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How would you go about to prove that the final digits of the Fibonacci numbers recur after a cycle of 60?

References:

The sequence of final digits in Fibonacci numbers repeats in cycles of 60. The last two digits repeat in 300, the last three in 1500, the last four in , etc. The number of Fibonacci numbers between and is either 1 or 2 (Wells 1986, p. 65).

http://mathworld.wolfram.com/FibonacciNumber.html

crazyGuy
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5 Answers5

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Since each term in the Fibonacci sequence is dependent on the previous two, each time a $0\pmod{m}$ appears in the sequence, what follows must be a multiple of the sequence starting at $F_0,F_1,\dots=0,1,\dots$ That is, a subsequence starting with $0,a,\dots$ is $a$ times the sequence starting with $0,1,\dots$

Consider the Fibonacci sequence $\text{mod }2$: $$ \color{red}{0,1,1,}\color{green}{0,1,1,\dots} $$ Thus, the Fibonacci sequence repeats $\text{mod }2$ with a period of $3$.

Consider the Fibonacci sequence $\text{mod }5$: $$ \color{red}{0,1,1,2,3,}\color{green}{0,3,3,\dots} $$ Thus, the Fibonacci sequence is multiplied by $3\pmod{5}$ each "period" of $5$. Since $3^4=1\pmod{5}$, the Fibonacci sequence repeats $\text{mod }5$ with a period of $20=4\cdot5$.

Thus, the Fibonacci sequence repeats $\text{mod }10$ with a period of $60=\operatorname{LCM}(3,20)$.

robjohn
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Notice that:

$F_{n+15} \equiv 7F_n \pmod{10}$ for $n\geq 1$.

Also the order of $7$ mod $10$ is $4$ so the repetition in the digits of the Fibonacci numbers begins after place $15\times 4 = 60$.

thepiercingarrow
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fretty
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  • I got that recursion by just writing out the first $16$ Fibonacci numbers mod $10$ and noticing that you get $1,1,...,7,7,...$, this dictates that that next $13$ numbers must be $7$ times their corresponding numbers (mod $10$) in the first lot of $15$ numbers. – fretty Feb 26 '12 at 10:03
  • How do we know that F_n+15 = 7F_n mod 10 ? and How do you get that 7 mod 10 is 4 ? thanks. – crazyGuy Feb 26 '12 at 10:07
  • Write out the first 16 Fibonacci numbers mod 10: 1,1,...,7,7,... Now by the recursion of the fibonacci numbers the fact that this "multiple-repeat" has happened now tells us that the next 13 numbers must be the same as the 3rd-15th Fibonacci numbers but multiplied by $7$ mod 10. So an actual repeat will happen as soon as we have multipled by enough $7$'s to get $1$ mod 10. This is the "order" of $7$ mod $10$. It is easy to work out here, $7^2 \equiv 4$ mod $10$ and $7^4 \equiv 1$ mod $10$ so the order is $4$. – fretty Feb 26 '12 at 10:13
  • @sic2 He's not saying that $7\mod10$ is $4$, but rather that $7^4\equiv 1\mod 10$ and that this is not true for any smaller power of $7$, which is a consequence of Euler's theorem. – Alex Becker Feb 26 '12 at 10:15
  • Also this method generalises to the Fibonacci numbers mod $n$. Find the first $1,1,...,a,a,...$ then work out the order of $a$ mod $n$ and you will have your first place where it recurs. – fretty Feb 26 '12 at 10:29
  • This is equivalent to finding the first occurrence of $0$, since the two numbers after this will be of the form $a,a$, where $a$ is the number before the $0$. – fretty Feb 26 '12 at 11:57
  • @sic2, when he says "the order of 7 mod 10", he's referring to the "multiplicative order." – senderle Feb 26 '12 at 14:46
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    In my answer, it is shown that $F_{n+5}\equiv3F_n\pmod{5}$ and $F_{n+3}\equiv F_n\pmod{2}$. Thus, we get that $F_{n+15}\equiv2F_n\pmod{5}$ and $F_{n+15}\equiv F_n\pmod{2}$. The Chinese Remainder Theorem then says that $F_{n+15}\equiv7F_n\pmod{10}$. That's a different way than computing $16$ terms. – robjohn Feb 26 '12 at 19:31
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Note that

$$ \begin{pmatrix} F_{n+1}\\ F_{n+2} \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ 1 & 1 \end{pmatrix} \begin{pmatrix} F_n \\ F_{n+1} \end{pmatrix} $$

and

$$ \begin{pmatrix} 0 & 1 \\ 1 & 1 \end{pmatrix}^{60} \equiv \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \mod 10. $$

One can verify that $60$ is the smallest power for which this holds, so it is the order of the matrix mod 10. In particular

$$ \begin{pmatrix} F_{n+60}\\ F_{n+61} \end{pmatrix} \equiv \begin{pmatrix} F_n \\ F_{n+1} \end{pmatrix} \mod 10 $$

so the final digits repeat from that point onwards.

WimC
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    The same matrix applies to the Lucas Numbers, however, $\text{ mod }10$, they have a period of $12$. The fact that $\begin{pmatrix} 0&1\1&1\end{pmatrix}^{60}=\begin{pmatrix}1&0\0&1\end{pmatrix}\text{ mod }10$, says that the period of a sequence satisfying $a_n=a_{n-1}+a_{n-2}$ divides $60$. – robjohn Feb 26 '12 at 19:21
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$F_{0}=F_1=F_{60}=F_{61}= 1 \mod10$

By inspection, these are the first two pairs of consecutive Fibonaccis for which this is true. Since the recurrence relation only takes into account the previous two terms and last digits only depend on previous last digits, this suffices to prove the claim.

Tom Boardman
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I have found that there is an 11 number grouping. You must reduce numbers to a single digit (13=4 or 55=10=1). After 24 numbers into the sequence it repeats. To go a little deeper, a 9 is in the twelfth and twenty- fourth sequence and the real number is also divisible by 12 in these positions (which coincidentally is every twelfth number in the sequence).

Now if you do the multiplication table of numbers 1 through 8 and reduce all numbers to a single digit, you will find that 1 and 8 correspond in reverse with each other. The same for 2 and 7, also 4 and 5. 3 and 6 are different. You will notice that for every number sequence it will repeat after a 9 appears. So, Fibonacci sequence would be this:

1,1,2,3,5,8,4,3,7,1,8,9,8,8,7,6,4,1,5,6,2,8,1,9,1,1,2,3,5,8,4,3,7,1,8,9,8,8,7,6,4,1,5,6,2,8,1,9......

Remember how I said 1 and 8, 2 and 7, 4 and 5 correspond with each other by reducing the multiplication table to single digits and all numbers repeat a sequence after a 9?

If you move the eleven number sequence between the nines and set 1,1,2,3... above the sequence 8,8,7,6... You will find that they all correspond with the reducing method afore mentioned.

It's very interesting.

David Andrei Ned
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Michael Bunton
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  • What do you mean $11$-number grouping? Why are you reducing mod $9$ instead of mod $10$? How does this prove that it repeats with cycle $60$? – Caleb Stanford Oct 24 '16 at 02:58
  • I apologize. This was not to prove or disprove a 60 cycle repeat. Maybe I posted in the wrong area? I just found this sequence repeat and I thought I would give it to bigger brains and see what they can make of it. – Michael Bunton Oct 24 '16 at 03:12
  • 11 number grouping means the first 11 numbers in the sequence. The twelfth number is 144 and that reduces to 9 i.e.(1+4+4=9). Then a second set of eleven numbers followed by a reduced 9. – Michael Bunton Oct 24 '16 at 03:20