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Consider set $A:= \{F_{n} \mid n, \, 1 \leq n \leq 1000002\}$ where $F_{n}$ denotes the $n$th Fibonacci number.

Prove that exist at least two numbers $F_a$ and $F_b$, such that $F_a,F_b \in A$ and $F_a$, $F_b$ are divisible by $1000$.

I trying to use the Dirichlet's box principle, but I have problem, how to define pigeonhole.

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  • $\begingroup$ Not sure this is clear. $F_0=1=F_1$ and clearly $1\,|\,1000$. But, then, I think you meant something else, no? $\endgroup$ – lulu Jan 11 at 0:09
  • $\begingroup$ Is it possible that you meant to require $1000\,|\,F_a$? If so, then note that $1000=2^3\times 5^3$ so try to find indices $i,j$ for which $8\,|\,F_i$ and $125\,|\,F_j$ and then use the divisibility properties of the Fibonacci numbers. (Note: the divisibility properties look much better if you use the convention $F_0=0,F_1=1$ instead of the convention I used earlier). $\endgroup$ – lulu Jan 11 at 0:19
  • $\begingroup$ Yes. I have already corrected. Sorry. $\endgroup$ – pawelK Jan 11 at 0:21
  • $\begingroup$ I don't see any correction. Anyway, assuming I guessed your intent correctly, my hint should lead to a solution pretty quickly. Not sure it's a pencil and paper solution, but it is close. $\endgroup$ – lulu Jan 11 at 0:23
  • $\begingroup$ Do you mean $1000$ divides $F_a,\ $ or $\ F_a$ divides $1000?\ \ $ $\endgroup$ – Bill Dubuque Jan 11 at 0:24
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Every sixth element is divisible by $8$ and every $125^{th}$ is divisible by $125$, just by computation. Then every $750^{th}$ is divisible by $1000$

Note that we just have to find the first element divisible by any prime power. The next number after that may not be $1$, but the recurrence is linear. In this example, $F_7\equiv 5 \pmod 8$, but the next block is just the first block multiplied by $5$ and $0 \cdot 5=0$

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  • $\begingroup$ "just by computation" hides a lot. Also, this does not use the pigeon hole principle. $\endgroup$ – marty cohen Jan 11 at 2:25
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    $\begingroup$ @martycohen: I don't think it hides anything. I just made a spreadsheet and I justified why once you find one zero it will repeat at that period. I agree it does not use pigeonhole. I don't have a good pigeonhole approach. It looks like we are supposed to use $1000^2=1000000$ but I don't see how to show there isn't a correlation that prevents a solution. $\endgroup$ – Ross Millikan Jan 11 at 3:02
  • $\begingroup$ I agree that is true. Look up divisibility sequences. $\endgroup$ – marty cohen Jan 11 at 3:04

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