Display of infinite isoperimetric series of natural numbers using non-adjacent Fibonacci numbers -1

1. what we know now　

From Zeckendorf's theorem, we know that we can uniquely represent a natural number by the sum of non-adjacent Fibonacci numbers.

Thanks to Euclid and Gauss, we know that we can uniquely represent a natural number by a product of prime numbers.

Wilson's theorem shows that Γ(n) (where n is a natural number) can be uniquely represented by a fraction of Γ(p) (where p is a prime number). 2.

2. what we found out

The number of non-adjacent Fibonacci numbers minus one can be uniquely expressed by the sum of the natural numbers as in Zeckendorf's theorem.

The natural numbers were expressed using infinite identity series.

The following thoughts gave me an idea.

Γ(p) is (p-1)! is (p-1)!

→Fibonacci numbers can be made only in the world of sums, so multiplication will not be used (*1)

→For Fibonacci numbers f, isn't f-1 also unique with respect to the decomposition of the sum of natural numbers?

→I actually tried to find out.

→I was able to do it. (Continued in Chapter 3)

→Thinking about the reciprocal, I noticed that it resembles the form of an infinite isoperimetric series.

→I was able to express the natural numbers using infinite identity series. (Continued from Chapter 4)

(*1 Likewise, there is a property that the square of any Fibonacci number is close to the product of the Fibonacci numbers before and after it (by 1 more or less). The general term also contains n squared. But even if you don't know that, you can make it with an asymptotic formula).

3. how to do

Think about 1) 18, 2) 50 3) 712

Step 1 Choose the smallest Fibonacci number minus 1 that is greater than or equal to the number you are considering.

1) 20 2) 52 3) 986

Step 2 Think about the absolute value of the difference between the number you found in Step 1 and the number you are considering.

1) 2 2) 2 3) 274

Step 3 Repeat Step 1 and Step 2, ending with 0 in Step 2.

Result

1) 18 = 20-2

2) 50 = 52-2

3) 712 = 986-(376-(143-(54-(20-7)))))

4. infinite identity series

The following image shows how to decompose 1/5 and 1/57. Please try to think of them with other numbers as well.

©︎2021 Yume Isioto