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This problem is giving me the hardest time:

Prove or disprove that there is a Fibonacci number that ends with 2014 zeros.

I tried mathematical induction (for stronger statement that claims that there is a Fibonacci number that ends in any number of zeroes), but no luck so far.


Related question: Fibonacci modular results

VividD
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    Where did you find this problem? – apnorton Jul 19 '14 at 22:07
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    Modular arithmetic. Also, $0$ is a fibonacci number. –  Jul 19 '14 at 22:07
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    @anorton In a high school math teacher's notes, a friend of mine. – VividD Jul 19 '14 at 22:16
  • @VividD Hm. I was hoping to find more of these. `:)` – apnorton Jul 19 '14 at 22:17
  • @Hurkyl I couldn't make a significant progress based on your hint. :( BTW, are you saying that there is no Fibonacci number ending in 2014 digits 6, because 6 is not a Fibonacci number? That's probably not what you meant, but I couldn't help asking you... – VividD Jul 19 '14 at 22:19
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    It seems that $F_{n}$ ends with $k$ zeroes ($k\ge 3$), where $n = 75 \cdot 10^{k-2}$. – Oleg567 Jul 19 '14 at 22:41
  • @Oleg567 this is so odd and unexpected. Why 75? What is so magic about 75? – VividD Jul 19 '14 at 22:59
  • @Oleg567 it would be fantastic if you devise a proof for your observation... – VividD Jul 19 '14 at 23:05
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    Just computations. One idea is to prove this statement (if it is true): $$k|F_n \Rightarrow k^d|F_{k^{d-1}n}.$$ But how to prove??? – Oleg567 Jul 19 '14 at 23:10
  • I have a simple proof when $k$ is odd, but I think this problem deserves a separate question. Just use the identity: $$\frac{F_{kn}}{F_n}=\left(5F_{\frac{k-1}{2}n}^2+2(-1)^{\frac{k-1}{2}n}+(-1)^n 5 F_{\frac{k-3}{2}n}^2+2 (-1)^{\frac{k-1}{2}n}\ldots\right)$$ and notice that the terms $2(-1)^{\frac{k-1}{2}}$ add to $\pm k$ while $k|F_n|F_{\nu n}^2$. – Jack D'Aurizio Jul 20 '14 at 00:41
  • @Oleg567: it is straightforward to complete the proof for the cases $k=2^h$ and put all together with the Chinese theorem, but please post this problem as a separate question, I will be glad to give a detailed answer. – Jack D'Aurizio Jul 20 '14 at 00:50

5 Answers5

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This is a classic. The Fibonacci sequence $\pmod{m}$ is periodic for any $m$, since there are only a finite number of elements in $\mathbb{Z}_m\times\mathbb{Z}_m$, so for two distinct integers $a,b$ we must have $\langle F_a,F_{a+1}\rangle\equiv\langle F_b,F_{b+1}\rangle \pmod{m}$ as a consequence of the Dirichlet box principle. However, the last condition implies $F_{a+2}\equiv F_{b+2}\pmod{m}$ and, by induction, $F_{a+k}\equiv F_{b+k}\pmod{m}$. Hence the period of the Fibonacci sequence $\pmod{m}$ is bounded by $m^2$ ($m^2-1$ if we are careful enough to notice that $\langle F_c,F_{c+1}\rangle\equiv\langle0,0\rangle\pmod{m}$ is not possible since two consecutive Fibonacci numbers are always coprime). Now it suffices to take $m=10^{2014}$ and notice that $F_0=0$ to prove that there exists an integer $u\leq 10^{4028}$ such that $F_u\equiv 0\pmod{m}$, i.e. $F_u$ ends with at least $2014$ zeroes.

It is also possible to give better estimates for $u$.

Since $F_k$ is divisible by $5$ only when $k$ is divisible by $5$ and:

$$ F_{5k} = F_k(25 F_k^4 + 25(-1)^k F_k^2 + 5),$$ it follows that: $$ \nu_5(F_k) = \nu_5(k), $$ so $$u\leq 2^{4028}\cdot 5^{2014}=20^{2014}$$ by the Chinese theorem. I put a proof of the Oleg567 conjecture: $$ k\mid F_n \quad\Longrightarrow\quad k^d\mid F_{k^{d-1}n} $$ in a separate question. Since $8=F_6\mid F_{750}$ (because $6\mid 750$) and $\nu_5(750)=3$, we have that $1000|F_{750}$ and through the Oleg567's lemma we get $$ u\leq \frac{3}{4}10^{2014}.$$

Jack D'Aurizio
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  • Wait, how does the congruence of the gcd's of two pairs of integers lead to the congruence of the sums of each pair? – Nishant Jul 19 '14 at 22:22
  • It is not the gcd, it is the couple of residue classes as an element of $\mathbb{Z}_m\times\mathbb{Z}_m$. So we say that $(2,11)\equiv (9,4)\pmod{7}$, for instance. – Jack D'Aurizio Jul 19 '14 at 22:33
  • Oh, so $a$ and $b$ are specially chosen integers, and they can't just be any integers? – Nishant Jul 19 '14 at 22:37
  • I am just saying that is we consider the function $f:\mathbb{N}\to\mathbb{Z}_m\times\mathbb{Z}_m$ defined by $f(n)=(F_n \pmod{m},F_{n+1}\pmod{m})$, for two distinct integers $a,b$ we must have $f(a)=f(b)$. – Jack D'Aurizio Jul 19 '14 at 23:13
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    Unless I'm missing something, your argument shows that the sequence $(F_n)$ is *ultimately* periodic mod $m$. I fail to see how you can conclude that $F_n = 0$ mod $m$ for some $n > 0$. – Yann Jul 19 '14 at 23:45
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    Since $F_0\equiv 0\pmod{m}$ for every $m$, given that $\phi(m)$ is the period of the Fibonacci sequence $\pmod{m}$, then $F_{\phi(m)}\equiv 0\pmod{m}$. It is more or less a proof of the Fermat little theorem in disguise. – Jack D'Aurizio Jul 20 '14 at 00:22
  • There is nothing strange since the period of the Fibonacci sequence $\pmod{10}$ is $\phi(10)=60$, given by $F_0\equiv F_{60}\pmod{10}, F_1\equiv F_{61}\pmod{10}$, or $f(0)=f(60)$. – Jack D'Aurizio Jul 20 '14 at 02:54
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    @Yann: the Fibonacci recurrence is reversible, so showing that the Fibonacci sequence is eventually periodic shows that it's always periodic. – Qiaochu Yuan Jul 20 '14 at 03:56
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    I've swapped out parentheses for angle brackets in your ordered pairs to try and ameliorate some of the confusion - if you'd rather they were parens then by all means feel free to swap them back, but I think it's a little clearer this way. – Steven Stadnicki Jul 20 '14 at 04:01
  • See also [this answer](http://math.stackexchange.com/a/744335/242) on *reinventing the wheel* (cyclical structure of permutation orbits). – Bill Dubuque May 04 '15 at 13:28
  • Since you have not used any property of `m`, it means it is possible to replace $10^{2014}$ with any positive number, isn't it? Thus, for any integer there are infinitely many Fibonacci numbers which are divisible by it... Sounds cool but strange. – antonpp May 20 '17 at 01:08
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We can do a little algebraic number theory. Let $\phi$ be a root of $X^2 - X - 1$ over $\mathbb{Q}$ ("golden ratio"), and let us work in the number field $\mathbb{Q}(\phi) = \mathbb{Q}(\sqrt{5})$ and its ring of integers $\mathbb{Z}[\phi]$: we call $v_2$ and $v_5$ the valuations of $\mathbb{Q}(\phi)$ which extend the usual $2$-adic and $5$-adic valuations on $\mathbb{Q}$.

The $n$-th Fibonacci number is

$$F_n = \frac{\phi^n - \phi^{\prime n}}{\phi - \phi'}$$

where $\phi' = 1-\phi$ is the conjugate of $\phi$. The question is to characterize the $n$ for which $v_2(F_n) \geq 2014$ and $v_5(F_n) \geq 2014$ (and, of course, show that such an $n$ exists). Now $\phi - \phi' = 2\phi - 1 = \sqrt{5}$, so clearly $v_2(\phi - \phi') = 0$ and $v_5(\phi - \phi') = \frac{1}{2}$. Also, since $\phi\phi' = 1$, it is clear that $\phi,\phi'$ are units, so $v_2(\phi) = v_2(\phi') = 0$ and $v_5(\phi) = v_5(\phi') = 0$.

Concerning $v_2$, we now have $v_2(F_n) = v_2(\phi^n - \phi^{\prime n}) = v_2(\lambda^n-1)$ where $\lambda = \phi'/\phi = \phi - 2$. Annoyingly, the $2$-adic exponential only converges (on unramified extensions of $\mathbb{Q}_2$, here the completion of $\mathbb{Q}(\phi)$ under $v_2$) for $v_2(z)>1$, and we have to go as far as $n=6$ to get $v_2(F_n) = 3 > 1$, after what it is clear that $v_2(\lambda^{6k}-1) = 3 + v_2(k)$ by proposition II.5.5 of Neukirch's Algebraic Number Theory (quoted below; here $e=1$ and $p=2$). For $n$ not congruent to $6$, it is then easy to see that $v_2(\lambda^n-1)$ is $1$ if $n$ is congruent to $3$ mod $6$ and $0$ if $n$ is congruent to $1,2,4,5$ mod $6$. So the $n$ for which $v_2(F_n) \geq 2014$ are the multiples of $2^{2011} \times 6 = 2^{2012} \times 3$.

Concerning $v_5$, have $v_5(F_n) = -\frac{1}{2} + v_5(\phi^n - \phi^{\prime n}) = -\frac{1}{2} + v_5(\lambda^n-1)$. This time, convergence of the exponential is unproblematic because ramification is tame (in the notation of Neukirch's above-quoted proposition, $e=2$ and $p=5$): we have $v_5(\lambda^n-1) = \frac{1}{2} + v_5(n)$, that is $v_5(F_n) = v_5(n)$, for every $n$. So the $n$ for which $v_5(F_n) \geq 2014$ are the multiples of $5^{2014}$.

So, putting this together, the $n$ for which $F_n$ ends with 2014 zeroes are the multiples of $75\times 10^{2012}$. The first one is $F_{n_0}$ where $n_0 = 75\times 10^{2012}$.

As a bonus, we could show that the last few digits of $F_{n_0}$ before the 2014 zeroes are: $177449$.


Edit: For convenience, here is the statement of the proposition in Neukirch's book that I quote above:

Let $K|\mathbb{Q}_p$ be a $\mathfrak{p}$-adic number field with valuation ring $\mathcal{O}$ and maximal ideal $\mathfrak{p}$, and let $p\mathcal{O} = \mathfrak{p}^e$ [Gro-Tsen's note: $p$ is the residual characteristic, so $e$ is the absolute ramification index]. Then the power series

$$\exp(x) = 1 + x + \frac{x^2}{2} + \frac{x^3}{3!} + \cdots$$

and

$$\log(1+z) = z - \frac{z^2}{2} + \frac{z^3}{3} - \cdots$$

yield, for $n > \frac{e}{p-1}$, two mutually inverse isomorphisms (and homeomorphisms)

$$\mathfrak{p}^n \overset{\exp}{\underset{\log}{\mathop{\rightleftarrows}}} U^{(n)}$$

[Gro-Tsen's note: $\mathfrak{p}^n$ is the set of elements with valuation $v(x) \geq n/e$ (normalized by $v(p) = 1$), and $U^{(n)} = 1 + \mathfrak{p}^n$ is the set of $z$ such that $v(z-1) \geq n/e$.]

We apply this to the completion of $\mathbb{Q}(\phi)$ under $v_2$ or $v_5$. The conclusion we use is that $v(\exp(x)-1) = v(x)$, or rather, $v(c^x - 1) = v(x) + v(\log c)$ (where $c$ is $\lambda^6$ in the case of $v_2$ and $\lambda$ in the case of $v_5$, and $v(\log c)$ is a constant we compute).

Gro-Tsen
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    Great answer! I am just not clear about digits 677449 - how exactly were they obtained? – VividD Jul 21 '14 at 00:23
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    @VividD: Concerning the digits before the zeros, the point is that $(c^x-1)/x$ tends to $\log c$ when $x\to0$ (if $c$ is a $\mathfrak{p}$-adic sufficiently close to $1$, as in the end of my answer). That and the expressions for $F_n$ easily imply that if $n=75\times 10^k$ then $F_n/10^{k+2}$ converges $2$-adically and $5$-adically (hence, "$10$-adically") when $k$ is an integer tending to $+\infty$. And we can compute their limits (something like $(\log(\lambda^6)/(\phi-\phi'))/8$). (continued) – Gro-Tsen Jul 21 '14 at 01:12
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    (cont.) Now of course, I cheated: knowing that the 10-adic limit exists, i.e., that the last digits of $F_n/10^{k+2}$ stabilize, I just computed the first few values and got the value from there. But it wouldn't be very difficult to work out the error bounds explicitly. (This is why I wrote "we could show".) Of course, one needs a computer algebra software or a great deal of patience to actually compute $F_{75000}$, if only in a $2$- and $5$-adic approximation (or some expression involving $p$-adic logs). – Gro-Tsen Jul 21 '14 at 01:19
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    All answers are good, but I liked this the most. – VividD Jul 23 '14 at 14:31
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    @VividD: Here's an extra comment you might like: the function $\mathbb{Z}\to\mathbb{Z}$ defined by $n\mapsto F_{3n}$ extends uniquely to a continuous and indeed analytic function $\mathbb{Z}_2\to\mathbb{Z}_2$. The same holds for $n\mapsto F_{4n}$ giving $\mathbb{Z}_5\to\mathbb{Z}_5$. I think it's an interesting exercise in $\mathfrak{p}$-adics to show this (and to consider where the $3$ and $4$ come from, and to compute their derivatives at $0$). – Gro-Tsen Jul 23 '14 at 15:11
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Just observation: $$F_{750}\equiv 0 \pmod{1000}$$ $$F_{7500}\equiv 0 \pmod{10000}$$ $$F_{75000}\equiv 0 \pmod{100000}$$ $$F_{750000}\equiv 0 \pmod{1000000}$$ $$F_{7500000}\equiv 0 \pmod{10000000}$$


Here (Pisano Period) is said, that the sequence of last $d$ digits of Fibonacci numbers has a period of $15·10^{d-1}$.
Our $75\cdot 10^{d-2}$ is a half-period (knowing that $F_0=0$).


I think must be some property of Fibonacci numbers: $$ \large{ k|F_n \quad \Rightarrow \quad k^d|F_{k^{d-1}n}. }$$

Lynn
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Oleg567
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    All of this must be known to Fib specialists, but that term doesn’t apply to me. By a cumbersome $p$-adic computation ($p=2,5$), I seem to have shown that for $m\ge3$, $2^m|F_k$ if and only if $3\cdot2^{m-2}|k$. And that $5^m|F_k$ if and only if $5^m|k$. This would show that your $750$, etc. are the smallest numbers satisfying the desired conditions. – Lubin Jul 20 '14 at 13:45
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Given any n (even one with 2014 trailing zeros) there has to be values b and k so that $F_b\equiv F_{b+k} \pmod{n}$ AND $F_{b+1}\equiv F_{b+k+1} \pmod{n}$ (which is equivalent to the claim the mod n residues have to eventually form a repeating cycle, and proven in other answers).

Once such b and k values are found, by the "two hop" identity using hops of 1 and k starting from index b we get $F_bF_{b+k+1} = F_{b+1}F_{b+k} - (-1)^{b}F_kF_1$.

Now, notice that $F_bF_{b+k+1}\equiv F_{b+1}F_{b+k} \pmod{n}$ by construction, so it must be $F_k\equiv 0 \pmod{n}$, that is, $F_k$ has 2014 trailing zeroes.

edit: corrected typo on exponent to (-1)

Random Excess
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    This is so far the shortest and the most elegant answer/proof. – VividD Jul 22 '14 at 07:21
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    I couldn't find what "two hop" identity is, however d'Ocagne's identity $F_nF_{m+1} = F_{n+1}F_{m} + (-1)^{m}F_{n-m}$ can be equivalently used. – VividD Jul 22 '14 at 07:38
  • Yes, that is just the special case where one of the hops is of length 1 and the other is n-m from a starting point of m and rearranged. $F[m]F[n+1] = F[m+1]F[n] - (-1)^{m}F[n-m]F[1]$ – Random Excess Jul 22 '14 at 08:35
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Let $\ell_n$ be the last $2014$ digits of of the $n$th of the Fibonacci sequence. Note if we know $\ell_{k+1}$ and $\ell_k$, then we know $\ell_{k-1}$. So if for two natural numbers $n,k$ we find that $\ell_n=\ell_{n+k}$ and $\ell_{k+1}=\ell_{n+k+1}$, then it follows that $$\begin{align*}\ell_{k-1}&=\ell_{n+k-1}\\\ell_{k-2}&=\ell_{n+k-2} \\ &\dots \\\ell_1&=\ell_{n+1} \end{align*}$$And since $\ell_1=0$, it follows that $\ell_{n+1}=0$ and hence the $(n+1)$st Fibonacci number will end in $2014$ zeroes.

Now let us prove that such a number exists. Consider the following pairs: $$\begin{align*} \ell_1&,\ell_2 \\ \ell_{2}&,\ell_3 \\ &\vdots \\ \ell_{10^{48}}&, \ell_{10^{48}+1} \\ \ell_{10^{48}+1}&, \ell_{10^{48}+2} \end{align*}$$Since there are $\left({10^{24}}\right)^2$ different values of the pairs, by the Pigeonhole Principle, there exists $i,j$ where $1<i<j\leq 10^{48}+1$ such $\ell_{i}=\ell_{j}$ and $\ell_{i+1}=\ell_{j+1}$. It follows that $(\ell_1,\ell_2)=(\ell_{j-i+1},\ell_{j-i})$ hence the $j-i+1$st Fibonacci number ends in $2014$ zeroes.

user574848
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