Fibonacci Numbers with a Prescribed Block of Digits

Abstract

In this paper, we prove that $F_{22}=17711$ is the largest Fibonacci number whose decimal expansion is of the form $ab\dots bc\dots c$. The proof uses lower bounds for linear forms in three logarithms of algebraic numbers and some tools from Diophantine approximation.

1. Introduction

Let ${\left(F_n\right)}_{n\ge 0}$ be the Fibonacci sequence given by second-order recurrence $F_{n+2}=F_{n+1}+F_n$, for $n\ge 0$, with initial conditions $F_0=0$ and $F_1=1$. A few terms of this sequence are

$$0,1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,4181,6765,10946,17711,\dots .$$

In the last decades, many results on Diophantine properties of Fibonacci numbers have been proved with the use of refined tools in number theory. For instance, Bugeaud, Mignotte, and Siksek [1] settled the problem of Fibonacci perfect power numbers (i.e., the equation $F_n=y^t$, for $t>1$) by combing the powerful Baker’s theory with the Modular method (used by Wiles in the proof of the Fermat Last Theorem). See a generalization of their result in [2].

We remark that digital problems involving Fibonacci numbers have received much attention in the literature. A first result in this direction was proved, in 2000, by F. Luca [3] who showed that the largest Fibonacci number with only one distinct digit is $F_{10}=55$. After this, many authors worked on repdigits (i.e., integers having only one distinct digit in its decimal expansion) as expressions related to sum, product of terms of binary recurrences (see [4,5,6,7,8,9,10,11,12,13] and references therein). However, the related problem of finding all Fibonacci numbers with only two distinct digits remains open.

The aim of this paper is to continue this program. In fact, our main result searches for all Fibonacci numbers of the form $ab\dots bc\dots c$, which provides a generalization for Luca’s result ($a=b=c$). More precisely, our main result is the following:

Theorem 1.

Let $a\in [1,9]$ and $b,c\in [0,9]$. The largest solution of the Diophantine equation

$$F_n=a\underset{m}{\underbrace{b\dots b}}\underset{\ell }{\underbrace{c\dots c}}=a·{10}^{m+\ell }+b·\left(\frac{{10}^m-1}{9}\right){10}^{\ell }+c·\left(\frac{{10}^{\ell }-1}{9}\right),$$

(1)

in positive integers $m,n$, and ℓ, with $2\le \ell \le m$, is $n=22$. Explicitly, $F_{22}=17711$.

Our proof combines two deep techniques in number theory, namely, the Baker’s theory on linear forms in logarithms and some tools from Diophantine approximation (a reduction method due to Baker and Davenport).

2. Auxiliary Results

In this section, we shall present some results which will be useful in the proofs.

Let $\alpha =(1+\sqrt{5})/2$ and $\beta =-1/\alpha$. The Binet’s formula asserts that $F_n=({\alpha }^n-{\beta }^n)/\sqrt{5}$. From this formula, it is possible to deduce that the estimates

$${\alpha }^{n-2}\le F_n\le {\alpha }^{n-1},$$

(2)

hold for all $n\ge 1$. In addition, this Binet’s formula allows us to manipulate our Diophantine equation to obtain upper bounds for some linear forms in three logarithms. Thus, in order to obtain lower bounds for these forms, we shall use the celebrated Baker’s theory. Among these lower bounds, we decided to use one which was proved in ([1], Theorem 9.4).

Lemma 1.

Let ${\gamma }_1,\dots ,{\gamma }_t$ be real algebraic numbers and let $b_1,\dots ,b_t$ be nonzero rational integer numbers. Let D be the degree of the number field $\mathbb{Q}({\gamma }_1,\dots ,{\gamma }_t)$ over $\mathbb{Q}$ and let $A_j$ be a positive real number satisfying

$$A_j\ge max\{Dh\left({\gamma }_j\right),|log{\gamma }_j|,0.16\},$$

where $j=1,\dots ,t$. Assume that

$$B\ge max\left\{\right|b_1|,\dots ,|b_t\left|\right\}.$$

If ${\gamma }_1^{b_1}\cdots {\gamma }_t^{b_t}\ne 1$, then

$$|{\gamma }_1^{b_1}\cdots {\gamma }_t^{b_t}-1|\ge exp(-1.4·{30}^{t+3}·t^{4.5}·D^2(1+logD)(1+logB)A_1\cdots A_t).$$

Here, the logarithmic height of an n-degree algebraic number $\alpha$ is defined as

$$h\left(\alpha \right)=\frac{1}{n}\left(log\left|a\right|+\sum _{j=1}^{n}logmax\{1,|{\alpha }^{\left(j\right)}\left|\right\}\right),$$

where a is the leading coefficient of the minimal primitive polynomial of $\alpha$ (over $\mathbb{Z}$) and ${\left({\alpha }^{\left(j\right)}\right)}_{1\le j\le n}$ are the (algebraic) conjugates of $\alpha$.

With these lower and upper bounds, we shall obtain an upper bound for n which is, in general, very large and then the next step is to reduce it. For that, we shall use a reduction method which is originated from Diophantine approximation. Here, we shall use a result due to Dujella and Pethö [14] (which is a variant of a famous method due to Baker–Davenport). For a real number x we use $\Vert x\Vert =min\left\{\right|x-n|:n\in \mathbb{Z}\}$ for the distance from x to the nearest integer (the so-called Nint function).

Lemma 2.

Let $M>0$ be an integer and let $\gamma ,\mu$ be real numbers, such that $\gamma \notin \mathbb{Q}$. Let $p/q$ be a convergent of the continued fraction expansion of γ such that $q>6M$ and $ϵ:=\Vert \mu q\Vert -M\Vert \gamma q\Vert$ > 0. Then there is no solution to the Diophantine inequality

$$0<m\gamma -n+\mu <A·B^{-m}$$

in positive integers $m,n$ with

$$\frac{log(Aq/ϵ)}{logB}\le m<M.$$

After presetting these tools, we can now prove our main result.

3. The Proof of The Theorem

3.1. Finding a Bound on N

By the Binet’s formula and the identity in Equation (1), we have

$${\displaystyle \frac{{\alpha }^n}{\sqrt{5}}}-\left(a+{\displaystyle \frac{b}{9}}\right){10}^{m+\ell }=\left({\displaystyle \frac{c-b}{9}}\right){10}^{\ell }+{\displaystyle \frac{{\beta }^n}{\sqrt{5}}}-{\displaystyle \frac{c}{9}}.$$

(3)

Thus,

$$\left|{\displaystyle \frac{{\alpha }^n}{\sqrt{5}}}-\left(a+{\displaystyle \frac{b}{9}}\right){10}^{m+\ell }\right|<2·{10}^{\ell }.$$

On dividing through by $(a+b/9){10}^{m+\ell }$, we obtain

$$\left|{\left(a+{\displaystyle \frac{b}{9}}\right)}^{-1}{\displaystyle \frac{{\alpha }^n}{\sqrt{5}}}{10}^{-(m+\ell )}-1\right|<{\displaystyle \frac{2}{{10}^m}},$$

(4)

where we used the fact that $a+b/9\ge 1$. Now, we are in a position to apply Lemma 1, but first we must prove that $|{\left(\sqrt{5}(a+b/9)\right)}^{-1}{10}^{-(m+\ell )}{\alpha }^n-1|\ne 0$. Indeed, in the contrary case, we would get that ${\alpha }^{2n}\in \mathbb{Q}$ which is an absurd. Thus, let us take

$${\gamma }_1:=\sqrt{5}(a+b/9),{\gamma }_2:=\alpha ,{\gamma }_3:=10,b_1:=-1,b_2:=n,b_3:=-(m+\ell ).$$

Note that $\mathbb{Q}({\gamma }_1,{\gamma }_2,{\gamma }_3)=\mathbb{Q}\left(\sqrt{5}\right)$ and then $D=2$. The conjugates of ${\gamma }_1,{\gamma }_2$, and ${\gamma }_3$ are ${\gamma }_1^{\prime }=-{\gamma }_1,{\gamma }_2^{\prime }=\beta ,{\gamma }_3^{\prime }={\gamma }_3$, respectively. Surely, ${\gamma }_2$ and ${\gamma }_3$ are algebraic integers, while the minimal polynomial of ${\gamma }_1$ is $(X-{\gamma }_1)(X-{\gamma }_1^{\prime })=X^2-5{(a+b/9)}^2$ which is a divisor of $81X^2-5{(9a+b)}^2$. Therefore,

$$h\left({\gamma }_1\right)\le {\displaystyle \frac{1}{2}}\left(log81+2log\left(10\sqrt{5}\right)\right)<5.4.$$

(A more relaxed upper bound for $h\left({\gamma }_1\right)$ could be found by using the well-known property that $h(x+yz)\le h\left(x\right)+h\left(y\right)+h\left(z\right)+log2$). In addition, $h\left({\gamma }_2\right)=(log\alpha )/2<0.75$ and $h\left({\gamma }_3\right)=log10<2.31$. Let us take $A_1:=10.9,A_2=1.5$, and $A_3=4.7$. Of course, we can assume that $n>14$. Thus,

$$\left|{\left(a+{\displaystyle \frac{b}{9}}\right)}^{-1}{\displaystyle \frac{{\alpha }^n}{\sqrt{5}}}{10}^{-(m+\ell )}-1\right|>exp\left(-7.6·{10}^{13}(1+logB)\right),$$

(5)

where $B\ge max\{n,m+\ell \}$

By combining the estimates in Equations (4) and (5), we get

$$m<3.4·{10}^{13}(1+logB).$$

Now, we have that $F_n$ has $m+\ell +1$ digits and so

$$m+\ell +1=⌊{\displaystyle \frac{log{F}_n}{log10}}⌋+1.$$

(6)

Since $m\ge \ell$ and $\lfloor x\rfloor +1>x$, we have

$$2m+1>{\displaystyle \frac{log{F}_n}{log10}}\ge (n-2){\displaystyle \frac{log\alpha }{log10}}>0.2(n-2)$$

so $10m+6\ge n$. Thus, we can take $B:=10m+6$, which yields to

$$m<3.4·{10}^{13}(1+log(10m+6))$$

and therefore $m\le 1.3·{10}^{15}$.

3.2. Reducing the Bound

Now, let us write $\Lambda :=nlog\alpha -(m+\ell )log10+log\left({(a+b/9)}^{-1}{\left(\sqrt{5}\right)}^{-1}\right)$. We know that $e^x-1>x$, for all $x\in {\mathbb{R}}^{+}$. By supposing that $\Lambda >0$ (the other case is completely similar, where we used the fact that $|e^x-1|=1-e^{-\left|x\right|}$, if $x<0$), we can rewrite Equation (4) as

$$0<nlog\alpha -(m+\ell )log10+log\left({(a+b/9)}^{-1}{\left(\sqrt{5}\right)}^{-1}\right)<1.8·{10}^{-m}.$$

Since $m>\left(0.1\right)n-0.7$ (because $2m+1>0.2(n-2)$), we can divide the previous inequality by $log10$, to obtain

$$0<n\gamma -(m+\ell )+\mu <4·{\left(1.2\right)}^{-n},$$

(7)

where $\gamma :=log\alpha /log10$ and $\mu :=log\left({(a+b/9)}^{-1}{\left(\sqrt{5}\right)}^{-1}\right)/log10$.

Clearly $\gamma$ is an irrational number (because ${\alpha }^k$ is irrational for any non-zero integer k). Let us denote $p_n/q_n$ as the nth convergent of its continued fraction.

In order to reduce our bound on m, we shall use Lemma 2. Now, since $n\le 10m+6<1.4·{10}^{16}$, we choose $M=1.4·{10}^{16}$. Thus,

$${\displaystyle \frac{p_{38}}{q_{38}}}={\displaystyle \frac{1426134855866370784}{6824015306170795931}},$$

then $q_{38}\ge 6824015306170795931>8.4·{10}^{16}=6M$. Furthermore, we have $M\Vert q_{38}\gamma \Vert <0.0013$. On the other hand, by computing $\Vert q_{38}\mu \Vert$, for $a\in [1,9]$ and $b\in [0,9]$, we have that the minimal value of this expression is obtained when $a=8,b=2$ and is $>0.0028$. Hence,

$$ϵ=\Vert q_{38}\mu \Vert -M\Vert q_{38}\gamma \Vert >0.0015.$$

We notice the all the hypotheses of the Lemma 2 are fulfilled, where $A=4$ and $B=1.2$, so, by that lemma, there is no solution of the inequality in Equation (7) (and then for the Diophantine Equation (1)) for n in the range

$$[⌊log(A{q}_{38}/ϵ)/logB⌋+1,M]=[281,1.4·{10}^{16}].$$

Since $n<M$, we get $n\le 280$. By using Equation (6), we deduce that

$$m+\ell \le (n-1){\displaystyle \frac{log\alpha }{log10}}-1<58.4$$

and so $m+\ell \le 58$. Since $m+\ell +1\ge 5$, it is seen that $F_n$ has at least 5 digits yielding $n\ge 21$. A simple search in the list of the Fibonacci numbers $F_n$ in the range $n\in [21,280]$ (see

Table A1. Values of $F\left(n\right)$ for n from 21 to 280.

n	${\mathit{F}}_{\mathit{n}}$	n	${\mathit{F}}_{\mathit{n}}$
21	10946	280	14691098406862188148944207245954912110548093601382197697835
22	17711	279	9079598147510263717870894449029933369491131786514446266146
23	28657	278	5611500259351924431073312796924978741056961814867751431689
24	46368	277	3468097888158339286797581652104954628434169971646694834457
25	75025	276	2143402371193585144275731144820024112622791843221056597232
26	121393	275	1324695516964754142521850507284930515811378128425638237225
27	196418	274	818706854228831001753880637535093596811413714795418360007
28	317811	273	505988662735923140767969869749836918999964413630219877218
29	514229	272	312718191492907860985910767785256677811449301165198482789
30	832040	271	193270471243015279782059101964580241188515112465021394429
31	1346269	270	119447720249892581203851665820676436622934188700177088360
32	2178309	269	73822750993122698578207436143903804565580923764844306069
33	3524578	268	45624969256769882625644229676772632057353264935332782291
34	5702887	267	28197781736352815952563206467131172508227658829511523778
35	9227465	266	17427187520417066673081023209641459549125606105821258513
36	14930352	265	10770594215935749279482183257489712959102052723690265265
37	24157817	264	6656593304481317393598839952151746590023553382130993248
38	39088169	263	4114000911454431885883343305337966369078499341559272017
39	63245986	262	2542592393026885507715496646813780220945054040571721231
40	102334155	261	1571408518427546378167846658524186148133445300987550786
41	165580141	260	971183874599339129547649988289594072811608739584170445
42	267914296	259	600224643828207248620196670234592075321836561403380341
43	433494437	258	370959230771131880927453318055001997489772178180790104
44	701408733	257	229265413057075367692743352179590077832064383222590237
45	1134903170	256	141693817714056513234709965875411919657707794958199867
46	1836311903	255	87571595343018854458033386304178158174356588264390370
47	2971215073	254	54122222371037658776676579571233761483351206693809497
48	4807526976	253	33449372971981195681356806732944396691005381570580873
49	7778742049	252	20672849399056463095319772838289364792345825123228624
50	12586269025	251	12776523572924732586037033894655031898659556447352249
51	20365011074	250	7896325826131730509282738943634332893686268675876375
52	32951280099	249	4880197746793002076754294951020699004973287771475874
53	53316291173	248	3016128079338728432528443992613633888712980904400501
54	86267571272	247	1864069667454273644225850958407065116260306867075373
55	139583862445	246	1152058411884454788302593034206568772452674037325128
56	225851433717	245	712011255569818855923257924200496343807632829750245
57	365435296162	244	440047156314635932379335110006072428645041207574883
58	591286729879	243	271964099255182923543922814194423915162591622175362
59	956722026041	242	168083057059453008835412295811648513482449585399521
60	1548008755920	241	103881042195729914708510518382775401680142036775841
61	2504730781961	240	64202014863723094126901777428873111802307548623680
62	4052739537881	239	39679027332006820581608740953902289877834488152161
63	6557470319842	238	24522987531716273545293036474970821924473060471519
64	10610209857723	237	15156039800290547036315704478931467953361427680642
65	17167680177565	236	9366947731425726508977331996039353971111632790877
66	27777890035288	235	5789092068864820527338372482892113982249794889765
67	44945570212853	234	3577855662560905981638959513147239988861837901112
68	72723460248141	233	2211236406303914545699412969744873993387956988653
69	117669030460994	232	1366619256256991435939546543402365995473880912459
70	190392490709135	231	844617150046923109759866426342507997914076076194
71	308061521170129	230	522002106210068326179680117059857997559804836265
72	498454011879264	229	322615043836854783580186309282650000354271239929
73	806515533049393	228	199387062373213542599493807777207997205533596336
74	1304969544928657	227	123227981463641240980692501505442003148737643593
75	2111485077978050	226	76159080909572301618801306271765994056795952743
76	3416454622906707	225	47068900554068939361891195233676009091941690850
77	5527939700884757	224	29090180355503362256910111038089984964854261893
78	8944394323791464	223	17978720198565577104981084195586024127087428957
79	14472334024676221	222	11111460156937785151929026842503960837766832936
80	23416728348467685	221	6867260041627791953052057353082063289320596021
81	37889062373143906	220	4244200115309993198876969489421897548446236915
82	61305790721611591	219	2623059926317798754175087863660165740874359106
83	99194853094755497	218	1621140188992194444701881625761731807571877809
84	160500643816367088	217	1001919737325604309473206237898433933302481297
85	259695496911122585	216	619220451666590135228675387863297874269396512
86	420196140727489673	215	382699285659014174244530850035136059033084785
87	679891637638612258	214	236521166007575960984144537828161815236311727
88	1100087778366101931	213	146178119651438213260386312206974243796773058
89	1779979416004714189	212	90343046356137747723758225621187571439538669
90	2880067194370816120	211	55835073295300465536628086585786672357234389
91	4660046610375530309	210	34507973060837282187130139035400899082304280
92	7540113804746346429	209	21327100234463183349497947550385773274930109
93	12200160415121876738	208	13180872826374098837632191485015125807374171
94	19740274219868223167	207	8146227408089084511865756065370647467555938
95	31940434634990099905	206	5034645418285014325766435419644478339818233
96	51680708854858323072	205	3111581989804070186099320645726169127737705
97	83621143489848422977	204	1923063428480944139667114773918309212080528
98	135301852344706746049	203	1188518561323126046432205871807859915657177
99	218922995834555169026	202	734544867157818093234908902110449296423351
100	354224848179261915075	201	453973694165307953197296969697410619233826
101	573147844013817084101	200	280571172992510140037611932413038677189525
102	927372692193078999176	199	173402521172797813159685037284371942044301
103	1500520536206896083277	198	107168651819712326877926895128666735145224
104	2427893228399975082453	197	66233869353085486281758142155705206899077
105	3928413764606871165730	196	40934782466626840596168752972961528246147
106	6356306993006846248183	195	25299086886458645685589389182743678652930
107	10284720757613717413913	194	15635695580168194910579363790217849593217
108	16641027750620563662096	193	9663391306290450775010025392525829059713
109	26925748508234281076009	192	5972304273877744135569338397692020533504
110	43566776258854844738105	191	3691087032412706639440686994833808526209
111	70492524767089125814114	190	2281217241465037496128651402858212007295
112	114059301025943970552219	189	1409869790947669143312035591975596518914
113	184551825793033096366333	188	871347450517368352816615810882615488381
114	298611126818977066918552	187	538522340430300790495419781092981030533
115	483162952612010163284885	186	332825110087067562321196029789634457848
116	781774079430987230203437	185	205697230343233228174223751303346572685
117	1264937032042997393488322	184	127127879743834334146972278486287885163
118	2046711111473984623691759	183	78569350599398894027251472817058687522
119	3311648143516982017180081	182	48558529144435440119720805669229197641
120	5358359254990966640871840	181	30010821454963453907530667147829489881
121	8670007398507948658051921	180	18547707689471986212190138521399707760
122	14028366653498915298923761	179	11463113765491467695340528626429782121
123	22698374052006863956975682	178	7084593923980518516849609894969925639
124	36726740705505779255899443	177	4378519841510949178490918731459856482
125	59425114757512643212875125	176	2706074082469569338358691163510069157
126	96151855463018422468774568	175	1672445759041379840132227567949787325
127	155576970220531065681649693	174	1033628323428189498226463595560281832
128	251728825683549488150424261	173	638817435613190341905763972389505493
129	407305795904080553832073954	172	394810887814999156320699623170776339
130	659034621587630041982498215	171	244006547798191185585064349218729154
131	1066340417491710595814572169	170	150804340016807970735635273952047185
132	1725375039079340637797070384	169	93202207781383214849429075266681969
133	2791715456571051233611642553	168	57602132235424755886206198685365216
134	4517090495650391871408712937	167	35600075545958458963222876581316753
135	7308805952221443105020355490	166	22002056689466296922983322104048463
136	11825896447871834976429068427	165	13598018856492162040239554477268290
137	19134702400093278081449423917	164	8404037832974134882743767626780173
138	30960598847965113057878492344	163	5193981023518027157495786850488117
139	50095301248058391139327916261	162	3210056809456107725247980776292056
140	81055900096023504197206408605	161	1983924214061919432247806074196061
141	131151201344081895336534324866	160	1226132595394188293000174702095995
142	212207101440105399533740733471	159	757791618667731139247631372100066
143	343358302784187294870275058337	158	468340976726457153752543329995929
144	555565404224292694404015791808	157	289450641941273985495088042104137
145	898923707008479989274290850145	156	178890334785183168257455287891792
146	1454489111232772683678306641953	155	110560307156090817237632754212345
147	2353412818241252672952597492098	154	68330027629092351019822533679447
148	3807901929474025356630904134051	153	42230279526998466217810220532898
149	6161314747715278029583501626149	152	26099748102093884802012313146549
150	9969216677189303386214405760200	151	16130531424904581415797907386349

in Appendix A), returns only $F_{22}=17711$ with the required properties. This completes the proof. □

4. Conclusions

In this paper we have been interested in finding all Fibonacci numbers which are special concatenation of digits. In particular, we show that $F_{22}=17711$ is the largest Fibonacci number whose decimal expansion is of the form $ab\dots bc\dots c$, where $a,b$, and c are decimal digits. Our approach to the proof is based on the combination of lower bounds for linear forms in logarithms (due to Baker) with reduction methods (due to Dujella–Pethö).