1. Cryptographer Wang Qi mentioned that if you count any long decimal sequence of π, you will find that the frequency of each number is basically the same, which shows that π is probably normal, that is, 0 to 9 are evenly distributed on every bit of π.

You take out a certain number in π, such as 12347, and ask which is the most likely next number from 0 to 9? As a result, the probability of each number appearing is similar. At present, the normality of the binary form of π has been proved, but so far no one has been able to prove the normality of the decimal form.

In addition, in 1897, an American amateur mathematician tried to get Indiana Parliament to pass the so-called Indiana PI bill, hoping to force π=3.2 by law, because it would skillfully solve a series of problems such as turning a circle into a square! Wonderful, really wonderful!

In the end, although the bill passed the vote of the Indiana House of Representatives, it was rejected by the Senate.

2. Euler proved that E is irrational in 1744, and Lambert proved that π is irrational in 176 1. More than a hundred years later, Hermite of France finally proved in 1873 that e is a transcendental number (that is, it is not the root of any rational coefficient polynomial).

So, excuse me, is e+π an irrational number? No one knows, even Wang Qi can't prove it!

Besides, Baidu knows what to write. How did 176 1 year become17th century? According to this logic, isn't the twentieth century the century of biology? ! !

Ah, I wrote a lot more.

3. Perfect cuboid problem

Is there a cuboid whose side length, face diagonal and body diagonal are integers?

Just ask

Positive integer solution of this system of equations. If there is, this cuboid composed of a, b and c is a perfect cuboid.

So far, the perfect cuboid has not been found, and no one has proved that the perfect cuboid does not exist.

4. Existence of odd perfect numbers

Is there a perfect number that is odd?

When the sum of all true factors of an integer (that is, divisors other than itself) is exactly equal to itself, we call this number a perfect number.

For example, the first perfect number is 6, which is about 1, 2, 3, 6. Except itself 6, the other three numbers add up, 1+2+3 = 6, which is exactly equal to itself. The second perfect number is 28, which is about 1, 2,4,7,14,28. Except itself 28, the other five numbers add up, 1+2+4+7+ 14 = 28, which is exactly equal to itself.

All the perfect numbers we have found so far are even numbers. Is it possible that an odd number is also a perfect number? I don't know

5. Twin prime conjecture

Is there an infinite number of prime numbers P, so that p+2 is also a prime number?

If both P and p+2 are prime numbers, they are collectively called twin prime numbers.

The most important progress of this problem was made by Zhang, who proved that there are infinitely many prime numbers P, so that p+c is also a prime number, among which C.

Of course, many people improved his method later, and it has been proved that C < =246.

Unfortunately, the effect of this improvement is limited. Various theoretical calculation results show that C can be improved.

Some people in the comment area said that I copied this part from others. I smiled coldly: Have you seen a favorable answer about Zhang and Nanke University in the last month or two? I didn't expect this. That's what I wrote.

Me? Copy? Me? Since? himself

6. Goldbach conjecture

Any even number greater than 2 can be expressed as the sum of two prime numbers.

As we all know, Chen Jingrun did the closest thing to this problem. He proved by screening that every even number E greater than 4 is equal to the sum of two odd prime numbers A+B or the sum of the product of two odd prime numbers and an odd prime number A * B+C ... (He is really not proving the kindergarten arithmetic problem of 1+2 = 3! )

No one has solved this problem for decades since then, because the screening method has been used to the extreme, and now a new method is needed. When will this new method appear? Maybe tomorrow, maybe 300 years later.

The method of expressing even numbers by the sum of two prime numbers is equal to the intersection of the blue line and the red line on the same horizontal line.

7. Hail conjecture

There are many aliases for this conjecture, such as 3n+ 1 conjecture, and corner-valley conjecture. It means: for every positive integer, if it is odd, multiply it by 3 and add 1, if it is even, divide it by 2, and so on, and finally get 1.

If n = 6, the sequence 6,3, 1 0,5,16,8,4,2,1can be obtained according to the above formula.

For example, n = 1 1, according to the above formula, the sequence 1 1, 34, 17, 52, 26, 13, 40, 20,/kloc-0 is obtained.

If n = 27, the sequence is obtained according to the above formula.

27, 82, 4 1, 124, 62, 3 1, 94, 47, 142, 7 1, 2 14, 107, 322, 16 1, 484, 242, 12 1, 364, 182, 9 1, 274, 137, 4 12, 206, 103, 3 10, 155, 466, 233, 700, 350, 175, 526, 263, 790, 395, 1 186, 593, 1780, 890, 445, 1336, 668, 334, 167, 502, 25 1, 754, 377, 1 132, 566, 283, 850, 425, 1276, 638, 3 19, 958, 479, 1438, 7 19, 2 158, 1079, 3238, 16 19, 4858, 2429, 7288, 3644, 1822, 9 1 1, 2734, 1367, 4 102, 205 1, 6 154, 3077, 9232, 46 16, 2308, 1 154, 577, 1732, 866, 433, 1300, 650, 325, 976, 488, 244, 122, 6 1, 184, 92, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1

The greatest charm of hail lies in its unpredictability. For example, although this 27 is an unremarkable natural number, if calculated according to the above method, its fluctuation is extremely intense: first, it takes 77 steps for 27 to reach its peak value, and then it takes 34 steps to reach the bottom value of 1. A * * * has11step, in which the peak value of 9232 is 342 times that of the original number 27!

Paul Erdos, a generation of arrogant people, said that he would give 500 yuan (well, although the money is not much ...) to those who solved this problem, and Jeffrey Lagarias even said on 20 10: "This problem is too difficult for modern mathematics to solve!"

By the end of 20 17, we calculated one by one, and it had reached 87 * 2 60, and no abnormality was found. But this does not prove that this conjecture can be established for numbers of any size.

I can't help but think that many years ago, some people spent their lives calculating Goldbach's conjecture, hoping to find an even number that could not be decomposed into the sum of two prime numbers and was greater than 2, but they didn't find it until their death ... I guess they all thought of Euler. How lucky do you think Euler is? When he counted fermat number, he only counted the sixth fermat number, and found that Fermat's formula for generating prime numbers was wrong. What if the tenth and twentieth are wrong? I wonder how long it will take!