How was Euler s theorem discovered? How to prove Euler s theorem?

Overview. 1) Background: Behind Euler's formula is a new geometry, which only studies the relative order of the positions of the parts of the figure, and does not consider the size of the figure, which is the "geometry on the rubber sheet" (positional geometry) founded by Leibniz and Euler, and now this discipline has developed into an important branch of mathematics - topology.

2) History: One of the most interesting theorems about convex polyhedra is Euler's formula "v-e f=2", which was discovered by Descartes around 1635. Euler independently discovered this formula in 1750 and published it in 1752.

Since Descartes' research was not discovered until 1860, the theorem was called Euler's formula rather than Descartes' formula.

Euler, born in Basel, Switzerland, entered the University of Basel at the age of 13 under the tutelage of Johann Bernoulli (1667-1748), the most famous mathematician of his time

Euler made many achievements in mathematics, and pioneered the study of graph theory for the solution of the famous Königsberg Seven Bridges problem. Euler also found that no matter what shape of convex polyhedra is, there is always a relationship between the vertices v, the edges e, and the faces f=2. V-e f is known as Euler's schematic number and becomes the fundamental concept of topology.

Mathematical formulas and theorems named after Euler can be found everywhere in mathematics books, and at the same time, he has also made brilliant achievements in physics, astronomy, architecture, and philosophy. Euler also created many mathematical symbols, such as (1736), I (1777), E (1748), Sin and COS (1748), TG (1753), X (1755), 1755), F(X) (1734), etc.

In 1733, at the age of 26, Euler became a professor of mathematics at the St. Petersburg Academy of Sciences In 1735, Euler solved an astronomical problem (calculating the orbit of a comet), which took several months of hard work by several famous mathematicians, but Euler used his own method to complete it in three days However, he was only 28 years old when he suffered from eye disease and unfortunately lost sight in his right eye due to overwork

Euler's life was a life of struggle for the development of mathematics, and his outstanding wisdom, tenacious perseverance, tireless fighting spirit and noble scientific ethics are always worthy of our learning

Euler's Tranquility Relaxation: E (ix) = Cosx + Isinx. Where: e is the base of the natural logarithm and i is the unit of imaginary numbers.

Replace x with -x in the formula to get :

e (-ix) = cosx-isinx, and then the addition and subtraction of the two formulas are used to obtain :

sinx=[e (ix)-e (-ix)] 2i), the free mode cosx=[e (ix)+e (-ix)] 2.

Accumulation and Difference Formula:

sin ·cos = (1 2) [sin( +sin( -cos ·sin =(1 2)[sin( +sin( -cos ·cos =(1 2)[cos( +cos( -sin ·sin =-1 2)[cos( +cos( - and the difference product formula:

Euler's theorem states that two positive integers a and n that are sensitive to each other and greater than 1 have the following relationship: (a f(n)) n = 1, where f(n) is Euler's function.

Let the set of numbers coprime with n in [1, n] be x, and each element is x, and let f=f(n), then the number of elements of x be f

Let r=(a f) %n, since the coarse swim is coprime between a and n, so a f is coprime with n, so r belongs to [1, n).

According to theorem , a*1, a*2, .,a*n is a complete residual system of n, so (a*x) %n is different for any two different x elements.

Since both a and x are coprime with n, a*x is coprime with n, and according to the theorem, a*x)%n is coprime with n.

In summary, (a*x) %n is f pairwise numbers in the range [1, n) that are different and coprime with n, so they are the set of numbers x, i.e.,

a*x1) %n) *a*xf) %n) =x1* .xf

If n is modulated on both sides, then ( (a*x1) %n) *a*xf) %n) )n= (x1* .).xf) %n, set to k

According to the product of the modulo theorem, i.e. ( a*x1) *a*xf) ) n = k

i.e. ( a f) *x1*xf) )n = k

Again, apply the product theorem of the module, i.e., ( (a f) %n) *x1*xf)%n) )n = k, i.e. (r*k)%n=k

Because x is coprime with n, (x1*.).xf) is coprime with n, so k is coprime with n, according to the complete remainder system theorem, 1*k, 2*k, .,n*k is a complete residual line of n.

Since r belongs to [1, n), and when r=1, (r*k)%n=k%n=k, so when r is not equal to 1, (r*k)%n is not equal to k according to the complete remainder system, which contradicts (r*k)%n=k, so it is impossible for r not to be equal to 1.

Therefore r=1, the proof is complete.

Euler's theorem and its proof ymzqwq's blog - csdn blog Euler's theorem proof.

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Euler's theorem, also known as Euler's function theorem or Euler's reed-void congruence theorem, is an important theorem in number theory. This theorem states the following relation: for any non-zero integer n when the modulus is prime, the value of the Euler function of modulo p (n) is equal to the sum of the multipliers required to divide n by a multiple of p.

That is, for the prime p and the integer n, there is (n) = n * 1 - 1 p) + n * n modulo p) p.

1.In number theory, Euler Theorem (also known as Fermat-Euler's theorem or Euler function theorem) is a property about congruence.

2.Euler's theorem in complex numbers, also known as Euler's formula, is considered one of the most wonderful theorems in the mathematical world.

3.Euler's theorem is actually a generalization of Fermat's minor theorem.

4.In addition, there are Euler's theorem and polyhedral Euler's theorem in plane geometry (in a convex polyhedron, the number of vertices - the number of edges + the number of faces = 2, that is, v-e + f = 2).

5.In Western economics, Euler's theorem, also known as the net exhaustion theorem of output distribution, refers to the fact that under the conditions of perfect competition, assuming that the long-term returns of scale remain unchanged, then all products are just enough to be allocated to various factors.

6.There is also Euler's formula.

Euler's Laws of Motion (Euler'S laws of motion) is an extension of Newton's laws of motion, which can be applied to the motion of a multi-particle system or the motion of a rigid body, describing the relationship between the motion of a multi-particle system or the translational motion and rotational motion of a rigid body and the force and moment it feels, respectively. It was not until 1750 that Leonhard Euler succeeded in formulating Newton's laws of motion, more than half a century after Isaac Newton published them. A rigid body is also a multi-particle system, but an ideal rigid body is a solid of finite size with negligible deformation.

Regardless of whether the force is felt or not, the distance between points does not change inside the rigid body. Euler's laws of motion can also be extended to the translational and rotational motion of any part of a deformable body. The density of the internal forces at any position inside the deformable body is not necessarily the same, that is, there is a stress distribution in the inner part of the body.

This internal force variation is governed by Newton's second law. In general, Newton's second law is applied to calculate the dynamic motion of particles or particles, but in continuum mechanics, it can be applied to calculate the motion behavior of objects with continuously distributed masses after being extended by nuclear storms. Suppose that an object is modeled as being composed of a group of discrete particles, each of which obeys Newton's second law, Euler's law of motion can be derived.

In any case, Euler's laws of motion can also be directly regarded as axioms that specifically describe the motion of large objects, regardless of the structure of the object.