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The classical closed form of the nth fibonacci number is

$$x_{n}=\frac{(1+5^{1/2})^{n} -(1-5^{1/2})^n}{\sqrt{5}2^n}$$

But is there any way to compute it with only integer operations? By that, I mean no intermediate result has a fractional part (e.g. 1.222). So this would rule out the square root of 5. But if a new formula involved square roots on square integers, then this is allowed.

I'm pretty sure there's not based on what I've read so far. But since this is math question, what is the proof that such a formula does not exist?

In additional, I'm aware of some ways to compute it with integer operations such as matrix exponentiation and fast doubling technique. I consider these techniques involve integer operations but aren't closed forms. It's kind of controversial though for the matrix exponentiation case because of the exponentiation. But let's assume that exponeniation on integers is O(1) (suggesting closed-form) and on matrix is non-constant time.

Since I think closed-form is kind of unspecific. Let me be specific for our ease, this means at the minimum*:

  1. Integer arithmetic (add, sub, mul, div)
  2. Integer roots
  3. Integer powers

And if it's not possible with just that, I feel like these additional operations would still fit albeit loosely:

  1. Factorial
  2. Binomial Coefficient

I am not trying to ask two questions. The real question is what's the "most closed-form" I could get, with the first three rules being a priority.

*In case I forgot some operations on integers that are necessary, please let me know.

Mercado
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    Does this count? https://math.stackexchange.com/q/61997/42969 – Martin R Mar 12 '20 at 04:20
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    Do you consider binomial expansions to be valid? Since you can just binomially expand this. I don't like questions of this form since the goal posts keep moving since OP never knows what they want. – Rushabh Mehta Mar 12 '20 at 04:23
  • @MartinR Edit the post to add some concerns on matrix expo. I consider closed form as one involving number operations only, so excluding matrices. I don't think there is a way to reduce the matrix expo to some closed form. – Mercado Mar 12 '20 at 04:24
  • @Martin R Interesting suggestion but I'm not sure how the matrix multiplication would be turned into a expression for individual Fn unlike the standard recursive definition of Fn. – Jam Mar 12 '20 at 04:25
  • @DonThousand That is interesting. Is there a formal definition of closed-form and would binomial coefficient be consider one? I would take an advice and I am really just curious. – Mercado Mar 12 '20 at 04:28
  • @Mercado There is no formal definition for closed form. At least not a general one. Within certain fields, there may be. – Rushabh Mehta Mar 12 '20 at 04:34
  • The n'th derivative of the generating function evaluated at 0 effectively gives you an expression that can be computed with only addition and multiplication. But I wouldn't personally call it a "closed form" in the strict sense since it's really a limit or recursion in disguise. – Jam Mar 12 '20 at 04:38
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    What is permitted as a "closed form" is context dependent but typically only includes finite expressions of elementary functions and excludes operations involving limits (derivatives, series, integrals). But what's included in "elementary functions" is fairly arbitrary imo. – Jam Mar 12 '20 at 04:42
  • @DonThousand I edited the post to be more specific on what I'll accept as an answer. It's in my best interest to know what's the most closed-form that's possible. – Mercado Mar 12 '20 at 04:42

1 Answers1

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$$F_n=\sum_{k=0}^{[(n-1)/2]}{n-k-1\choose k}$$ is a closed form expression using only integer operations (unless one objects to $[(n-1)/2]$, the integer part of $(n-1)/2$).

Gerry Myerson
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