Fibonacci number

The first 100 Fibonacci numbers are shown in this table below.

n	F(n)	n	F(n)	n	F(n)	n	F(n)
1	1	31	8110275	61	7655814421X301	91	7034384X03093XXXX0575
2	1	32	11110575	62	10243EX10286901	92	E444E61X4E394E4XX1275
3	2	33	1922082X	63	178997E544X5002	93	16479326852468X398182X
4	3	34	2X3311X3	64	27E21796476E903	94	259022886X184198862XX3
5	5	35	47551X11	65	437EE38E9054905	95	4017E5E33340XX80624711
6	8	36	75882EE4	66	6E720E661804608	96	65X8187EX15930592875E4
7	11	37	101214X05	67	E3320335X859311	97	X6041273149X1E198E0105
8	19	38	176X979E9	68	162X412X00461919	98	14EE02E32E6374E76E776E9
9	2X	39	2780E0802	69	256161615E0EE02X	99	235E441X60E156X94867802
X	47	3X	432E885EE	6X	3E8EX28E5E560947	9X	385X471190550EX4E8232EE
E	75	3E	6XE079201	6E	65314430EX65E975	9E	5EE98E3031466692448XE01
10	100	40	E22045800	70	X50127005X000700	X0	98581642019E767740E2200
11	175	41	1611102X01	71	14X326E3158660475	X1	13855X572332621498581101
12	275	42	2533148601	72	233339631E6660E75	X2	214E1EEE4350598050673301
13	42X	43	3E4424E402	73	3816645635310142X	X3	35147X566682EE9529034402
14	6X3	44	6477397X03	74	5E49X1E95497623X3	X4	56639X55XX135955796X7703
15	E11	45	X3EE627205	75	976446538X0863811	X5	8E7858E05496592XX671EE05
16	15E4	46	14876X03008	76	136E2285122X405EE4	X6	12620374642X9E68464207608
17	2505	47	2307642X211	77	2125672X4E0E069805	X7	1E5989436978453E34X927511
18	3XE9	48	37931231219	78	349489E361394737E9	X8	31EE90E811X724X77E2E32E19
19	6402	49	5X9X765E42X	79	55EX3521E048521402	X9	51595X3E7E636X26E4185X42X
1X	X2EE	4X	96718890647	7X	8X9303155185994EEE	XX	83592E37914X9312734791347
1E	14701	4E	13550432EX75	7E	12491383742122E6401	XE	114E6897750E2413967642E775
20	22X00	50	210021000500	80	1E3643E50939808E400	E0	19853E8E3224114501XE000E00
21	37501	51	345525330375	81	31835778815XX385801	E1	2E14X866X73335589865430675
22	5X301	52	555546330875	82	50E99E718X986455001	E2	489X2836195746X19X54431575
23	95802	53	89XX6E66102X	83	8281372X5037481X802	E3	77E314X1048X803X76E986202X
24	133E03	54	12343E59918X3	84	1137E16X01E13E073803	E4	1049141172226072055520935X3
25	209705	55	1E13265432911	85	19640520X6E4E3892405	E5	1808455E826E4875E104E935611
26	341608	56	314765E2045E4	86	2X9EE68XX8X632946008	E6	285559713491X927E65X1X08EE4
27	54E111	57	505X904637305	87	4843EEXE939E26618411	E7	4461X310E74135X1X7631742605
28	890719	58	81X636383E8E9	88	7723E67X808559362419	E8	70E7408230132309X201354E5E9
29	121E82X	59	11245068277002	89	10367E66X54648397X82X	E9	E5592393275458XE89645092002
2X	1XE0347	5X	1942E40EXE68EE	8X	17X8EE129152X21121047	EX	16654645557677EE96E6586215EE
2E	310EE75	5E	2X674478171901	8E	28237X7976992X4X9E875	EE	25EE18828830018X9390X16E3601
30	5000300	60	47XX3888068600	90	441079904830106000900	100	4064630821X6798X6X873X115000

F(n) is (probable) prime for n = 3, 4, 5, 7, E, 11, 15, 1E, 25, 37, 3E, 6E, XE, E5, 25E, 2EE, 301, 315, 365, 3E5, 3E7, 1877, 2897, 314E, 547E, 5725, 8427, 12961, 15971, 189EE, 1985E, 25501, 3E43E, 50867, 632E1, 7184E, 9846E, 171E6E, 18E045, 245561, 2476X1, 251E47, 38E0X5, 4276E5, 51EE9E, 66X91E, 72E52E, 7XE381, E80915, 1105641, 11510EE, ...

If F(n) is prime and n ≠ 4, then n is also prime, but the converse is not true: 17 is prime, but F(17) = 2505 = 31 × 95 is not prime.

Note that dozenal (base 10) is the only base such that 100 is a Fibonacci number (since 1 cannot be a base of a numeral system). 100 is indeed F₁₀, and 10 is the square root of 100.

A Fibonacci number can end with any digit but 6, and if a Fibonacci number ends with 0, then it must end with 00.

The period of the final digit of Fibonacci number is 20, that of the final two digits is also 20 (dozenal is the largest base such that the period of the final digit of Fibonacci number is the same as that of the final two digits of Fibonacci number, if there are no Wall-Sun-Sun primes). For n >= 2, the period of the final n digits of Fibonacci number is 2*10^(n-1). (2 followed by n-1 zeros, which is an n-digit number 200...000)

Since 175 and 100 are two consecutive Fibonacci numbers, the golden ratio (=(-1+sqrt(5))/2 = 1.74EE6772802X...) is very close to 175/100 = 1.75, and thus very close to 175%

n	F(n+1)	F(n)	F(n+1)/F(n)
1	1	1	1
2	2	1	2
3	3	2	1.6
4	5	3	1.8
5	8	5	1.72497249724972497249724X
6	11	8	1.76
7	19	11	1.747474747474747474747475
8	2X	19	1.75186X35186X35186X35186X
9	47	2X	1.74E36429X708579214E3642X
X	75	47	1.750275027502750275027503
E	100	75	1.74EX470174EX470174EX4702
10	175	100	1.75
11	275	175	1.74EE470074EE470074EE4701
12	42X	275	1.74EE74EE74EE74EE74EE7500
13	6X3	42X	1.74EE6401X7E45426271E1743
14	E11	6X3	1.74EE68E808906694E1083963
15	15E4	E11	1.74EE670E330627E0560065XX
16	2505	15E4	1.74EE67975739E04XX21X35XE
17	3XE9	2505	1.74EE67638024130246264642
18	6402	3XE9	1.74EE6776X53584854X684003
19	X2EE	6402	1.74EE67710916859X700544XE
1X	14701	X2EE	1.74EE677334621038707X41E3
1E	22X00	14701	1.74EE67725262E37E68694450
20	37501	22X00	1.74EE67729114EE13256229EX

F(n+1)/F(n) --> (-1+sqrt(5))/2 (≈ 1.74EE6772802X46X6X1865307) if n --> infinity.

Note that the recurring dozenal of F(n+1)/F(n) terminates if and only if n is a divisor of 10.

Factors of the Fibonacci numbers:

F(10)   =         100 = 10²

(F(10) is exactly the square of 10, and F(10) is also the largest Fibonacci number which is square)

F(10−1) = F(E)  =  75
F(10+1) = F(11) = 175

75 × 175 = 10001 = 10⁴ + 1

(10±1 are both primes, and F(10±1) are also both primes)
(10⁴ + 1 = 10001 is a semiprime, and 10⁴ − 1 is a member of betrothed number pair, together with 5600)

F(10−2) = F(X)  =  47 =  5 × E (exactly the largest Fibonacci number which is also triangular number)
F(10+2) = F(12) = 275 = 25 × 11 (exactly the largest Fibonacci number which is also generalized pentagonal number)

5 × 25 = 101 = 10² + 1
E × 11 =  EE = 10² − 1

(10±2 are both semiprimes, and F(10±2) are also both semiprimes)
(10² ± 1 are both semiprimes, and 5, E, 25, and 11 are all primes)

Fibonacci number

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