Keith Number
A Keith number is an 
-digit integer 
such that if a Fibonacci-like sequence (in which each term in the sequence is the sum of the 
previous terms) is formed with the first 
terms taken as the decimal digits of the number 
, then 
itself occurs as a term in the sequence. For example, 197 is a Keith number since it generates the sequence 1, 9, 7, 
, 
, 
, 
, 
, ... (Keith). Keith numbers are also called repfigit (repetitive fibonacci-like digit) numbers.
There is no known general technique for finding Keith numbers except by exhaustive search. Keith numbers are much rarer than the primes, with only 84 Keith numbers with 
digits. The first few are 14, 19, 28, 47, 61, 75, 197, 742, 1104, 1537, 2208, 2580, 3684, 4788, 7385, 7647, 7909, ... (OEIS A007629). As of Mar. 31, 2006, there are 95 known Keith numbers (Keith). The number of Keith numbers having 
, 2, ... digits are 0, 6, 2, 9, 7, 10, 2, 3, 2, 0, 2, 4, 2, 3, 3, 3, 5, 3, 5, 3, 1, 1, 3, 1, 1, 3, 7, 1, 2, 5, 2, 4, 6, 3, ... (OEIS A050235), as summarized in the following table.
 |
 -digit Keith numbers |
| 2 |
14, 19, 28, 47, 61, 75 |
| 3 |
197, 742 |
| 4 |
1104, 1537, 2208, 2580, 3684, 4788, 7385, 7647, 7909 |
| 5 |
31331, 34285, 34348, 55604, 62662, 86935, 93993 |
| 6 |
120284, 129106, 147640, 156146, 174680, 183186, 298320, 355419, 694280, 925993 |
| 7 |
1084051, 7913837 |
| 8 |
11436171, 33445755, 44121607 |
| 9 |
129572008, 251133297 |
| 10 |
(none) |
| 11 |
24769286411 96189170155 |
| 12 |
171570159070, 202366307758, 239143607789, 296658839738 |
| 13 |
1934197506555, 8756963649152 |
| 14 |
43520999798747, 74596893730427, 97295849958669 |
| 15 |
120984833091531, 270585509032586, 754788753590897 |
| 16 |
3621344088074041, 3756915124022254, 4362827422508274 |
| 17 |
11812665388886672, 14508137312404344, 16402582054271374, 69953250322018194, 73583709853303061 |
| 18 |
119115440241433462, 166308721919462318, 301273478581322148 |
| 19 |
1362353777290081176, 3389041747878384662, 5710594497265802190, 5776750370944624064, 6195637556095764016 |
| 20 |
12763314479461384279, 27847652577905793413, 45419266414495601903 |
| 21 |
855191324330802397989 |
| 22 |
7657230882259548723593 |
| 23 |
26842994422637112523337, 36899277593852609997403, 61333853602129819189668 |
| 24 |
229146413136585558461227 |
| 25 |
9838678687915198599200604 |
| 26 |
18354972585225358067718266, 19876234926457288511947945, 98938191214220718050301312 |
| 27 |
153669354455482560987178342, 154677881401007799974564336, 133118411174059688391045955, 154140275428339949899922650, 295768237361291708645227474, 956633720464114515890318410, 988242310393860390066911414 |
| 28 |
9493976840390265868522067200 |
| 29 |
41796205765147426974704791528, 70267375510207885242218837404 |
| 30 |
127304146123884420932123248317, 389939548933846065763772833753, 344669719564188054170496150677, 756672276587447504826932994366, 534139807526361917710268232010 |
| 31 |
1598187483427964679092074853838, 2405620130870553672640058975437 |
| 32 |
41030306579725050560909919549414, 47824404246899742508216679149392, 42983394195992223818343905028410, 89980815134051887612993101615858 |
| 33 |
172451142646837728336412943204299, 193962639439026709638083447831059, 381933008901296879565658130750756, 359253598248137147666007355623218, 303294117104027490007126494842828, 312736110821858321305917486145434 |
| 34 |
1876178467884883559985053635963437, 2787674840304510129398176411111966, 5752090994058710841670361653731519 |
It is not known if there are an infinite number of Keith numbers.
The known prime Keith numbers are 19, 47, 61, 197, 1084051, 74596893730427, ... (OEIS A048970).
The 26-digit Keith number 98938191214220718050301312 was found in 2004 by D. Lichtblau using integer linear programming to solve the relevant Diophantine equations in the Wolfram Language. D. Lichtblau found all 30- and 31-digit Keith numbers on Jun. 23, 2009, and all 32-, 33-, and 34-digit Keith numbers on Aug. 26, 2009. The largest of these is 5752090994058710841670361653731519, which is the largest Keith number known as of August 2009.
Similarly, a reverse Keith, or revrepfigit (reverse replicating Fibonacci-like digit) number, is an 
-digit integer 
such that if a Fibonacci-like sequence (in which each term in the sequence is the sum of the previous terms) is formed with the first terms taken as the decimal digits of the number 
, then the reversal of 
occurs as a term in the sequence. For example, 341 is a revrepfigit since it generates the sequence 3, 4, 1, 
, 
, 
, 
, 
, 
.
The currently known revrepfigits are 12, 24, 36, 48, 52, 71, 341, 682, 1285, 5532, 8166, 17593, 28421, 74733, 90711, 759664, 901921, 1593583, 4808691, 6615651, 6738984, 8366363, 8422611, 26435142, 54734431, 57133931, 79112422, 351247542, 428899438, 489044741, 578989902, 3207761244, 4156222103, 5426705064, 5785766973, 6336657062, 48980740972, 51149725354, 83626284302, 94183600081, 98665175305, 1935391095868, 6002181268035, 6334708806271, 12348924235856, 27488180694681, 76365591939888, 309217509306732, 352062080376812, 714692062325732, 723735537269331, 2437358882180001, 6792079280704301, 62244424802562056, 203414193894268461, 217049132946408803, 415499563488189604, 561624665953167171, ... (OEIS A097060; A. Vrba, pers. comm., Dec. 28, 2006). Notice there are no numbers ending with zeros; they are not permitted since the zeros would be dropped upon reversal. But terms with internal zeros such as 90711 are allowed. The known prime revrepfigits are 71, 1593583, and 54734431 (A. Vrba, pers. comm,., Dec. 28, 2006).
It is not known if there are infinitely many revrepfigit numbers.
REFERENCES:
Esche, H. A. "Non-Decimal Replicating Fibonacci Digits." J. Recr. Math. 26, 193-194, 1994.
Heleen, B. "Finding Repfigits--A New Approach." J. Recr. Math. 26, 184-187, 1994.
Keith, M. "Repfigit Numbers." J. Recr. Math. 19, 41-42, 1987.
Keith, M. "All Repfigit Numbers Less than 100 Billion (
)." J. Recr. Math. 26, 181-184, 1994.
Keith, M. "Keith Numbers." https://users.aol.com/s6sj7gt/mikekeit.htm.
Keith, M. "Determination of All Keith Numbers Up to 
." https://users.aol.com/s6sj7gt/keithnum.htm.
Lichtblau, D. "Solving Knapsack and Related Problems." International Mathematica Symposium 2004. Banff, Canada, 2004.
Pickover, C. "All Known Replicating Fibonacci Digits Less than One Billion." J. Recr. Math. 22, 176, 1990.
Piele, D. "Mathematica Pearls: Keith Numbers." Mathematica Res. Educ. 6, No. 3, 50-52, 1997.
Piele, D. "Mathematica Pearls: Keith Numbers." Mathematica Res. Educ. 7, No. 1, 44-45, 1998.
Robinson, N. M. "All Known Replicating Fibonacci Digits Less than One Thousand Billion (
)." J. Recr. Math. 26, 188-191, 1994.
Sherriff, K. "Computing Replicating Fibonacci Digits." J. Recr. Math. 26, 191-193, 1994.
Sloane, N. J. A. Sequences A007629, A048970, A050235, and A097060 in "The On-Line Encyclopedia of Integer Sequences."
"Table: Repfigit Numbers (Base 
) Less than 
." J. Recr. Math. 26, 195, 1994.
