The "problem of geometric ruler drawing" mentioned here refers to the restriction that only rulers and compasses can be used in drawing. The ruler here refers to a ruler that can only draw straight lines without scale. The problem of geometric ruler drawing includes the following four problems.

1. Turn a circle into a square-find a square with an area equal to a known circle;

2. Divide any corner into three equal parts;

3. Double Cube-Find a cube, and make its volume twice that of the known cube.

4. Make a regular heptagon.

The above four problems have puzzled mathematicians for more than two thousand years, but in fact, the first three problems have been proved impossible to be solved with a ruler and compass in a limited number of steps. The fourth problem was solved by Gauss in algebra. He also regarded it as a masterpiece of his life and told him to carve the regular heptagon on his tombstone. But later, his tombstone was not engraved with a heptagon, but with a 17 star, because the sculptor in charge of carving the monument thought that the heptagon was too similar to the circle, so everyone would be confused.

2. Honeycomb conjecture

Pepos, an ancient Greek mathematician in the 4th century, pointed out that the beautiful shape of beehives is the representative of the most efficient labor in nature. He guessed that the honeycomb with hexagonal cross section was built by bees with minimal beeswax. His conjecture is called the hive conjecture, but no one can prove it. 1943, the Hungarian mathematician Taos skillfully proved that the perimeter of a regular polygon is the smallest of all regular polygons connected end to end. 1943, the Hungarian mathematician Taos skillfully proved that the perimeter of a regular polygon is the smallest of all regular polygons connected end to end. But what if the edges of a polygon are curves? Taos thinks that a regular hexagon has the smallest circumference compared with other figures, but he can't prove it. However, when considering that the periphery is a curve, whether the curve is convex or concave, Hale proved that the circumference of a figure composed of many regular hexagons is the most correct. He has put the proof process of page 19 on the internet, and many experts have seen this proof and think that Hale's proof is correct.

3. Twin prime conjecture

1849, Polinak put forward the conjecture of twin prime numbers, that is, he guessed that there were infinite pairs of twin prime numbers. Twin prime numbers are a pair of prime numbers with a difference of 2. For example, 3 and 5, 5 and 7, 1 1 3, …, 100 16957 and 100 16959 are all twin prime numbers. 1966, Chen Jingrun, a mathematician in China, got the best result in this respect: there are infinitely many prime numbers P, so that p+2 is the product of no more than two prime numbers. The conjecture of twin prime numbers has not been solved yet, but it is generally considered to be correct.

4. Fermat's last theorem

One day more than 360 years ago, Fermat wrote a seemingly simple theorem in the margin of the page on a whim. The content of this theorem is about an equation xn +yn = zn.

When n=2, the problem of positive integer solution is the well-known Pythagorean Theorem (also called Pythagorean Theorem in ancient China).

Fermat claims that when you are n>2, you can't find satisfaction.

xn +yn = zn

For example, the integer solution of the equation.

x3 +y3 = z3

You can't find an integer solution.

Fermat, the initiator, thus left an eternal problem. For more than 300 years, countless mathematicians have tried in vain to solve this problem. This Fermat's last theorem, known as the century's difficult problem, has become a big worry in mathematics and is extremely eager to solve it.

However, this 300-year-old math unsolved case has finally been solved. Andrew wiles, an English mathematician, solved this mathematical problem. In fact, Willis used the achievements of the development of abstract mathematics in the last 30 years of the twentieth century to prove it.

5, four-color conjecture

1852, when Francis guthrie, who graduated from London University, came to a scientific research institute to do map coloring, he found an interesting phenomenon: "It seems that every map can be colored with four colors, which makes countries with the same border painted with different colors."

1872, Kelly, the most famous mathematician in Britain at that time, formally put forward this question to the London Mathematical Society, so the four-color conjecture became a concern of the world mathematical community. Many first-class mathematicians in the world have participated in the great battle of four-color conjecture.

1976, American mathematicians Appel and Harken spent 1200 hours on two different computers at the University of Illinois in the United States, made 1000 billion judgments, and finally completed the proof of the four-color theorem. The computer proof of the four-color conjecture has caused a sensation in the world.

6. Goldbach conjecture

1742 on June 7, Goldbach wrote to the great mathematician Euler at that time, and put forward the following conjecture:

(a) Any > even number =6 can be expressed as the sum of two odd prime numbers.

(b) Any odd number > 9 can be expressed as the sum of three odd prime numbers.

Since then, this famous mathematical problem has attracted the attention of thousands of mathematicians all over the world. 200 years have passed and no one has proved it. Goldbach conjecture has therefore become an unattainable "pearl" in the crown of mathematics.