (Usually, the explanation somehow relates 108 to mystical intuitions about the number 9.) Some claim that Sanskrit was at one time spoken using exactly 2 * 54 basic phonetic elements, and regardless of whether this is true, the numerological significance is seen in the belief that it was so. (The same is true of the claim that there are or were originally exactly 108 Upanishads.) Also, there are most commonly 108 beads on a Hindu or Buddhist mala (rosary).

According to an article in the Buddhist magazine _Tricycle_, the West is heir to this numerology through the game of baseball, whose inventor had Buddhist inclinations and so made the standard baseball with 108 stitches, as well as the 9 innings and 3 strikes, etc.

From Properties of the number 72 : The Chinese astrology has 36 beneficial stars and 72 malevolent stars, their sum
constitutes the sacred number 108.


back [107] F_ba 3249909720631025 = 5 x 5 x 47 x 2765880613303 F_pt 5280956596126015106216956590825 = 3 x 3 x 3 x 6991 x 41141 x 209249 Date October 11, 1990 back [106] F_ba 362785412063728 = 2 x 2 x 2 x 2 x 269 x 929 x 2699 x 33617 F_pt 65806627603124642130672660856 = 29 x 307 x 2341 x 17406523 Date February 16, 1990 back [105] F_ba 353520620692923 = 3 x 37 x 167 x 563 x 33874033 F_pt 62488414627554945572641488426 = 2 x 47 x 4637 x 405527029 Date February 12, 1990 back [104] F_ba 326825312860029 = 3 x 19 x 5733777418597 F_pt 53407392563028082036529370435 = 5 x 32682531286003 Date January 31, 1990
The number 949 pops up at the start and end of basenumber [104] 326.***.***.***.029 Add their reversals 326 + 623 = 949 = 029 + 920 Consider the following basenumbers [104]
***.***.***.860.029 [103] 173.008.***.***.*** Now 173 + 029 = 202 and 008 + 860 = 868 Both numbers are palindromic AND their difference is 868 - 202 = 666 again The Number of the Beast !
back [103] F_ba 173008538941238 = 2 x 7 x 11 x 11 x 13 x 7856168329 F_pt 14965977273291019237277956941 = 3 x 3 x 349 x 21317 x 2583887 Date January 16, 1990
Consider the following basenumber [103]
173.***.538.***.238 ***.008.***.941.*** Now it happens that 173 + 538 + 238 = 949 and palindromic but at the same time is 949 = 008 + 941 ! The same technique but applied to the first and last numbers of each block of three produces a similar finding : 1*3.***.5*8.***.2*8 13 + 58 + 28 = 99 ***.0*8.***.9*1.*** 08 + 91 = 99 Nice coincidence.
back [102] F_ba 164402190438221 = 36319 x 4526616659 F_pt 13514040110442624401104041531 = 3 x 11 x 139 x 3121 x 5741893 Date January 14, 1990
[102]
164.***.190.***.221 ***.402.***.438.*** Notice that 164 + 190 + 221 = 575 is palindromic.
back [101] F_ba 109237730189290 = 2 x 5 x 1503113 x 7267433 F_pt 5966440848454114548480446695 = 11 x 19 x 73 x 389 x 1409 x 13063
The prime number 101 can be written as the sum of five consecutive primes 101 = 13 + 17 + 19 + 23 + 29
back [100] F_ba 50364608806281 = 3 x 7 x 11 x 37 x 103 x 57210341 F_pt 1268296910104884010196928621 = 349 x 12899 x 5593891
A palindromic subgroup of 6 digits exists in the basenumber
50364608806281
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