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.
Hypertext links to related topics in number theory
Comm Comments about the palindromic triangular number.
Factorization
[195]
F_ba 11604355897766717137264209814751 =
11 * 1054941445251519739751291801341
F_pt 67330537901016595837936819195144159191863973859561010973503376 =
2 * 2 * 2 * 2 * 3 * 3 * 13 * 13 * 71 * 151391 * 22181118452937982831
Date September 18, 2023 ( Patrick De Geest / CUDA program by Robert Xiao )
Note the absence of the digit 2 in the palindrome expansion.
[194]
F_ba 11551324884361124513629290544188 =
2 * 2 * 3 * 3 * 11 * 23 * 43 * 61 * 83339 * 658688593 * 8808096391
F_pt 66716553292030273308573626471522517462637580337203029235561766 =
3797 * 5730317 * 530899790335043948461
Date September 20, 2023 ( Patrick De Geest / CUDA program by Robert Xiao )
[193]
F_ba 11425320268440457835417867182496 =
2 * 2 * 2 * 2 * 2 * 509363 * 700956407098207579578431
F_pt 65268971618218167746296167721899812776169264776181281617986256 =
3 * 11 * 1597 * 216795132320837514191720597
Date September 21, 2023 ( Patrick De Geest / CUDA program by Robert Xiao )
Note the absence of the digit 0 in the palindrome expansion.
[192]
F_ba 7864314421857712727542017000682 =
2 * 13 * 37 * 720793 * 11341623762278641530077
F_pt 30923720662919605193151763743388334736715139150691926602732903 =
11 * 2503 * 3919 * 62475154307 * 1166607416147
Date September 19, 2023 ( Patrick De Geest / CUDA program by Robert Xiao )
[191]
F_ba 5956055302210614567120330592493 =
7 * 11 * 77351367561176812560004293409
F_pt 17737297381495587611197659145355354195679111678559418379273771 =
3 * 13 * 408872239 * 186756830320472175007
Date September 18, 2023 ( Patrick De Geest / CUDA program by Robert Xiao )
Note the absence of the digit 0 in the palindrome expansion.
[190]
F_ba 5783475877417359746201161921358 =
2 * 11 * 1378190070973 * 190746742914595193
F_pt 16724296612334249588772356940422404965327788594243321669242761 =
3 * 151 * 32411 * 393911169573173127334073
Date September 21, 2023 ( Patrick De Geest / CUDA program by Robert Xiao )
[189]
F_ba 5510313150359237826498809954773 =
Prime!
F_pt 15181775507510974169398336861133116863389396147901570557718151 =
11 * 146669 * 260453060311 * 6556707410563
Date September 18, 2023 ( Patrick De Geest / CUDA program by Robert Xiao )
Note the absence of the digit 2 in the palindrome expansion.
[188]
F_ba 3488109207628045100943025672428 =
2 * 2 * 3 * 301619 * 30075474103 * 32043330532517
F_pt 6083452922169774323707832554844484552387073234779612292543806 =
29383 * 3401301745943 * 34901876422541
Date September 14, 2023 ( Patrick De Geest / CUDA program by Robert Xiao )
[187]
F_ba 2649878227057025949437140416142 =
2 * 37 * 564367 * 48976381 * 1295525288454329
F_pt 3510927309115443586289107273332333727019826853445119037290153 =
1511 * 4349 * 39749 * 481489 * 21069754133017
Date September 13, 2023 ( Patrick De Geest / CUDA program by Robert Xiao )
[p186]
F_ba 1782781186350933790009742990098 =
2 * 89 * 31663 * 4813568209 * 65714143059823
F_pt 1589154379203421456422621821928291281262246541243029734519851 =
29 * 257 * 19753 * 15429024683 * 784865760517
Date September 16, 2023 ( Patrick De Geest / CUDA program by Robert Xiao )
[p185]
F_ba 1523763959995218243970890589666 =
2 * 13 * 47 * 263 * 7190213 * 659400067260773737
F_pt 1160928302890154532497341802232322081437942354510982038290611 =
59 * 179 * 1091 * 51061 * 1055899783 * 2452877659
Date September 14, 2023 ( Patrick De Geest / CUDA program by Robert Xiao )
The base ends with the number of the beast!
Its vertical mirror value 999 is also present.
1523763959995218243970890589666
[184]
F_ba 344598058975690735641865358690 =
2 * 5 * 34459805897569073564186535869
F_pt 59373911124906815186764391910901919346768151860942111937395 =
3 * 71 * 3745694071753 * 431917618418719
Date September 7, 2023 ( Patrick De Geest / CUDA program by Robert Xiao )
Note the absence of the digit 2 in the base expansion.
[183]
F_ba 332687791755510481729984663485 =
3 * 3 * 5 * 17 * 2311 * 16097 * 7502681 * 1558167200687
F_pt 55340583391578954029492500160806100529492045987519338504355 =
67 * 2482744714593361803955109429
Date September 6, 2023 ( Patrick De Geest / CUDA program by Robert Xiao )
[182]
F_ba 83852501297050811722296086142 =
2 * 3 * 29 * 37579 * 12823942281448279490027
F_pt 3515620986885954031114236657997566324111304595886890265153 =
11 * 19 * 19 * 61 * 346167506624052296040953
Date June 26, 2023 ( Patrick De Geest / CUDA program by Robert Xiao )
[181]
F_ba 61861419564584047527727330613 =
11 * 23 * 1873 * 15566521 * 20908981 * 401085677
F_pt 1913417615272750984206227878228787226024890572725167143191 =
3 * 3 * 91183 * 37690638949867633420781
Date June 26, 2023 ( Patrick De Geest / CUDA program by Robert Xiao )
[180]
F_ba 36097712499280391828621137096 =
2 * 2 * 2 * 23 * 17374519 * 11291433167432418601
F_pt 651522423840351916117173996292699371711619153048324225156 =
61 * 26177508257 * 22605886429946461
Date June 26, 2023 ( Patrick De Geest / CUDA program by Robert Xiao )
[179]
F_ba 32972288624426519581946337409 =
7 * 21211 * 91733 * 2420830108182706049
F_pt 543585908566243233447813416222614318744332342665809585345 =
5 * 13 * 13 * 17 * 223087 * 5144452996386259291
Date June 26, 2023 ( Patrick De Geest / CUDA program by Robert Xiao )
[178]
F_ba 26703535701134165007419065382 =
2 * 7 * 19 * 197 * 509590009944928915068491
F_pt 356539409470873460765948942434249849567064378074904935653 =
3 * 31 * 71 * 27855293 * 145184328408801977
Date June 26, 2023 ( Patrick De Geest / CUDA program by Robert Xiao )
In the palindrome not a single digit is repeated ie. is followed by itself!
[177]
F_ba 19232217442397011118856549521 =
446231 * 1356337 * 14790001 * 2148492343
F_pt 184939093875819915846819675717576918648519918578390939481 =
3 * 7 * 7 * 163 * 90134333 * 4452502545902597
Date June 26, 2023 ( Patrick De Geest / CUDA program by Robert Xiao )
[176]
F_ba 4171522634325685120423923187 =
17 * 8513 * 9324115133 * 3091401562559
F_pt 8700800544345751829458446650566448549281575434450080078 =
2 * 1042880658581421280105980797
Date June 14, 2023 ( Patrick De Geest / CUDA program by Robert Xiao )
[175]
F_ba 3479635046544710084611458996 =
2 * 2 * 3 * 13 * 17333 * 22541 * 249589 * 228737168623
F_pt 6053930028571103358472012555552102748533011758200393506 =
226127976707 * 15387901564490471
Date June 14, 2023 ( Patrick De Geest / CUDA program by Robert Xiao )
[174]
F_ba 1712322112672987930004654686 =
2 * 13 * 65858542795114920384794411
F_pt 1466023508774442385882534859584352885832444778053206641 =
7 * 11 * 29 * 73 * 24118573 * 435534209879291
Date June 14, 2023 ( Patrick De Geest / CUDA program by Robert Xiao )
[173]
F_ba 1694934457715605943771327513 =
14048291 * 120650580039636561043
F_pt 1436401407975847596488073464643708846957485797041046341 =
3 * 11 * 409 * 32917 * 2898503 * 658099655831
Date June 14, 2023 ( Patrick De Geest / CUDA program by Robert Xiao )
[172]
F_ba 1608843601060583225861746481 =
1447 * 3121 * 51343031 * 6938570803873
F_pt 1294188866336792535757443701073447575352976336688814921 =
3 * 7 * 139838765279 * 273928334169899
Date June 14, 2023 ( Patrick De Geest / CUDA program by Robert Xiao )
[171]
F_ba 778304443341202475722787762 =
2 * 114298534813 * 3404700001686637
F_pt 302878903262329527375854916619458573725923262309878203 =
3 * 3 * 7 * 11 * 13 * 31 * 71 * 4097447 * 9579427075181
Date November 21, 2022 ( David Griffeath / CUDA program by Robert Xiao )
[170]
F_ba 527704544889979086555637821 =
3 * 3 * 161057597 * 364055092173694777
F_pt 139236043348769976436410774477014634679967843340632931 =
11 * 23 * 704441 * 2028239 * 729922184813
Date November 21, 2022 ( David Griffeath / CUDA program by Robert Xiao )
[169]
F_ba 464762829353941815880187201 =
Prime!
F_pt 108002243774540620212510333333015212026045477342200801 =
3 * 11 * 486924413011 * 14461918241627
Date November 21, 2022 ( David Griffeath / CUDA program by Robert Xiao )
[168]
F_ba 317588942343943926619604585 =
5 * 7 * 31 * 67 * 1471 * 119191 * 24917513135623
F_pt 50431368149572469821594569096549512896427594186313405 =
3 * 3 * 11 * 11 * 241 * 479 * 20809 * 60702102169987
Date November 21, 2022 ( David Griffeath / CUDA program by Robert Xiao )
[167]
F_ba 194997423092985883324586013 =
3 * 64999141030995294441528671
F_pt 19011997506452472127996132423169972127425460579911091 =
31 * 103 * 491 * 15061 * 4129188007270849
Date November 21, 2022 ( David Griffeath / CUDA program by Robert Xiao )
[166]
F_ba 193338224725418777596313553 =
3 * 13 * 7243 * 11657443117 * 58712602217
F_pt 18689834569988266379838452825483897366288996543898681 =
7 * 281 * 81799 * 600807542987076769
Date November 21, 2022 ( David Griffeath / CUDA program by Robert Xiao )
[165]
F_ba 173886091636007644064387226 =
2 * 3 * 3 * 9660338424222646892465957
F_pt 15118186432223023774728508780582747732032223468181151 =
11 * 233 * 27010158827 * 2511823400027
Date November 21, 2022 ( David Griffeath / CUDA program by Robert Xiao )
[164]
F_ba 62665727903192949362246738 =
2 * 11 * 11 * 167 * 9601 * 1943141 * 83114628187
F_pt 1963496726818507700938130110318390077058186276943691 =
3 * 19 * 1099398735143735953723627
Date November 21, 2022 ( David Griffeath / CUDA program by Robert Xiao )
[163]
F_ba 47790468351311396682904866 =
2 * 3 * 3 * 8159951 * 325372789556861287
F_pt 1141964432618848122919519889159192218488162344691411 =
11 * 409 * 10622464625763813443633
Date November 21, 2022 ( David Griffeath / CUDA program by Robert Xiao )
[162]
F_ba 34925848025358339344648596 =
2 * 2 * 2063 * 10258067 * 412593336100969
F_pt 609907430145213505805139181931508505312541034709906 =
29 * 14873101 * 115551731 * 700762703
Date November 9, 2022 ( David Griffeath / Rust program by Robert Xiao )
[161]
F_ba 34570045466623693915618365 =
3 * 3 * 5 * 37 * 20762790070044260609981
F_pt 597544021782214705597988959889795507412287120445795 =
7 * 13 * 474779 * 400070990422214647
Date November 9, 2022 ( David Griffeath / Rust program by Robert Xiao )
[160]
F_ba 16113599345002654884269393 =
47 * 2851 * 12791 * 9401409868006259
F_pt 129824041925634994253924353429352499436529140428921 =
3 * 3 * 557 * 9717919 * 165383274260251
Date November 9, 2022 ( David Griffeath / Rust program by Robert Xiao )
[159]
F_ba 4063560947310268109225792 =
2 * 2 * 2 * 2 * 2 * 2 * 29 * 571 * 13271957 * 288906778631
F_pt 8256263786252561776297593957926771652526873626528 =
3 * 7 * 13 * 389 * 9403 * 54492547 * 74677709
Date November 6, 2022 ( David Griffeath / Rust program by Robert Xiao )
[158]
F_ba 4052563875295474515566672 =
2 * 2 * 2 * 2 * 232207 * 1090773500393903531
F_pt 8211636981674937160312274722130617394761896361128 =
3 * 19 * 55487 * 1281338184570963047
Date November 6, 2022 ( David Griffeath / Rust program by Robert Xiao )
[157]
F_ba 3283130995819972903686354 =
2 * 3 * 92227 * 1516000579 * 3913627723
F_pt 5389474567856923467988997998897643296587654749835 =
5 * 11 * 47 * 211 * 6019289183532360233
Date November 6, 2022 ( David Griffeath / Rust program by Robert Xiao )
[156]
F_ba 3281940831768274049845645 =
5 * 656388166353654809969129
F_pt 5385567811613915254381275721834525193161187655835 =
7 * 656479 * 888157 * 402061168363
Date November 6, 2022 ( David Griffeath / Rust program by Robert Xiao )
[155]
F_ba 2669765477898748990170342 =
2 * 3 * 47 * 79 * 167 * 460039 * 1559860610753
F_pt 3563823853489967791349267629431977699843583283653 =
110323 * 24199536614293927741
Date November 6, 2022 ( David Griffeath / Rust program by Robert Xiao )
[154]
F_ba 1972088846281593888294413 =
11 * 11 * 47 * 53 * 1871 * 3496983566911873
F_pt 1944567208814134024246830386424204314188027654491 =
3 * 328681474380265648049069
Date November 6, 2022 ( David Griffeath / Rust program by Robert Xiao )
[153]
F_ba 1920508288374901008727646 =
2 * 64399 * 14911010173876154977
F_pt 1844176042858345966511280821156695438582406714481 =
3 * 3 * 3 * 11 * 11 * 32189 * 18262472141290969
Date November 5, 2022 ( David Griffeath / Rust program by Robert Xiao )
[152]
F_ba 1535048277308111618202058 =
2 * 107487646451 * 7140580001479
F_pt 1178186606833300573192218122913750033386066818711 =
7 * 13 * 139 * 2521 * 26029 * 220469 * 8388571
Date November 6, 2022 ( David Griffeath / Rust program by Robert Xiao )
[151]
F_ba 1425999292822499645424473 =
13 * 8744159989 * 12544630179689
F_pt 1016736991565134544383803083834454315651996376101 =
3 * 3 * 373 * 39283579 * 5406633426979
Date November 6, 2022 ( David Griffeath / Rust program by Robert Xiao )
[150]
F_ba 128184152897963669861552 =
2 * 2 * 2 * 2 * 11 * 17 * 127 * 1873 * 3581 * 50295245131
F_pt 8215588527084263951180440811593624807258855128 =
3 * 3243912979 * 13171762387769
Date January 5, 2008 ( Feng Yuan )
[149]
F_ba 115945380367510626966648 =
2 * 2 * 2 * 3 * 193 * 8017 * 3122288398366517
F_pt 6721665614283359365218338125639533824165661276 =
7 * 11 * 131 * 1543 * 169633 * 43915229833
Date January 5, 2008 ( Feng Yuan )
[148]
F_ba 114860741505066911342296 =
2 * 2 * 2 * 7 * 7 * 7 * 11 * 31 * 468151 * 262208746399
F_pt 6596494969546900318903773098130096459694946956 =
101 * 479 * 15420611 * 153961859113
Date January 5, 2008 ( Feng Yuan )
[147]
F_ba 26539182748774333465497 =
3 * 3 * 3 * 677 * 19661 * 141101 * 523358663
F_pt 352164110486420593101030101395024684011461253 =
11 * 509 * 123553 * 19181995591667
Date January 5, 2008 ( Feng Yuan )
[146]
F_ba 24864166542722584070757 =
3 * 1297 * 128393 * 49770422637839
F_pt 309113388932122569558171855965221239883311903 =
11083 * 48857 * 22959360170209
Date December 6, 2022 ( Robert Xiao )
[145]
F_ba 14170433757425168663998 =
2 * 3007159 * 2356116480276761
F_pt 100400596436787391913333319193787634695004001 =
13 * 71 * 6469 * 2373254385391177
Date December 6, 2022 ( Robert Xiao )
[144]
F_ba 5805651147054009150041 =
1429 * 153395789 * 26485322761
F_pt 16852792620644766088388388066744602629725861 =
3 * 3 * 3 * 11 * 11 * 13 * 17 * 23 * 281 * 153469 * 4053449
Date January 5, 2008 ( Feng Yuan )
[143]
F_ba 3630238456595636157728 =
2 * 2 * 2 * 2 * 2 * 113444951768613629929
F_pt 6589315625872933253747473523392785265139856 =
3 * 23 * 29 * 17234557 * 105265956197
Date January 2, 2008 ( Feng Yuan )
[142]
F_ba 3261059106801402665754 =
2 * 3 * 3 * 3 * 3 * 3 * 23 * 23 * 149 * 4477841 * 19011299
F_pt 5317253249026181079052509701816209423527135 =
5 * 23 * 23 * 149 * 652211821360280533151
Date January 2, 2008 ( Feng Yuan )
[141]
F_ba 775781766082836455602 =
2 * 11 * 337817 * 104384348772323
F_pt 300918674293302389819918983203392476819003 =
167 * 4645399796903212309
Date May 31, 2001
Note the large palindromic substring in the largest prime factor
4645399796903212309
[140]
F_ba 585863634063453017106 =
2 * 3 * 19 * 83 * 61917526322495563
F_pt 171618098859017793192291397710958890816171 =
11 * 53260330369404819737
Date April 18, 2001
[139]
F_ba 333741662509416495210 =
2 * 3 * 3 * 5 * 97 * 127 * 227 * 4523 * 293183831
F_pt 55691748647274629891519892647274684719655 =
17 * 37 * 530590878393348959
Date April 18, 2000
[138]
F_ba 333470345690102634085 =
5 * 23 * 21661 * 303959 * 440418821
F_pt 55601235727338276212021267283372753210655 =
100045471 * 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 !
[137]
F_ba 191600462227318472121 =
2 * 3 * 17 * 47 * 197 * 503 * 8693 * 92795011
F_pt 18355368562861046305450364016826586355381 =
2477 * 38675910825054193
Date April 22, 2000
[136]
F_ba 35369824079822102851 =
53 * 667355171317398167
F_pt 625512227718781732353237187817722215526 =
2 * 30253 * 8817409 * 33148469
Date October 28, 1999 (after a two-year period of nonactivity...)
[135]
F_ba 32637734114649927570 =
2 * 3 * 3 * 5 * 37 * 113 * 86735587219033
F_pt 532610844069291845737548192960448016235 =
7 * 11 * 47 * 193 * 241027 * 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
[134]
F_ba 19123354745855372721 =
3 * 17209 * 1199089 * 308912707
F_pt 182851348367914603505306419763843158281 =
9561677372927686361
Date August 2, 1997
The following basenumbers start and end with the same digits :
[122] 19 35755375665009 721
[134] 19 123354745855372 721
A lot of palindromic subgroups can be extracted from this basenumber
19123354745855372721
19123354745855372721
19123354745855372721
19123354745855372721
19123354745855372721
19123354745855372721
19123354745855372721
19123354745855372721
Confert [131] the 'universal' number 42 is present :
Add the digits of the above basenumber together and divide by 2 !
[133]
F_ba 16175904024612952346 =
2 * 3187 * 5347 * 474620309557
F_pt 130829935506744754616457447605539928031 =
3 * 7 * 13 * 746939 * 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)
Sum up all the digits of the basenumber to arrive at a palindrome namely 77 .
Note that 77 x 88 = 6776 !
The summation of the digits at either side of the middle number 1 of the palindromic
triangular number is the palindrome 88 .
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 !
[132]
F_ba 11665833272979576316 =
2 * 2 * 9091 * 17761 * 18062451829
F_pt 68045832976478686977968687467923854086 =
11 * 844511 * 1255792165577
Date January 29, 1997
The number 131 can be written as the sum of three consecutive primes
131 = 41 + 43 + 47
[131]
F_ba 3654345456545434563 = PALINDROMIC
3 * 11 * 11 * 1429 * 9091 * 774923959
F_pt 6677120357887130286820317887530217766 =
2 * 17 * 101 * 149 * 849217 * 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
[130]
F_ba 3652242206567626036 =
2 * 173 * 2957 * 1784851760869
F_pt 6669436567716980985890896177656349666 =
43 * 285451 * 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
[129]
F_ba 3637582781740258588 =
2 * 2 * 19 * 37 * 3930491 * 329117339
F_pt 6616004247006598875788956007424006166 =
7 * 277 * 320741 * 5848986211
Date May 16, 1996
[128]
F_ba 3405603489389985674 =
2 * 7 * 43 * 467 * 29027 * 417328993
F_pt 5799067563472623134313262743657609975 =
3 * 5 * 5 * 11 * 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 [22]!
[127]
F_ba 3333229904907177885 =
3 * 3 * 5 * 29531 * 2508271838563
F_pt 5555210799483757064607573849970125555 =
13 * 109 * 761389 * 1544752211
Date April 25, 1996
[126]
F_ba 3271889143987456170 =
2 * 3 * 5 * 79 * 83 * 421 * 24109 * 1638743
F_pt 5352629285271484348434841725829262535 =
239 * 2023027 * 6767044007
Date April 22, 1996
[125]
F_ba 3237532871306389774 =
2 * 293 * 1483 * 1097909 * 3393197
F_pt 5240809546394698286828964936459080425 =
5 * 5 * 7 * 107 * 172898951738659
Date April 20, 1996
[124]
F_ba 2672899778869085022 =
2 * 3 * 83 * 241 * 499 * 727 * 61390523
F_pt 3572196613939201806081029393166912753 =
359 * 5941253 * 1253170549
Date April 12, 1996
[123]
F_ba 2470152923949718922 =
2 * 311 * 26293 * 151040465207
F_pt 3050827733848672937392768483377280503 =
3 * 73 * 571 * 19753480027427
Date April 6, 1996
[122]
F_ba 1935755375665009721 =
233 * 4721 * 1759788812897
F_pt 1873574437207991455541997027344753781 =
3 * 103 * 3132290251885129
Date March 25, 1996
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
[121]
F_ba 775527779120670322 =
2 * 1153 * 336308663972537
F_pt 300721668093919607706919390866127003 =
11 * 70502525374606393
Date November 17, 1995
[120]
F_ba 352465914177746748 =
2 * 2 * 3 * 3 * 3 * 271967 * 11999887043
F_pt 62116110328577368186377582301161126 =
107 * 413461 * 7967072987
Date December 22, 1995
[119]
F_ba 352449681516778876 =
2 * 2 * 19 * 30338993 * 152855957
F_pt 62110389000639430803493600098301126 =
43 * 8196504221320439
Date December 22, 1995
[118]
F_ba 247184344095525797 =
913687 * 270535034531
F_pt 30550049982967649594676928994005503 =
3 * 13 * 78367 * 40438322923
Date March 29, 1992
[117]
F_ba 112540786741633183 =
17 * 67 * 173 * 22277 * 25637957
F_pt 6332714340212879669782120434172336 =
2 * 2 * 2 * 2 * 11 * 81239 * 3935525353
Date October 11, 1995
[116]
F_ba 109113947229482909 =
7 * 41 * 47 * 843559 * 9589259
F_pt 5952926739999190550919999376292595 =
3 * 3 * 5 * 11 * 13 * 17 * 498715422229
Date September 30, 1995
[115]
F_ba 108100203429839929 =
7 * 11 * 17 * 43 * 367 * 6343 * 825007
F_pt 5842826990786388228836870996282485 =
5 * 10810020342983993
Date September 27, 1995
[114]
F_ba 78358935395500837 =
7 * 11 * 719 * 29501 * 47976899
F_pt 3070061378158136996318518731600703 =
13 * 17 * 89 * 769 * 4679 * 553601
Date October 2, 1995
[113]
F_ba 26555908318527497 =
7 * 13 * 3931 * 74236369457
F_pt 352608133311018969810113331806253 =
3 * 73 * 60629927667871
Date April 24, 1992
[112]
F_ba 10686132500837474 =
2 * 5351 * 165569 * 6030823
F_pt 57096713912727488472721931769075 =
3 * 5 * 5 * 11 * 181 * 14591 * 4904593
Date October 13, 1993
Test for divisibility by 111
[111]
F_ba 4500990532341401 =
7 * 12163 * 219311 * 241051
F_pt 10129457886113466431168875492101 =
3 * 11 * 29 * 223 * 10545357391
Date July 5, 1991
[110]
F_ba 4062262265478592 =
2 * 2 * 2 * 2 * 2 * 2 * 7 * 365201 * 24828929
F_pt 8250987356765633365676537890528 =
11 * 13513 * 19853 * 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
[109]
F_ba 3729035890325943 =
3 * 3 * 43 * 811 * 26951 * 440849
F_pt 6952854335669501059665334582596 =
2 * 2 * 11 * 4106521 * 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.
[108]
F_ba 3249909720631025 =
5 * 5 * 47 * 2765880613303
F_pt 5280956596126015106216956590825 =
3 * 3 * 3 * 6991 * 41141 * 209249
Date October 11, 1990
[107]
F_ba 362785412063728 =
2 * 2 * 2 * 2 * 269 * 929 * 2699 * 33617
F_pt 65806627603124642130672660856 =
29 * 307 * 2341 * 17406523
Date February 16, 1990
[106]
F_ba 353520620692923 =
3 * 37 * 167 * 563 * 33874033
F_pt 62488414627554945572641488426 =
2 * 47 * 4637 * 405527029
Date February 12, 1990
[105]
F_ba 326825312860029 =
3 * 19 * 5733777418597
F_pt 53407392563028082036529370435 =
5 * 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
[105] ***.***.***.860.029
[104] 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 !
[104]
F_ba 173008538941238 =
2 * 7 * 11 * 11 * 13 * 7856168329
F_pt 14965977273291019237277956941 =
3 * 3 * 349 * 21317 * 2583887
Date January 16, 1990
Consider the following basenumber
[104] 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.
[103]
F_ba 164402190438221 =
36319 * 4526616659
F_pt 13514040110442624401104041531 =
3 * 11 * 139 * 3121 * 5741893
Date January 14, 1990
[102] 164.***.190.***.221
***.402.***.438.***
Notice that 164 + 190 + 221 = 575 is palindromic.
[102]
F_ba 109237730189290 =
2 * 5 * 1503113 * 7267433
F_pt 5966440848454114548480446695 =
11 * 19 * 73 * 389 * 1409 * 13063
The prime number 101 can be written as the sum of five consecutive primes
101 = 13 + 17 + 19 + 23 + 29
[101]
F_ba 50364608806281 =
3 * 7 * 11 * 37 * 103 * 57210341
F_pt 1268296910104884010196928621 =
349 * 12899 * 5593891
A palindromic subgroup of 6 digits exists in the basenumber
50364608806281
[100]
F_ba 41801907377572 =
2 * 2 * 83 * 22769 * 5529859
F_pt 873699730201575102037996378 =
7 * 31 * 89 * 223 * 617 * 15731
[
TOP OF PAGE]
Patrick De Geest - Belgium 
- Short Bio - Some Pictures
E-mail address : pdg@worldofnumbers.com
Details of Palindromic Triangulars
The Details [1] to [49] 
