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196 AND OTHER LYCHREL NUMBERS

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Welcome to p196.org!

If more people are working on the 196 quest, I thought I would provide a location for someone to come to and verify the numbers that they have found. If yours and mine don't match, LET ME KNOW!!

Eric Goldstein has pointed out that some of the files are not the first iteration of the dataset. (See the Blackboard entry for 8/6/02) For example, the 2,000,244 and 14,000,008 files shown below are in fact, the second iteration of each. I originally thought that all of these files were the first iteration to contain the stated digit length, but I guess not... I do still believe that they are all correct for the given iteration!!

In addition, Beth Cooper and Chun Yi asked to see the spreadsheet that I mentioned somewhere on the site, that I use to track the status of my progress. I don't know if anyone else is interested, and I don't even know what value it could be to someone, but here it is in a zipped Excel file. Maybe there is value to people like Eric Goldstein, who was verifying some of my work, by running 196 also. I don't intend to update it except maybe rarely, but it is current as of this afternoon (10/7/04). Anyway..... It is there if someone wants it. If nothing else, it shows a really nice "time line" of how I have moved forward!!

Towards the bottom of the page, is a program that you can download, that was written by Ben Despres. With some formatting, you can load your number into his program, and verify that any iteration you are on, is giving you the correct answer.

Here are the first 25 digits of every data file that I have..... I hope it is useful to someone. :-)

NOTE: As I've fallen behind in updates, this page has gotten a bit burdensome to keep updated. In addition, I really don't think adding every milestone's data is of any use to anyone, since I don't know of anyone duplicating my work this far out. As a result, I've discontinued keeping it totally current. If someone decided that they really need to know the first 25 digits of a dataset not listed here, let me know, and I will email you the information directly. I'm sure everyone understands.

Digit Count Iteration First 25 Digits
1,000,000 2,415,836 1321620866345900792403087
2,000,000 4,830,292 1771479103435526006981020
2,000,244 4,830,868 4966682241339127981797944
3,000,000 7,245,928 1572127370436337959801040
4,000,000 9,662,071 1189951505503730299942650
5,000,000 12,078,865 1786916673224171246235654
6,000,000 14,495,294 1717694931286918098671266
7,000,000 16,911,863 1370104827081045647878991
8,000,000 19,327,667 1019762622265770924957665
9,000,000 21,744,278 1877698672190182856774072
10,000,000 24,159,531 1570638871513424055539043
11,000,000 26,574,642 1552863515777339147880472
12,000,000 28,991,159 1094658057189398961997993
13,000,000 31,408,295 1570104209676429293120872
14,000,000 33,824,775 1760057992641613493083452
14,000,008 33,824,793 4479163772166847272965861
15,000,000 36,240,602 1031240279852766564723228
16,000,000 38,655,227 1765950614893173230506692
17,000,000 41,070,365 1326956429587373980563323
18,000,000 43,487,182 1023601015155152788222220
19,000,000 45,901,512 1061077962735256900716595
20,000,000 48,316,988 1055357468774267123532056
21,000,000 50,732,545 1366299557692634038868234
22,000,000 53,148,015 1618818677775920355359715
23,000,000 55,565,809 1314877753155237116425525
24,000,000 57,982,436 1011377187109780706245147
25,000,000 60,398,130 1090994049526433659630228
26,000,000 62,816,046 1117977825788778867038244
27,000,000 65,231,462 1076857885074947084574730
28,000,000 67,648,377 1823526487861564244724874
29,000,000 70,067,270 1751451457695893086640504
30,000,000 72,482,579 7983997334581381745393030
31,000,000 74,899,358 1782282447580070383893664
32,000,000 77,314,955 1758023338115480171010379
33,000,000 79,731,034 1191494564496990603108460
34,000,000 82,145,364 1776180000353277502063208
35,000,000 84,563,227 9667719580493752405737223
36,000,000 86,979,674 1260718990353737727166287
37,000,000 89,397,247 1713947301231045001039844
38,000,000 91,812,894 1083367793408406027652822
39,000,000 94,228,899 1723587855017460734098176
40,000,000 96,645,213 1796080271215504475068003
41,000,000 99,061,141 1312344045189134863351509
41,388,546 100,000,000 9974438904743885498437291
42,000,000 101,476,346 1323537379287526648818853
43,000,000 103,893,138 1435691352112910044161238
44,000,000 106,309,010 1782430225531195634814511
44,444,444 107,382,665 1834222638993065199417095
45,000,000 108,723,539 1754201184575564515873759
46,000,000 111,138,308 1031943459984297079060569
47,000,000 113,554,695 1714140231595266216147023
48,000,000 115,969,543 1027319277035757930071344
49,000,000 118,387,016 1682327227245818701228607
50,000,000 120,802,305 1132236530787968938016671
51,000,000 123,218,044 7189775322254585819022781
51,737,154 125,000,000 1407502882820027975246439
52,000,000 125,634,154 1328004039944465751897404
53,000,000 128,050,160 1359977662868610427305071
54,000,000 130,463,975 1881923366313273415179428
55,000,000 132,880,223 1033298508834332407949161
56,000,000 135,295,252 1753668819826260460568993
57,000,000 137,712,060 1618620689808652775577199
58,000,000 140,128,242 1775871269524132894299290
59,000,000 142,544,153 1609879637228067420870014
60,000,000 144,958,406 1960882487804145715718858
61,000,000 147,375,122 1727971809395357592464645
62,000,000 149,790,895 1715004244814853908992584
62,086,375 150,000,000 1642624515892626147533628
63,000,000 152,207,835 1776989222706458077880113
64,000,000 154,623,289 1051257477717223642530126
65,000,000 157,038,002 1780395318130583334332747
66,000,000 159,455,103 1511850227165859502646376
67,000,000 161,871,411 1595889760479982515515654
68,000,000 164,287,044 1771960332000080473579801
69,000,000 166,704,687 1757016083265992389585326
70,000,000 169,121,766 1315049565144120454520311
71,000,000 171,538,507 1809597642702333754908891
72,000,000 173,953,453 1015802779684755714656655
72,433,237 175,000,000 1316559584457038932346512
73,000,000 176,368,856 1632106585426253096555012
74,000,000 178,784,747 1986111000106825979146786
75,000,000 181,199,797 1008705138345884691746272
76,000,000 183,616,094 1198699207272219566300138
77,000,000 186,032,643 1352672193204568014634580
78,000,000 188,447,228 1776756206228864215556798
79,000,000 190,862,368 1721124162365259992368277
80,000,000 193,277,613 1276307469099717799141965
81,000,000 195,693,825 1808312984821215632848674
82,000,000 198,109,015 1012480217638681141201751
83,000,000 200,524,477 1319596114795897939644010
84,000,000 202,939,323 1475989142103872062172123
85,000,000 205,355,672 1751939135396043222508585
86,000,000 207,771,197 1838830663332309393661252
87,000,000 210,186,316 1776793449035037308767206
88,000,000 212,602,940 1763659752403662711071184
89,000,000 215,018,872 1175499636497806735606789
90,000,000 217,433,686 1315668188291812932328716
91,000,000 219,850,418 1818340855113748165713548
92,000,000 222,266,315 1028371706886347137074883
93,000,000 224,680,750 1440330481696796849948050
94,000,000 227,095,084 1046866858222464987946767
95,000,000 229,512,704 1766913757711631595955714
96,000,000 231,928,518 1368479950294739386391187
97,000,000 234,344,371 1362828111646940179935559
98,000,000 236,759,404 1881507588454484016602013
99,000,000 239,175,318 1032406741781114441023479
100,000,000 241,590,277 1768299281546132721740051
101,000,000 244,006,839 1592228315604702996112807
102,000,000 246,423,282 1032307672289521340027980
103,000,000 248,840,359 1728363664385978446379558
104,000,000 251,254,880 1371628576057332728611441
105,000,000 253,670,404 1408171282423797037195446
106,000,000 256,084,270 1010961355482533511759009
107,000,000 258,498,530 1775655571279791294879229
108,000,000 260,914,435 1034968781871203163914679
109,000,000 263,328,945 1834854968067461127800545
110,000,000 265,745,475 1276319349159512469971672
111,000,000 268,160,697 1636030357158473358172278
112,000,000 270,575,863 1778781020534052343845183
113,000,000 272,991,794 1776694520310345481883637
114,000,000 275,405,890 1838339385684762471833877
115,000,000 277,820,730 1062513163198909662417864
116,000,000 280,236,256 1342599540844045251109166
117,000,000 282,651,618 1726560261621573127603656
118,000,000 285,066,604 1065645149596777550426168
119,000,000 287,481,429 1369987370648582204968231
120,000,000 289,896,907 1364754677893412956617791
121,000,000 292,311,641 1077725969274592498575304
122,000,000 294,727,521 1021367585335991684990409
123,000,000 297,143,533 1357593504958409900797997
124,000,000 299,557,657 1764378472069655752133500
125,000,000 301,974,027 1884349195617554876361307
126,000,000 304,389,137 1814691856257280526407595
127,000,000 306,805,299 1309531209941831133045217
128,000,000 309,221,256 1189989613884366553921768
129,000,000 311,637,141 1395051195770820383642774
130,000,000 314,052,527 1771135333660318106892108
131,000,000 316,468,413 1778391702927079758653571
132,000,000 318,884,030 1012725947322424967634762
133,000,000 321,299,093 1121041007140108077870210
134,000,000 323,715,600 1847779323392555637004235
135,000,000 326,131,076 1280083196671448796316209
136,000,000 328,547,998 1729238936136697427548393
137,000,000 330,963,887 1597118719384752284695424
138,000,000 333,379,370 1774619990337551728717581
139,000,000 335,795,486 1617678254614088001034677
140,000,000 338,210,775 1644699407125444079404186
141,000,000 340,626,948 1327486016146262830536853
142,000,000 343,043,659 1327710752846177644901575
143,000,000 345,459,815 1596411840598771306167516
144,000,000 347,875,673 1730214275086980347571340
145,000,000 350,292,487 1718265558795616271668352
146,000,000 352,708,290 1312106668717122045265595
147,000,000 355,125,177 1767142633789771140100238
148,000,000 357,540,909 1600598556023084120548792
149,000,000 359,956,876 1642997003809356893180691
150,000,000 362,372,813 1314637152473183654990226
151,000,000 364,788,891 1188332537874075530304529
152,000,000 367,204,161 1073006369604535918485472
153,000,000 369,619,640 1859555702340902960335797
154,000,000 372,035,743 1760875634779847532377788
155,000,000 374,450,389 1771357398392101622859349
156,000,000 376,866,208 1023524464013606297309847
157,000,000 379,281,305 1142611035475515082947139
158,000,000 381,697,126 1638277710993139992025743
159,000,000 384,113,355 1142471351629698087522809
160,000,000 386,529,610 1681205753431630755947393
161,000,000 388,944,754 1025169762868649578809387
162,000,000 391,360,944 1056390811637837662190790
163,000,000 393,776,305 1818281510468184428404022
164,000,000 396,192,233 1017374370556308805985779
165,000,000 398,607,648 1992009903767606779381369
166,000,000 401,023,403 1329054188054981518112269
167,000,000 403,439,651 1316910951280828179910916
168,000,000 405,856,956 1327045841005308772401100
169,000,000 408,272,435 1076970507213999804808448
170,000,000 410,688,792 1321151759791130796049814
171,000,000 413,104,065 1198549725898726826143301
172,000,000 415,517,439 1325272168128847516405635
173,000,000 417,933,403 1589743019357954569195326
174,000,000 420,349,873 1790003513535215165183808
175,000,000 422,765,244 1312399913323475917294279
176,000,000 425,181,055 1769432898013976506388766
177,000,000 427,599,027 1818626607326632254471135
178,000,000 430,016,536 1326521712418739845238528
179,000,000 432,433,414 1756008571102755130842700
180,000,000 434,849,911 1080033386138916259428564
181,000,000 437,267,293 1329090739338784333341325
182,000,000 439,682,132 1133666960819913715481109
183,000,000 442,097,346 1387429023786513990346910
184,000,000 444,512,168 1321031053840463298205400
185,000,000 446,927,453 1841359440807157555745530
186,000,000 449,343,164 1436621873243350049254447
187,000,000 449,343,164 1760077886420128171435125
188,000,000 454,174,655 1645558013977187718195235
189,000,000 456,588,593 1634325347132490436204748
190,000,000 459,004,983 1343550229273330059631588
191,000,000 461,419,218 1059776527743508311624176
192,000,000 463,835,090 1050534741172605814636542
193,000,000 466,250,388 1029883359151196593470782
194,000,000 468,667,675 1778455050891958552858248
195,000,000 471,085,391 1099325204644556770554785
196,000,000 473,503,339 1153735022700273560598532
197,000,000 475,919,151 1724946300386006686906998
198,000,000 478,335,425 1714396635836795679955072
199,000,000 480,750,860 1331657446855898449041806
200,000,000 483,166,859 1799071225594998429967931
201,000,000 485,581,911 1108284260244640194542287
202,000,000 487,996,904 1196020372202958470929067
203,000,000 490,413,021 1046351701504472273576819
204,000,000 492,829,843 1644462350518600021444956
205,000,000 495,245,922 1646167982208731045769174
206,000,000 497,662,056 1877755165049243939821476
206,967,875 500,000,000 8918589423163331686495122
207,000,000 500,077,852 1797488733775233079132274
208,000,000 502,495,484 1744941635998996581169339
209,000,000 504,913,664 1792907222802001951438933
210,000,000 507,329,950 1771175645046732089097139
211,000,000 509,745,400 1843704293845284058481146
212,000,000 512,162,233 1982904778200990006745751
213,000,000 514,577,796 1846165739251682244770519
214,000,000 516,992,665 1512971247633930532350888
215,000,000 519,408,582 1725317666128036787316284
216,000,000 521,825,691 1360831732483998591800659
217,000,000 524,242,175 1816927139005667363497058
218,000,000 526,658,409 1561620443180033117533292
219,000,000 529,075,350 1319808959325670088638232
220,000,000 531,491,865 1634273529758445307189566
221,000,000 533,907,350 1110877014860707713184428
222,000,000 536,322,341 1135599357769221630012377
223,000,000 538,737,263 1314487905309579712064310
224,000,000 541,153,446 1776752824611635706720478
225,000,000 543,569,184 1386065459714198997653566
226,000,000 545,986,095 1752395420852108596738179
227,000,000 548,402,519 1014834823371790173254625
228,000,000 550,818,415 1016277544726554817059913
229,000,000 553,233,019 1019406864798804620082115
230,000,000 555,650,203 1706148909500958140200122
231,000,000 558,065,511 1844448649515508246479390
232,000,000 560,480,471 1192405340671479434412558
233,000,000 562,896,706 1778095873670909753059975
234,000,000 565,311,914 1370675801717563608722225
300,000,000 724,756,966 1026597650727484099914982
325,000,000 785,147,039 1776200194178200876739710
350,000,000 845,541,500 1810506203693027164318537
375,000,000 905,939,525 1634027200677038929973552

NOTE: On December 29th, 2010, I completed everything through the 234 million data set a second time using Pierre's app on an different machine. This allowed a proof that the verification digits above are correct. If you get different results for any of the above datasets, let me know about it, but I think we'll find a mistake occured somewhere other than the table above.

Thanks.

******************************************************************

CHECKER APPLICATION FROM BEN DESPRES

******************************************************************

The file that you can download HERE is a program that Ben Despres wrote. It runs under windows. When it starts, it will display a file select window, where you can choose which file it is, that you want to check. It's easy enough, even for me.... :-)

He has included a text file with the zip, that goes into some detail of how his program works. I think the descriptions below are a faithful paraphrase of his text file, but if anyone finds a discrepancy between what is here, and what is on the text file with the program, trust the text file, NOT my writeup!! :-)

It works with numbers other than 196. I have tested it on my 879, 1997, 7059, 10553, 10563, 32419, 36973, 43844 files. Turns out that my five million digit 7059 file is wrong. :-( O well......

His MOD-9 discovery is something like this:

I discovered a really neat fact about the checksums of the digits generated by flip-and-add... At any given step, the checksum of the result will equal twice the checksum of the last result minus a multiple of nine (that multiple just happens to equal the number of carries). At first I thought this may lead to a "proof" of Lychrelness, but I have given up that idea for now. *BUT*, it *does* lead to a really simple check of whether or not a given number could have resulted on a given iteration, including in some cases whether or not a number could ever have resulted from a given starting number.

Basically, if Cx equals the MOD-9 sum at iteration X, with C0 meaning the same for the starting value:

Cx = (C0*(2^(x MOD 6))) MOD 9.


Actually, three distinct cycles exist, with only the longest using MOD 6... {0}, {3,6}, and {1,2,4,8,7,5}. 196 itself has the longest of those cycles (and no, unfortunately I could not find any patterns between cycle and converging-vs-conconverging).

So, although the CRC will only detect *new* errors, the MOD-9 checksum will (possibly) detect errors that have already occurred.

A couple of notes:

His application is written to look at a file which has been formatted in what has become known as "Istvan-Standard Formatting". :-) Anyway, the file should be in a format that looks like this:

Automatic save #12345 (CRC, optional, but there must be a number on position 17)
Initial value: 196 (number must begin on position 19)
Iteration: 2415836 (number must begin on position 19)
Number of digits: 1000000 (number must begin on position 19)
1321620866345900792403087149937072597108944871590062627103371677853101
0218789552032457507537977475995640190839955855884253327321850902244094
0920987126363322689017611006006500875888569961960897043759484747753087
......

The automatic save line: If I understand correctly, there MUST be a number on that line, starting on position 17. My thanks to Eric Goldstein for figuring this out. His app was generating the correct numbers, but his files were failing Ben's Checker program. This was why...

The initial value line is obviously just that. What number did you start with? 196, 879, 1997 etc. This will be used by the program for it's calculations. The number MUST begin on position 19.

The Iteration and Digits line are also used by the program to test the data. The number MUST begin on position 19.

The data itself is 70 characters wide, followed by an End of Line. This makes the file much nicer to look at, even if it adds little or no value. It was what Istvan decided on, and Ben's program used Istvan's formatting. (Hence the name "Istvan-Standard Formatting".) :-) He's pretty sure that nothing bad will happen to the file if you do not format it like the one above, but he is sure that it will crash in an interesting, unexpected way. My opinion is to watch out, and test a backup copy of the file!!

So far, the worst thing I have seen is a message that says the number generated is not correct, even though it is.

Any file not generated by Ben's program, (which is just about anything anyone else has!!) will always get a message that reads something like this:

Warning: Incorrect CRC-16 (0000) in file, should have B56F

In Ben's own words: I use a non-mirrored, non-reflected, 0XFFFF-in with no XOR-out, standard CRC-16 with a generator polynomial of 0x1021, in case anyone really needs to duplicate my numbers. That's part of his internal checking for the files written by his application. It's O.K. if this warning appears. In fact, you should expect it.

Then it will report on the actual length compared to reported length. (These two numbers should match!! :-) )

Then, if all is good, you will get a message that reads:

MOD-9 checksum (0) passed.

If not, it will read something like what I saw with some of my own previous files:

Warning: MOD-9 checksum (1) failed - The stored 23000000-digit number cannot occur on the stored iteration (55561532).

Ben's app will also display a "waterfall" image of the number, as well as some interesting statistic information about the number. I'm not going to go into it all here. Grab his file if you're interested.