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In mathematics, when an important mathematical definition has been proposed and an important mathematical theorem has been proved, things are far from over. No matter how clear a mathematical work is, there is always room for more understanding of it, and one of the most common ways is to state it as a special case of a broader thing (promotion). There are different kinds of promotion, and only a few of them are discussed here.

Weakening hypothesis and strengthening conclusion the number 1729 discovered by the talented Indian mathematician Ramanujin is famous because it can be written as the complete cubic sum of two positive integers in two different ways, that is.

And 1729 is the smallest of its kind. Let's try to see if there is a number that can be written as the sum of four complete cubes in four different ways.

At first glance, it seems that this question is rare and surprising. If there is such a number, it must be very large, and if you want to try it one by one, it must be extremely tedious. So, is there any clever way?

The answer is that the hypothesis must be weakened. The problem we want to solve belongs to the following general type. In this paper, we give a sequence of positive integers, a sequence of positive integers, a sequence of positive integers. And tells us that the sequence has some property. Then we have to prove that there must be a positive integer so that it can be written as the sum of four terms in this sequence in ten different ways. Thinking about the problem in this way may be a bit artificial, because it is assumed that the sequence is a "sequence of complete cubes", because the sequence of this property is too special (compared to the so-called "sequence with a certain property"). So the more natural idea is to regard this problem as a specific sequence identification problem. However, this way of thinking encourages us to consider the possibility that this conclusion may still be true for a much broader sequence, and it does.

There are 1 000 complete cubes less than or equal to 1 000 000 000. We will see that this fact is sufficient to ensure that "there is an integer, and it can be written as the sum of four complete cubes in ten different ways." Specifically, our problem becomes to prove that ∶

In order to prove this, we should first notice that there are 24 ways to take any four items from the sequence, which is less than 40 billion, and the sum of any four items in this sequence must not be greater than 4 billion. So now there are 40 billion numbers not greater than 4 billion, among which there must be repeated numbers. on average, there should be more than ten numbers with the same value. Therefore, in 40 billion numbers, at least one of the 4 billion values will be taken more than ten times, and the certificate is completed.

Why does popularizing the problem in this way help solve the problem? One might think that in proving a result, the fewer assumptions there are, the more difficult it is to prove. However, this is often not the case. The fewer assumptions you have, the fewer choices you need to make when using this hypothesis to prove it, which sometimes speeds up the search for proof. If this problem is not promoted as above, there will be too many choices. For example, you might try to solve the very difficult Diophantine equation with a cube term, instead of doing a simple counting problem as it is now.

We can also think that the generalization of the above is the reinforcement of the conclusion. ∶ the original problem is only a proposition about the cube, but we have much more proof. There is no clear difference between weakening hypothesis and strengthening conclusion.

It is proved that there is a famous result in modular arithmetic of a more abstract result, called Fermat's Little Theorem ∶. If p is a prime and a positive integer an is not a multiple of p, then a ^ (pmae1) divided by p, the remainder must be 1. That is to say, a ^ (pMel 1) mod p must be congruent in 1.

There are several proofs of this result, one of which is a good example of seeking promotion. The following is a summary of his argument.

The first step is to prove that the number 1, 2, … , PMI 1 forms a group under the multiplication of mod p.

The multiplication of mod p means dividing by p and taking the rest after multiplication. For example, if you take paired 7, the product "mod7" of 3 and 6 is 4, because 4 is the remainder of 3 × 6 × 18 divided by 7.

In the second step, we notice that if 1 ≤ a ≤ PMI 1, then the power modp of a constitutes a subgroup of this group, and the size of this subgroup is the smallest integer m such that a ^ m = 1 m mod p. Then we apply Lagrange's theorem, that is, the size of a group must be divisible by the size of the subgroup. Now the size of the group is pmur1, so pmae1 can be divisible by m, but a ^ m = 1 Magi mod p, so a ^ (p Mel 1) = 1 M M mod p. The reason is over.

This argument shows that, if properly viewed, Fermat's theorem is only a special case of Lagrange's theorem (however, it is not entirely obvious that the integer modp becomes a group. This fact can be proved by Euclid algorithm.

Fermat himself could not have seen his theorem in this way, because when he proved it, the concept of group had not been invented. Therefore, the abstract concept of groups helps people to look at Fermat Little Theorem in a new way. ∶ can regard it as a special case of a more general result, but it is not even possible to state this more general result until a new abstract concept is developed.

This abstraction process has many benefits, the most obvious being that it gives a more general theorem, a theorem with many other interesting applications. Once you see this, you can prove the general results at once, without having to prove each special result separately. One of the benefits associated with it is that it enables us to see that many seemingly unrelated results are linked. Finding connections between different areas of mathematics will almost certainly affect the significant progress of the subject.

There is an obvious contrast between the way that the characteristic property defines the root sign 2 and the way of defining the imaginary number I. The way to define the root sign 2 is to prove that there is a positive real number and its square is 2. Then, define this number as the root sign 2.

For the imaginary number I, the proof of this style is impossible, because no real number squared is later equal to-1. So, let's replace it with another question: ∶, if there is a number squared that will later be equal to-1, what are the properties about this number? Such a number cannot be a real number, but this does not rule out the possibility of extending the real number system into a larger number system so that it can contain a square root of-1.

At first glance, it seems that we happen to know one thing about I, that is, I ^ 2 =-1. But if it is also assumed that I obeys the normal laws of arithmetic, more interesting calculations can be done, such as

This means that (1 square I) / the root sign 2 is a square root of I.

From these two simple assumptions (that is, I ^ 2 =-1 and I obey the general law of arithmetic), the whole theory of complex numbers can be developed without worrying about what I is. In fact, if you think about the existence of the root sign 2, you will see that the existence of the root sign 2 is not as important as its definition property, and this definition property is very similar to the definition property of I. the definition property is that 2 is given after the square and obeys the general law of arithmetic.

Many important generalizations in mathematics act in this way. Another important example is the definition of x ^ a when x and an are real numbers and x is positive. It is hard to see the meaning of the expression x ^ a unless an is a positive integer. However, no matter what value a takes, mathematicians hold the expression as if there is nothing wrong with it. What is going on? The answer is that what really matters about x ^ an is not what value it takes, but what its characteristic properties are when it is regarded as a function of a.

The so-called characteristic nature is not only the nature it has, but also as long as it has this property, that is, it, that is, the property that only it has.

The most important characteristic of x ^ an is

With this property and several other properties, the function x ^ an is completely determined.

There is an interesting relationship between abstraction and classification. in mathematics, the word "abstract" often refers to a part of mathematics, where the characteristic properties of an object are more often discussed than demonstrated directly from the definition of the object. The ultimate purpose of abstraction is to explore its inference from a set of axioms, such as the axioms of groups or vector spaces. However, sometimes it is helpful to classify these algebraic structures in order to reason about them, and the result of classification is to make them more specific. For example, every finite dimensional real vector space V is isomorphic to some R ^ n, and n is a non-negative integer. It is often helpful to think of V as a specific R ^ n, rather than an algebraic structure that satisfies some axioms. Therefore, in a certain sense, classification is the opposite of abstraction.

Restate it and then promote it later.

Dimension is a mathematical concept that is also familiar in everyday language. for example, a picture of a chair is a two-dimensional representation of a 3D object, because the chair has height, width and depth, but its image is only height and width. Roughly speaking, the dimension of a figure is the number of independent directions that can move freely along it and always stay in the graph. This rough concept can be mathematically defined precisely (using the concept of vector space).

If a graph is given, its normal dimension should be a non-negative integer. It is pointless to say that we can move in, for example, 1.4 independent directions. however, there is a strict mathematical theory of fractal dimension, in which any non-negative real number d can find a d-dimensional figure.

How do mathematicians do this seemingly impossible thing? The answer is to restate the concept, and only in this way can it be popularized. This sentence means to give dimension a new definition of ∶ with the following properties

For all "simple" graphics, the new definition of dimension is consistent with the old definition. For example, under the new definition, the straight line is still 1-dimensional, the square is still 2-dimensional, and the cube is still 3-dimensional.

Under the new definition, it is no longer obvious that the dimension of each graph must be a positive integer.

There are several ways to do this, but most of them focus on the differences in the concepts of length, area, and volume. Note that a straight line segment of length 2 can be divided into the union of two non-overlapping line segments of length 1; a square with side length of 2 can be divided into the union of four non-overlapping squares with side length of 1; and a cube with side length of 2 can be divided into the union of eight non-overlapping cubes with side length of 1. Therefore, if a d-dimensional figure is enlarged by the factor r, its d-dimensional "volume" will be multiplied by the factor r ^ d. Suppose you now want to show a 1.4-dimensional graph. One of the ways is to take

Then find a figure X and enlarge it by factor r so that the enlarged figure can be divided into two disjoint copies of X. The volume of the two copies of X should be twice the volume of X, so the dimension d of X should satisfy the equation r ^ d = 2. According to our choice of r, the dimension of X is 1.4.

Another concept that seems meaningless at first glance is noncommutative geometry. The word "commutative" is originally used only for binary operations, so it belongs to algebra rather than geometry, so what might noncommutative geometry mean?

But now, the answer is no longer surprising ∶ people use some algebraic structure to restate part of geometry, and then generalize the algebra here. This algebraic structure involves a commutative binary operation, so this algebra is generalized by allowing the binary operation to be non-commutative.

Part of the geometry we are talking about here is the study of manifolds. Related to the manifold X is the set C (X) of complex-valued continuous functions defined on the manifold X. Two functions f and g and two complex numbers λ and μ in C (X) are given, then the linear combination λ f + μ g is still a complex continuous function and therefore still in C (X), so C (X) is a vector space. However, you can also multiply f by g. This multiplication has a variety of natural properties (for example, for all functions f (g) g and h have f (gendh) = fg+fh), which makes C (X) an algebra, or even a C-algebra. It was later found that a considerable part of the geometry on the compact manifold X can be restated purely in the language of the Cauchy-algebra C (X). The word "purely" here means that there is no need to talk about manifold X, and C (X) is originally defined with reference to manifold X. all we need is that C (X) is an algebra. This means that it is possible to have such non-geometrically generated algebras, but for them, restated geometric concepts are still available.

There are two binary operations in algebra, ∶ vector space operations and multiplication. Vector space operations are always assumed to be commutative, but multiplication is not necessarily ∶ if multiplication is also commutative, this algebra is said to be commutative. Because fg and gf are obviously the same function, the algebra C (X) is a commutative Cumulative-algebra, so the algebra generated from geometry is always commutative. However, after many geometric concepts have been restated in algebraic language, they still have meaning for noncommutative Cumulative-algebras, and the word "noncommutative geometry" has been used in this way.

Such a program that is restated and popularized later has been found in many of the most important developments in mathematics. Now let's look at the third example in this article, the basic theorem of ∶ arithmetic. It is one of the cornerstones of number theory, which points out that every positive integer in ∶ can be written as the product of primes in a unique way. However, experts in number theory always have to look at the expanded number system, in the vast majority of this kind of number system, the obvious similar theorem of the basic theorem of arithmetic is not valid.

However, there is a natural way to generalize the concept of "number" to include ideal numbers, so that a version of the basic theorem of arithmetic can be proved in a ring such as the one just described. First of all, the problem is restated as follows: ∶, for each number γ, do the set of all multiples δ γ, where δ is the element in the ring. Note that this set is (γ) and has the following closed properties ∶ if α, β belong to (γ), and δ, ε are all elements of this ring, then

The subset of a ring with the above closed properties is called an ideal. If an ideal has the shape of (γ) and γ is a number, then the ideal is called a principal ideal. However, existence is not an ideal of the principal ideal, so the set of ideals can be regarded as a set of elements that generalize the original ring. The result is that there are natural concepts of addition and multiplication that can be applied to every ideal. In addition, it makes sense to define an ideal as a "prime" ideal. Here, it is said that ideal I is a prime ideal, that is, the only way to write I as the product of two ideals J _ () ~ () K is that one of J _ () ~ K is a "unit element". The unique decomposition theorem of factors on this extended set is true. These concepts give a very useful yardstick for "measuring the failure degree of the uniqueness theorem of factorization" in the original ring.

With higher dimensions and multiple variables, we have seen that the study of polynomial equations becomes much more complicated when we consider not only one equation with a single variable, but a system of equations with many variables. For example, partial differential equations can be regarded as differential equations involving multiple variables. typically, it is much more difficult to analyze them than ordinary differential equations. The system of polynomial equations of multivariables and partial differential equations are two noteworthy examples of a process, which is extended from univariate to multivariate, resulting in many of the most important mathematical problems and results, especially since the 20th century.

There is an equation involving three real variables ZBI y and z. It is often useful to think of triples as a single object rather than a set of three numbers. In addition, this object has a natural interpretation of ∶ which represents a point in three-dimensional space. This geometric explanation is important and largely helps to explain why it is so interesting to extend many definitions and theorems from one variable to multiple variables. If the work of an algebra is extended from a single variable to a multivariable, it can be considered that it is extended from an one-dimensional background to a high-dimensional background. This idea leads to many connections between algebra and geometry, so that techniques from one field can be used in other fields.

This article comes from the official account of Wechat: Lao Hu Shuo Science (ID:LaohuSci). Author: I am Lao Hu.

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