It is a very famous series, and the discussions around it are endless, with many interesting and profound conclusions.

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(Source: Mathematical Olympiad Series Second Edition High School Volume Series and Mathematical Induction, Feng Zhigang Fibonacci Series P057 Case 0 1)

Prove everything, that is, to find the greatest common factor of the terms of Fibonacci sequence can be converted into subscripts.

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It was obviously established at that time. Considering the situation, you might as well set it.

Using the recurrence formula of Fibonacci sequence, we can know that

So (used here, it can be proved by mathematical induction).

In the above conclusion, the discussion continues with substitution, which shows that the process of the greatest common factor of summation is essentially the division and tossing of subscripts.

It shows that the following proposition can be proved by the conclusion of this question: if it is a prime number, then it is a prime number.

In fact, if it is not a prime number, it can be written as,, and. At this point, sum is derived as a composite number.

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(Source: Mathematical Olympiad Series Second Edition High School Volume Series and Mathematical Induction, Case 02 of Fibonacci Series P058 in Feng Zhigang)

It is proved that every positive integer can be uniquely expressed in the following form.

Here is still, and there is no subscript, so here is the first term of Fibonacci sequence.

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A positive integer in the form of (1) represents what can be called a representation, similar to binary. This conclusion is the famous Seckendorf theorem.

Prove induction.

If, this proposition holds. Now it applies to all positive integer propositions less than.

Because there is uniqueness, if, then it has been expressed in the form of (1), if, then it is assumed that there is a representation in the form of (1) by induction.

That's among them. So, now it is contradictory, so there is a satisfactory appearance (1).

The form of latent syndrome (1) is unique.

In fact, if

There are more here.

If there is no subscript, then the definition of combination can be known.

Therefore, (2) can't take all equations.

So there are two appearances, which are inconsistent with the inductive hypothesis. Therefore, the representation of is unique.

To sum up, according to the second mathematical induction, the proposition holds.

2022-0 1-27-03

(Source: Mathematical Olympiad Series Second Edition High School Volume Series and Mathematical Induction, Case 03 of Fibonacci Series P058 in Feng Zhigang)

As we all know, the product of any continuous integer is a multiple of the product of the previous positive integer. Fibonacci series has similar properties. Please prove that the product of any continuous item in the series is a multiple of the product of the previous item.

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Introduce tags and specify. Then write.

In order to prove the proposition, it is only necessary to prove that there is any,, and.

Using a similar derivation process in the example 1, we know that

So, we have

The above formula holds for both, and it can be proved to be a positive integer by combining the initial situation (for both) and mathematical induction.

Therefore, the proposition holds.