# The First 200 Lucas numbers and their factors

This page follows on from an Introduction to the Lucas Numbers. They are a variation on The Fibonacci Numbers.

The Lucas numbers are defined very similarly to the Fibonacci numbers, but start with 2 and 1 (in this order) rather than the Fibonacci's 0 and 1:

L0 = 2
L1 = 1
Ln = Ln-1 + Ln-2 for n>1
This Maple program was used to produce the table below:
```lucas:=proc(n) option remember;
# this OPTION makes it very fast even though defined
# by using an inefficient form of recursion
if n=0 then  2
elif n=1 then  1
else lucas(n-1)+lucas(n-2)
fi
end;

seq(lprint(i,`:`,lucas(i),`=`,ifactor(lucas(i))),i=1..200);
```
and here is the output - a table of the first 200 Lucas numbers and their factors, where the prime numbers are indicated:

Each prime factor that is not a factor of any earlier Lucas number, is shown like this.
Index numbers that are prime are shown like this. n : Ln = Factors of Ln
0 : 2
1 : 1
2 : 3
3 : 4 = 22
4 : 7
5 : 11
6 : 18 = 2 x 32
7 : 29
8 : 47
9 : 76 = 22 x 19
10 : 123 = 3 x 41
11 : 199
12 : 322 = 2 x 7 x 23
13 : 521
14 : 843 = 3 x 281
15 : 1364 = 22 x 11 x 31
16 : 2207
17 : 3571
18 : 5778 = 2 x 33 x 107
19 : 9349
20 : 15127 = 7 x 2161
21 : 24476 = 22 x 29 x 211
22 : 39603 = 3 x 43 x 307
23 : 64079 = 139 x 461
24 : 103682 = 2 x 47 x 1103
25 : 167761 = 11 x 101 x 151
26 : 271443 = 3 x 90481
27 : 439204 = 22 x 19 x 5779
28 : 710647 = 72 x 14503
29 : 1149851 = 59 x 19489
30 : 1860498 = 2 x 32 x 41 x 2521
31 : 3010349
32 : 4870847 = 1087 x 4481
33 : 7881196 = 22 x 199 x 9901
34 : 12752043 = 3 x 67 x 63443
35 : 20633239 = 11 x 29 x 71 x 911
36 : 33385282 = 2 x 7 x 23 x 103681
37 : 54018521
38 : 87403803 = 3 x 29134601
39 : 141422324 = 22 x 79 x 521 x 859
40 : 228826127 = 47 x 1601 x 3041
41 : 370248451
42 : 599074578 = 2 x 32 x 83 x 281 x 1427
43 : 969323029 = 6709 x 144481
44 : 1568397607 = 7 x 263 x 881 x 967
45 : 2537720636 = 22 x 11 x 19 x 31 x 181 x 541
46 : 4106118243 = 3 x 4969 x 275449
47 : 6643838879
48 : 10749957122 = 2 x 769 x 2207 x 3167
49 : 17393796001 = 29 x 599786069
50 : 28143753123 = 3 x 41 x 401 x 570601
51 : 45537549124 = 22 x 919 x 3469 x 3571
52 : 73681302247 = 7 x 103 x 102193207
53 : 119218851371
54 : 192900153618 = 2 x 34 x 107 x 11128427
55 : 312119004989 = 112 x 199 x 331 x 39161
56 : 505019158607 = 47 x 10745088481
57 : 817138163596 = 22 x 229 x 9349 x 95419
58 : 1322157322203 = 3 x 347 x 1270083883
59 : 2139295485799 = 709 x 8969 x 336419
60 : 3461452808002 = 2 x 7 x 23 x 241 x 2161 x 20641
61 : 5600748293801
62 : 9062201101803 = 3 x 3020733700601
63 : 14662949395604 = 22 x 19 x 29 x 211 x 1009 x 31249
64 : 23725150497407 = 127 x 186812208641
65 : 38388099893011 = 11 x 131 x 521 x 2081 x 24571
66 : 62113250390418 = 2 x 32 x 43 x 307 x 261399601
67 : 100501350283429 = 4021 x 24994118449
68 : 162614600673847 = 7 x 23230657239121
69 : 263115950957276 = 22 x 139 x 461 x 691 x 1485571
70 : 425730551631123 = 3 x 41 x 281 x 12317523121
71 : 688846502588399
72 : 1114577054219522 = 2 x 47 x 1103 x 10749957121
73 : 1803423556807921 = 151549 x 11899937029
74 : 2918000611027443 = 3 x 11987 x 81143477963
75 : 4721424167835364 = 22 x 11 x 31 x 101 x 151 x 12301 x 18451
76 : 7639424778862807 = 7 x 1091346396980401
77 : 12360848946698171 = 29 x 199 x 229769 x 9321929
78 : 20000273725560978 = 2 x 32 x 90481 x 12280217041
79 : 32361122672259149
80 : 52361396397820127 = 2207 x 23725145626561
81 : 84722519070079276 = 22 x 19 x 3079 x 5779 x 62650261
82 : 137083915467899403 = 3 x 163 x 800483 x 350207569
83 : 221806434537978679 = 35761381 x 6202401259
84 : 358890350005878082 = 2 x 72 x 23 x 167 x 14503 x 65740583
85 : 580696784543856761 = 11 x 3571 x 1158551 x 12760031
86 : 939587134549734843 = 3 x 313195711516578281
87 : 1520283919093591604 = 22 x 59 x 349 x 19489 x 947104099
88 : 2459871053643326447 = 47 x 93058241 x 562418561
89 : 3980154972736918051 = 179 x 22235502640988369
90 : 6440026026380244498 = 2 x 33 x 41 x 107 x 2521 x 10783342081
91 : 10420180999117162549 = 29 x 521 x 689667151970161
92 : 16860207025497407047 = 7 x 253367 x 9506372193863
93 : 27280388024614569596 = 22 x 63799 x 3010349 x 35510749
94 : 44140595050111976643 = 3 x 563 x 5641 x 4632894751907
95 : 71420983074726546239 = 11 x 191 x 9349 x 41611 x 87382901
96 : 115561578124838522882 = 2 x 1087 x 4481 x 11862575248703
97 : 186982561199565069121 = 3299 x 56678557502141579
98 : 302544139324403592003 = 3 x 281 x 5881 x 61025309469041
99 : 489526700523968661124 = 22 x 19 x 199 x 991 x 2179 x 9901 x 1513909
100 : 792070839848372253127 = 7 x 2161 x 9125201 x 5738108801
101 : 1281597540372340914251 = 809 x 7879 x 201062946718741
102 : 2073668380220713167378 = 2 x 32 x 67 x 409 x 63443 x 66265118449
103 : 3355265920593054081629 = 619 x 1031 x 5257480026438961
104 : 5428934300813767249007 = 47 x 3329 x 106513889 x 325759201
105 : 8784200221406821330636 = 22 x 11 x 29 x 31 x 71 x 211 x 911 x 21211 x 767131
106 : 14213134522220588579643 = 3 x 1483 x 2969 x 1076012367720403
107 : 22997334743627409910279 = 47927441 x 479836483312919
108 : 37210469265847998489922 = 2 x 7 x 23 x 6263 x 103681 x 177962167367
109 : 60207804009475408400201 = 128621 x 788071 x 593985111211
110 : 97418273275323406890123 = 3 x 41 x 43 x 307 x 59996854928656801
111 : 157626077284798815290324 = 22 x 4441 x 146521 x 1121101 x 54018521
112 : 255044350560122222180447 = 223 x 449 x 2207 x 1154149773784223
113 : 412670427844921037470771
114 : 667714778405043259651218 = 2 x 32 x 227 x 26449 x 29134601 x 212067587
115 : 1080385206249964297121989 = 11 x 139 x 461 x 1151 x 5981 x 324301 x 686551
116 : 1748099984655007556773207 = 7 x 299281 x 834428410879506721
117 : 2828485190904971853895196 = 22 x 19 x 79 x 521 x 859 x 1052645985555841
118 : 4576585175559979410668403 = 3 x 15247723 x 100049587197598387
119 : 7405070366464951264563599 = 29 x 239 x 3571 x 10711 x 27932732439809
120 : 11981655542024930675232002 = 2 x 47 x 1103 x 1601 x 3041 x 23735900452321
121 : 19386725908489881939795601 = 199 x 97420733208491869044199
122 : 31368381450514812615027603 = 3 x 19763 x 21291929 x 24848660119363
123 : 50755107359004694554823204 = 22 x 4767481 x 370248451 x 7188487771
124 : 82123488809519507169850807 = 7 x 743 x 467729 x 33758740830460183
125 : 132878596168524201724674011 = 11 x 101 x 151 x 251 x 112128001 x 28143378001
126 : 215002084978043708894524818 = 2 x 33 x 83 x 107 x 281 x 1427 x 1461601 x 764940961
127 : 347880681146567910619198829 = 509 x 5081 x 487681 x 13822681 x 19954241
128 : 562882766124611619513723647 = 119809 x 4698167634523379875583
129 : 910763447271179530132922476 = 22 x 6709 x 144481 x 308311 x 761882591401
130 : 1473646213395791149646646123 = 3 x 41 x 3121 x 90481 x 42426476041450801
131 : 2384409660666970679779568599 = 1049 x 414988698461 x 5477332620091
132 : 3858055874062761829426214722 = 2 x 7 x 23 x 263 x 881 x 967 x 5281 x 66529 x 152204449
133 : 6242465534729732509205783321 = 29 x 9349 x 10694421739 x 2152958650459
134 : 10100521408792494338631998043 = 3 x 6163 x 201912469249 x 2705622682163
135 : 16342986943522226847837781364 = 22 x 11 x 19 x 31 x 181 x 271 x 541 x 811 x 5779 x 42391 x 119611
136 : 26443508352314721186469779407 = 47 x 562627837283291940137654881
137 : 42786495295836948034307560771 = 541721291 x 78982487870939058281
138 : 69230003648151669220777340178 = 2 x 32 x 4969 x 16561 x 162563 x 275449 x 1043766587
139 : 112016498943988617255084900949 = 30859 x 253279129 x 14331800109223159
140 : 181246502592140286475862241127 = 72 x 2161 x 14503 x 118021448662479038881
141 : 293263001536128903730947142076 = 22 x 79099591 x 6643838879 x 139509555271
142 : 474509504128269190206809383203 = 3 x 283 x 569 x 2820403 x 9799987 x 35537616083
143 : 767772505664398093937756525279 = 199 x 521 x 1957099 x 2120119 x 1784714380021
144 : 1242282009792667284144565908482 = 2 x 769 x 2207 x 3167 x 115561578124838522881
145 : 2010054515457065378082322433761 = 11 x 59 x 19489 x 120196353941 x 1322154751061
146 : 3252336525249732662226888342243 = 3 x 29201 x 37125857850184727260788881
147 : 5262391040706798040309210776004 = 22 x 29 x 211 x 65269 x 620929 x 8844991 x 599786069
148 : 8514727565956530702536099118247 = 7 x 10661921 x 114087288048701953998401
149 : 13777118606663328742845309894251 = 952111 x 4434539 x 3263039535803245519
150 : 22291846172619859445381409012498 = 2 x 32 x 41 x 401 x 601 x 2521 x 570601 x 87129547172401
151 : 36068964779283188188226718906749 = 1511 x 109734721 x 217533000184835774779
152 : 58360810951903047633608127919247 = 47 x 562766385967 x 2206456200865197103
153 : 94429775731186235821834846825996 = 22 x 19 x 919 x 3469 x 3571 x 13159 x 8293976826829399
154 : 152790586683089283455442974745243 = 3 x 43 x 281 x 307 x 15252467 x 900164950225760603
155 : 247220362414275519277277821571239 = 11 x 311 x 3010349 x 29138888651 x 823837075741
156 : 400010949097364802732720796316482 = 2 x 7 x 23 x 103 x 1249 x 102193207 x 94491842183551489
157 : 647231311511640322009998617887721 = 39980051 x 16188856575286517818849171
158 : 1047242260609005124742719414204203 = 3 x 21803 x 5924683 x 14629892449 x 184715524801
159 : 1694473572120645446752718032091924 = 22 x 785461 x 119218851371 x 4523819299182451
160 : 2741715832729650571495437446296127 = 641 x 1087 x 4481 x 878132240443974874201601
161 : 4436189404850296018248155478388051 = 29 x 139 x 461 x 1289 x 1917511 x 965840862268529759
162 : 7177905237579946589743592924684178 = 2 x 35 x 107 x 11128427 x 1828620361 x 6782976947987
163 : 11614094642430242607991748403072229 = 1043201 x 6601501 x 1686454671192230445929
164 : 18791999880010189197735341327756407 = 7 x 2684571411430027028247905903965201
165 : 30406094522440431805727089730828636 = 22 x 112 x 31 x 199 x 331 x 9901 x 39161 x 51164521 x 1550853481
166 : 49198094402450621003462431058585043 = 3 x 6464041 x 245329617161 x 10341247759646081
167 : 79604188924891052809189520789413679 = 766531 x 103849927693584542320127327909
168 : 128802283327341673812651951847998722 = 2 x 47 x 1103 x 10745088481 x 115613939510481515041
169 : 208406472252232726621841472637412401 = 521 x 596107814364089 x 671040394220849329
170 : 337208755579574400434493424485411123 = 3 x 41 x 67 x 1361 x 40801 x 63443 x 11614654211954032961
171 : 545615227831807127056334897122823524 = 22 x 192 x 229 x 9349 x 95419 x 162451 x 1617661 x 7038398989
172 : 882823983411381527490828321608234647 = 7 x 126117711915911646784404045944033521
173 : 1428439211243188654547163218731058171 = 78889 x 6248069 x 16923049609 x 171246170261359
174 : 2311263194654570182037991540339292818 = 2 x 32 x 347 x 97787 x 528295667 x 1270083883 x 5639710969
175 : 3739702405897758836585154759070350989 = 11 x 29 x 71 x 101 x 151 x 911 x 54601 x 560701 x 7517651 x 51636551
176 : 6050965600552329018623146299409643807 = 1409 x 2207 x 6086461133983 x 319702847642258783
177 : 9790668006450087855208301058479994796 = 22 x 709 x 8969 x 336419 x 10884439 x 105117617351706859
178 : 15841633607002416873831447357889638603 = 3 x 5280544535667472291277149119296546201
179 : 25632301613452504729039748416369633399 = 359 x 1066737847220321 x 66932254279484647441
180 : 41473935220454921602871195774259272002 = 2 x 7 x 23 x 241 x 2161 x 8641 x 20641 x 103681 x 13373763765986881
181 : 67106236833907426331910944190628905401 = 97379 x 21373261504197751 x 32242356485644069
182 : 108580172054362347934782139964888177403 = 3 x 281 x 90481 x 232961 x 6110578634294886534808481
183 : 175686408888269774266693084155517082804 = 22 x 14686239709 x 5600748293801 x 533975715909289
184 : 284266580942632122201475224120405260207 = 47 x 367 x 37309023160481 x 441720958100381917103
185 : 459952989830901896468168308275922343011 = 11 x 54018521 x 265272771839851 x 2918000731816531
186 : 744219570773534018669643532396327603218 = 2 x 32 x 15917507 x 3020733700601 x 859886421593527043
187 : 1204172560604435915137811840672249946229 = 199 x 1871 x 3571 x 905674234408506526265097390431
188 : 1948392131377969933807455373068577549447 = 7 x 18049 x 100769 x 153037630649666194962091443041
189 : 3152564691982405848945267213740827495676 = 22 x 19 x 29 x 211 x 379 x 1009 x 5779 x 31249 x 85429 x 912871 x 1258740001
190 : 5100956823360375782752722586809405045123 = 3 x 41 x 2281 x 4561 x 29134601 x 782747561 x 174795553490801
191 : 8253521515342781631697989800550232540799 = 22921 x 395586472506832921 x 910257559954057439
192 : 13354478338703157414450712387359637585922 = 2 x 127 x 383 x 5662847 x 6803327 x 19073614849 x 186812208641
193 : 21607999854045939046148702187909870126721 = 303011 x 76225351 x 935527893146187207403151261
194 : 34962478192749096460599414575269507712643 = 3 x 195163 x 4501963 x 5644065667 x 2350117027000544947
195 : 56570478046795035506748116763179377839364 = 22 x 11 x 31 x 79 x 131 x 521 x 859 x 1951 x 2081 x 2731 x 24571 x 866581 x 37928281
196 : 91532956239544131967347531338448885552007 = 73 x 14503 x 3016049 x 6100804791163473872231629367
197 : 148103434286339167474095648101628263391371 = 31498587119111339 x 4701907222895068350249889
198 : 239636390525883299441443179440077148943378 = 2 x 33 x 43 x 107 x 307 x 261399601 x 11166702227 x 1076312899454363
199 : 387739824812222466915538827541705412334749 = 2389 x 4503769 x 36036960414811969810787847118289
200 : 627376215338105766356982006981782561278127 = 47 x 1601 x 3041 x 124001 x 6996001 x 3160438834174817356001

## Rules for Primes and Factors of the Fibonacci Numbers

The table of the first 300 Fibonacci numbers had some very interesting properties such as:
Fnk is a multiple of Fk
For example:
2 and 4 are both factors of 8:
so F2=1 and F4=3 should be factors of F8=21

We also saw that, for the Fibonacci numbers,

the Fibonacci number Fn is prime only if n is prime.
apart from F4 which is prime!
[But remember the converse is not always true - just because n is prime does not mean that Fn must be prime!]

## Do the Fibonacci Rules apply to the Lucas Numbers?

The same rules do not seem to apply to the Lucas numbers above!
For example:
2 and 4 are factors of 8:
but L2=3 and L4=7 but L8=47 is prime
so cannot have factors 3 and 7!

So the big question is:

Can you find some other rules that apply to Lucas numbers and their factors?

To help with your investigations, here are the results of a search for prime number among the first 1000 Lucas numbers:

The only Lucas number which are prime up to L(1000) are L(i) where i=
2, 4, 5, 7, 8, 11, 13, 16, 17, 19, 31, 37, 41, 47, 53, 61, 71, 79, 113, 313, 353, 503, 613, 617, 863.
( Lucas(1000) has 209 digits!)

## Cycles in the Lucas numbers?

On the The Mathematics of the Fibonacci Series we saw that the units digits of the Fibonacci numbers repeat in a cycle of length 60 (so that the units digits of F60 = the units digits of F0, and so on for following digits).
• For the Lucas numbers, there is also a cycle of 60 - which is when the last two digits repeat in a cycle.
There is a cycle of units digits in the Lucas numbers, which is much shorter. What is it? How long is it?