Prove:

First, prove the following proposition:

For the set Zn={x 1, x2, ..., xφ(n)}, consider the set.

S = {ax 1 mod n，ax2mod n，...，axφ(n)mod n}

Then S = Zn

1) Since A, N are coprime, xi and N are coprime, axi must be coprime with P, so

Any xi, axi mod n must be Zn element

2) For two elements xi and xj in Zn, if xi ≠ xj

Then axi mod n ≠ axi mod n, which can be obtained from the coprime and elimination method of a and p.

So, obviously, S=Zn

In this case, then

(ax 1 × ax2×...×axφ(n))mod n

= (ax 1 mod n × ax2mod n ×...× axφ(n)mod n)mod n

= (x 1 × x2 ×...× xφ(n))mod n

Consider that left and right side of the above equation.

The left side is equal to (a φ (n) × (x1× x2×) ... × xφ (n)) mod n) mod n.

The right side is equal to x1× x2× ... × xφ (n)) mod n.

And x 1 × x2 ×...× xφ(n))mod n and p are coprime.

According to the elimination method, we can subtract from both sides of the equation and get:

aφ(n) ≡ 1 mod n

Inference: For coprime numbers A and N, aφ(n)+ 1 ≡ a mod n is satisfied.

Fermat theorem

A is a positive integer that cannot be divisible by prime number p, so there is AP-1≡1mod p.

It is very simple to prove this theorem, because φ(p) = p- 1 can be proved by substituting euler theorem.

It is also inferred that there is ap ≡ a mod p for a positive integer a that cannot be divisible by prime number p.

[Edit this paragraph] Euler formula

There is a relationship between the number of vertices v, the number of faces f and the number of edges e of a simple polyhedron.

V+F-E=2

This formula is called Euler formula. This formula describes the unique laws of the number of vertices, faces and edges of a simple polyhedron.

[Edit this paragraph] Understanding Euler

Euler, a Swiss mathematician, went to university of basel to study at the age of 13, and was carefully guided by Bernoulli, a famous mathematician. Euler is the most prolific outstanding mathematician in the history of science. He started publishing papers at the age of 19 until he was 76. In his tireless life, * * * wrote 886 books and papers, of which more than 700 papers were written before his death. In order to organize his works, the Petersburg Academy of Sciences spent 47 years.

It is no accident that Euler's works are surprisingly numerous. His tenacious perseverance and tireless academic spirit enable him to work in any harsh environment: he often kneels with his children in his arms to finish his papers. Even during the 17 years after his blindness, he did not stop studying mathematics, and dictated several books and more than 400 papers. He died while writing the calculation essentials for calculating Uranus' orbit. Euler will always be our respected teacher.

Euler's research works involve almost all branches of mathematics, including physical mechanics, astronomy, ballistics, navigation, architecture and music! There are many formulas, theorems, solutions, functions, equations and constants named after Euler. The mathematics textbook written by Euler was always regarded as a standard course at that time. 1Gauss, a great mathematician in the 9th century, once said that "studying Euler's works is always the best way to understand mathematics". Euler was also the inventor of mathematical symbols. Many mathematical symbols he created, such as π, that is, sin, cos, tg, ∑, f (x), are still in use today.

Euler not only solved the problem of calculating the trajectory of comets, but also solved the problem of moon deviation, which was a headache for Newton. The perfect solution of the famous "Seven Bridges in Konigsberg" opened the research of "Graph Theory". Euler found that no matter what shape the convex polyhedron is, there is always a relationship among the number of vertices V, the number of edges E and the number of faces F. V+F-E=2, which is the so-called Euler formula. V+F-E, that is, Euler characteristics, has become the basic concept of "topology". So what is "topology"? How did Euler discover this relationship? How did he study it? Today, let's follow the footsteps of Euler and explore this formula with reverence and appreciation. ......

[Edit this paragraph] The significance of euler theorem

(1) Mathematical Law: The formula describes the unique law among the number of vertices, faces and edges of a simple polyhedron.

(2) Innovation of ideas and methods: In the process of theorem discovery and proof, it is conceptually assumed that its surface is a rubber film, which can be stretched at will; Methods Cut off the bottom surface and turn it into a plane figure (three-dimensional figure → plan).

(3) Introduction of topology: From three-dimensional graph to open graph, the shape, length, distance and area of each face have changed, while the number of vertices, faces and edges remain unchanged.

Theorem leads us into a new field of geometry: topology. We use a material (such as rubber wave), which can be deformed at will, but it can't be torn or stuck. Topology is to study the unchangeable properties of graphics in this deformation process.

(4) Propose a polyhedron classification method:

In Euler's formula, f (p)=V+F-E is called Euler's characteristic. Euler theorem told us that simple polyhedron f (p)=2.

In addition to simple polyhedrons, there are non-simple polyhedrons. For example, dig a hole in a cuboid and connect the corresponding vertices at the bottom to get a polyhedron. Its surface cannot be transformed into a sphere by continuous deformation, but it can be transformed into a torus. Its Euler characteristic f (p)= 16+ 16-32=0, that is, the Euler characteristic of polyhedron with holes is 0.

(5) euler theorem can solve some practical problems.

For example, why are there only five regular polyhedrons? What's the relationship between football and C60? Is there a regular polyhedron with 7 sides? wait for

[Edit this paragraph] euler theorem's certificate

Method 1: (Using the Geometry Sketchpad)

Gradually reduce the number of edges of polyhedron and analyze v+f-e.

Firstly, the simple tetrahedron ABCD is taken as an example to analyze the proof method.

When a face is removed, it becomes a plane figure, and the number of tetrahedral vertices e, the number of edges v and the number of remaining faces F 1 remain unchanged after deformation. Therefore, in order to study the relationship between V, E and F, we only need to remove one surface and turn it into a plane figure, and prove that V+F 1-E= 1.

(1) If one edge is removed and one face is reduced, V+F 1-E remains unchanged. Remove all faces in turn and become a "tree".

(2) Every time an edge is removed from the remaining tree, a vertex is reduced, and V+F 1-E remains unchanged until only one edge remains.

In the above process, V+F 1-E remains unchanged, and V+F 1-E= 1, so a removed surface is added, and V+F-E =2.

For any simple polyhedron, this method has only one line segment left. Therefore, this formula is correct for any simple polyhedron.

Method 2: Calculate the sum of internal angles of polyhedron.

Let the number of vertices v, faces f and edges e of a polyhedron. Cut off a face to make it a plane figure (open graph), and find the sum of all angles in the face ∑ α.

On the one hand, the sum of internal angles is obtained by using all the faces in the original image.

There are f faces, the number of sides of each face is n 1, n2, …, nF, and the sum of the internal angles of each face is:

∑α=[(n 1-2) 1800+(N2-2) 1800+……+(nF-2) 1800]

=(n 1+N2+…+nF-2F) 1800

=(2E-2F) 1800 = (English-French) 3600 (1)

On the other hand, the sum of interior angles is obtained by using vertices in an open graph.

Let a cutting surface be an N-polygon, and the sum of its internal angles is (n-2 n-2) 1800, then among all V vertices, there are N vertices on the side and V-n vertices in the middle. The sum of the internal angles of the V-n vertices in the middle is (V-N) 3600, and the sum of the internal angles of the N vertices on the side is (N-2 n-2) 1800.

So, the sum of the internal angles of each face of a polyhedron:

∑α=(V-n)3600+(n-2) 1800+(n-2) 1800

=(V-2) 3600。 (2)

from( 1)(2):(e-f)3600 =(v-2)3600。

So V+F-E=2.

Method 3 Prove Euler formula by topological method.

This paper tries to prove the Euler formula about the number of faces, edges and vertices of polyhedron by topological method.

Euler formula: For any polyhedron (that is, a solid with a plane and no holes), assuming that F, E and V represent the number of faces, edges (or edges) and angles (or tops) respectively, then

F-E+V=2 .

The proof is shown in the figure (the figure is a cube, but the proof is general and "topological"):

(1) The polyhedron (① in the figure) is regarded as a hollow solid with thin rubber on its surface.

(2) If one face of the polyhedron is removed, it can be completely unfolded on the plane and a straight line can be obtained on the plane, as shown in Figure 2. Assuming that f ′, e ′ and v ′ respectively represent the number of (simple) polygons, edges and vertices of this planar graph, we only need to prove that f ′-e ′+v ′ =1.

(3) For this plane figure, triangle division is carried out, that is to say, diagonal lines are introduced into polygons that are not triangles until they become triangles, as shown in Figure 3. Every time a diagonal is introduced, f ′ and e ′ increase by 1 respectively, while v ′ remains unchanged, so f ′-e ′+v ′ remains unchanged. So when it is completely divided into triangles, the value of f'-e'+v' remains unchanged. Some triangles have one or two sides on the boundary of a plane figure.

(4) If a triangle has one side on the boundary, as shown in Figure ④, remove the side of the triangle that does not belong to other triangles, that is, AC, thus removing the △ABC. In this way, f ′ and e ′ are reduced by 1 and v ′ is unchanged, so f ′-e ′+v ′ is unchanged.

(5) If a triangle has two sides on the boundary, as shown in △DEF in Figure 5, remove the sides of the triangle that do not belong to other triangles, namely DF and EF, thus removing △DEF. So f ′ minus 1, e ′ minus 2, v ′ minus 1, so f ′-e ′+v ′ remains unchanged.

(6) Continue this way until only one triangle remains, as shown in Figure 6. At this time, f ′ =1,e ′ = 3 and v ′ = 3, so f ′-e ′+v ′ =1-3+3 =1.

(7) Because the original graphics are connected together, the various changes introduced in the middle will not destroy this fact, so the final graphics are still connected together, so the final graphics will not be a few triangles scattered outward, like ⑦ in the figure.

(8) If it looks like the end of the picture, we can remove one of the triangles, namely 1 triangle, 3 sides and 2 vertices. So F'-E'+V' remains the same.

That is, f ′-e ′+v ′ =1.

Is established, so Euler formula:

F-E+V=2

Get a license.

[Edit this paragraph] The application method of euler theorem

(1) score:

a^r/(a-b)(a-c)+b^r/(b-c)(b-a)+c^r/(c-a)(c-b)

When r=0, 1, the value of the formula is 0.

When r=2, the value is 1.

When r=3, the value is a+b+CB+C.

(2) Complex number

From e I θ = cos θ+isinθ, we get:

sinθ=(e^iθ-e^-iθ)/2i

cosθ=(e^iθ+e^-iθ)/2

(3) Triangle

Let r be the radius of the circumscribed circle of the triangle, r be the radius of the inscribed circle, and d be the distance from the outer center to the inner center, then:

d^2=R^2-2Rr

(4) polyhedron

Let v be the number of vertices, e be the number of edges and f be the number of faces, then

v-e+f=2-2p

For example, p is an Euler feature.

A polyhedron with p=0 is called a zero-class polyhedron.

A polyhedron with p= 1 is called the first polyhedron.

(5) Polygon

Let the number of vertices of a two-dimensional geometric figure be v, the number of divided regions be ar, and the number of strokes be b, then there are:

V+Ar-B= 1

(For example, a graph consisting of a rectangle and two diagonals, v = 5, ar = 4, b = 8)

(6) euler theorem

In the same triangle, its circumscribed circle, center of gravity, center of nine o'clock and vertical line.

In fact, there are many Euler formulas, and the above are just a few commonly used ones.

[Edit this paragraph] Calculate the number of football pentagons and hexagons with euler theorem.

Q: The football surface is made of pentagonal and hexagonal leather. A * * *, how many such pentagons and hexagons are there?

A: Football is a polyhedron, which satisfies the Euler formula F-E+V = 2, where F, E, V E and V respectively represent the number of faces, edges and vertices.

Suppose that there are regular pentagons (black leather) and hexagons (white leather) with X and Y on the football surface, then

Number of faces f = x+y

The number of sides e = (5x+6y)/2 (each side consists of a black leather and a white leather * * *).

The number of vertices v = (5x+6y)/3 (each vertex is used by three skins * * *).

According to Euler formula, x+y-(5x+6y)/2+(5x+6y)/3 = 2,

The solution is x = 12. So * * * has 12 pieces of black leather.

So the black leather * * * has 12× 5 = 60 sides, all sewn together with the white leather.

White leather: Of the six sides of each white leather, three sides are sewn together with the edge of the black leather, and the other three sides are sewn together with the edges of other white leather.

So half of all sides of the white leather are sewn together with the black leather.

Then a piece of white leather should have 60× 2 = 120 sides, 120 ÷ 6 = 20.

So * * * has 20 pieces of white skin

(Or, six sides of each hexagon are connected with three sides of other three hexagons and three sides of three pentagons; The five sides of each pentagon are connected with the five sides of the other five hexagons.

Therefore, the number of pentagons x=3y/5.

X= 12, so y=20).

"euler theorem" in Economics

In western economics, the relationship between output and production factors L and K is expressed as Q = Q (L, K). If the concrete function form is homogeneous, then there is: Q = L (? Q/？ L)+K(？ Q/？ K), in other words, the net distribution of products depends on whether q can be expressed as a homogeneous function.

Because? Q/？ L = mpl = w/p is regarded as the contribution of labor to output. Q/？ K = MPK = R/P is considered as the contribution of capital to output. Therefore, this formula is interpreted as "product distribution network theorem", that is, all products are only distributed by all elements without surplus. Because it conforms to the mathematical euler theorem in form, it is called euler theorem.

"euler theorem" in congruence theory

Let a, m∈N, (a, m)= 1, then a (f (m) ≡1(mod m).

(Note: f(m) refers to the number of simple systems with module m)

[Edit this paragraph] Euler formula

In the history of mathematics, many formulas were discovered by Euler (leonhard euler AD 1707- 1783). They are all called Euler formulas, which are scattered in various branches of mathematics.

1, Euler formula in complex variable function theory;

E ix = cosx+isinx, e is the base of natural logarithm, and I is the imaginary unit.

It extends the definition domain of trigonometric function to complex number, and establishes the relationship between trigonometric function and exponential function, which occupies a very important position in the theory of complex variable function.

Replace x in the formula with -x to get:

E-IX = cosx-isinx, and then we add and subtract the two formulas to get:

sinx=(e^ix-e^-ix)/(2i),cosx=(e^ix+e^-ix)/2.

These two are also called Euler formulas. Let x in e ix = cosx+isinx be ∏, and you get:

e^i∏+ 1=0.

This identity, also known as Euler's formula, is the most fascinating formula in mathematics, which connects several most important mathematics in mathematics: two transcendental numbers: the base of natural logarithm e, pi ∏, two units: imaginary number unit I and natural number unit 1, and the common 0 in mathematics. Mathematicians evaluate it as a "formula created by God", which we can only look at but can't understand.

2. Euler formula in topology;

V+F-E=X(P), v is the number of vertices of polyhedron p, f is the number of faces of polyhedron p, e is the number of edges of polyhedron p, and X(P) is the Euler characteristic of polyhedron p.

If P can be homeomorphic on a sphere (which can be understood as expanding into a sphere), then X(P)=2, and if P is homeomorphic on a sphere with H ring handles, then X(P)=2-2h.

X(P) is called the topological invariant of p, which is the research scope of topology.

3. Euler formula in elementary number theory;

Euler φ function: φ(n) is the integer number of n coprime in all positive integers less than n, and n is a positive integer.

Euler proved the following formula:

If the factorization of the standard prime factor of n is p1a1* p2a2 * ... * pm * am, all PJ (j = 1, 2, ..., m) are prime numbers and are not equal to each other. Then there is

φ(n)= n( 1- 1/p 1)( 1- 1/p2)……( 1- 1/pm)

It can be proved by the principle of inclusion and exclusion.

Theorem: Positive integers A and N are coprime, then a φ (n) is divided by n, and the remainder is 1.

It is proved that let {A 1, A2, ..., Am} be the contraction system of module n (if the integer A 1, A2, ..., Am module n corresponds to all m natural elements in 0, 1, 2, ..., n- 1.

Then {a A 1, a A2, ..., a Am} is also a contraction system of module n (if a Ax and a Ay (x is not equal to y) are divided by n and the remainder is the same, then a(Ax-Ay) is a multiple of n, which is obviously impossible).

That is a1* a2 * a3 * ... am aa1* aa2 * ... am (mod n) (where m=φ(n)).

Cut off A 1 * A2 * A3 *...AM to get 1 ≡ A φ (n) (mod n).

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