Before starting, thank you dear teacher jk for his great teaching!!!!!!
The work of this article: I have refined the ppt and made some supplementary explanations.
bijective proof : translated into one-to-one mapping method/bijective proof. The purpose is to construct a kind of problem that has a one-to-one mapping relationship with the original problem and is easy to solve when the original problem is difficult to solve. By solving the class problem, the original problem is solved,
·Description of the topic:
Proof method: use the uniqueness of power-square decomposition to construct a one-to-one mapping.
Method description:
In odd number decomposition, some odd numbers are repeated, such as 21=3+3+3+3+3+1+5, in which 3 is repeated 5 times. Then we need to convert these five 3s into two different numbers to achieve the purpose of "two different splits".
Let's take this decomposition as an example to talk about the conversion method:
① We use a pair of two-tuples to describe the "five 3s". The two-tuples are defined as (multiplicity k, odd number a)=(5,3)
②According to the principle of power square decomposition: any positive integer can be decomposed into a unique power of 2 with different items
That is, k= , where
Then . NOTE: This decomposition is unique! ! ! This ensures that our transformations are a one-to-one mapping.
③Since each item in the power decomposition above is unique, then they can be combined with an odd number a to obtain different numbers
For example 3+3+3+3+3 =3*5=1*3+4*3 = 3 + 12
Then convert these 5 repeated 3s into two non-repeated numbers 3 and 12
In this way, "odd numbers with repetitions" can be converted into unique "numbers without repetitions"
it's not over yet
Question 1: Is it possible for the converted non-repeated numbers to conflict with other numbers and cause new repetitions?
Answer: impossible.
Analysis: There are two possibilities for duplication: 1. Duplication with the number existing in the original sequence; 2. Conflict with other converted numbers that have duplicated odd numbers
For the time being, name the repeated number as a, then a is obtained by conversion, and can be written as , where 2k+1 represents an odd number that decomposes a
Possibility ①: When a is an even number, it is impossible because the original sequence is an odd sequence.
When a is an odd number, t can only be 0 (otherwise a is an even number), then a is the odd number itself that should be decomposed. Then it will naturally not repeat with other numbers in the original sequence.
Possibility ②: Duplicate with other converted odd numbers. This problem is equivalent to: can be obtained by decomposing two numbers. This may obviously not be true. There is one and only one way to decompose any number into an odd number*2 power! In other words, the odd number 2k+1 is unique!
Discussion results: This method can uniquely convert the odd-numbered splits with repetitions into non-repeated splits
it's not over yet
Question 2: The above results only show that there is a one-way mapping from "odd split" to "two-two non-repeated split" in this method. Is there a certain "two-two non-repeated split" in "odd split" Can't find the original image?
Answer: no!
Analysis: Prove that any "two-two non-repeated split" can be uniquely mapped back to "odd split"
Purpose: For an even number a in a certain "two-by-two non-repeating split", convert it back into some odd sums, and this conversion is unique. (Note that it is reverse : return along the original path, so Only then can it be proved that it is a bijective relationship, otherwise it is a random shot)
①Assume that a is converted from "odd split", then a=
②Since the splitting of any even number is unique (proved in number theory), then both 2^t (multiplicity) and 2k+1 (the odd number in the original image) can be uniquely determined.
③Write a in the form of an odd number combination: a= (2k+1) + (2k+1) +....+(2k+1) There are 2^t in total.
Conclusion : Prove the unique mapping relationship from "two-two non-repeated split" to "odd split" along the original path.
In summary:
This method constructs a bijective bridge between two sets of splits, and proves that the two splits are in one-to-one correspondence.
Below is the modified ppt.
Conclusion: There may be flaws in the logic of the above discussion, so let's just look at it. Think of the loopholes later and fill them up.