Question 1: What is a prime number?

A prime number is an integer that cannot be expressed as the product of any other two integers except itself and 1.

For example, 15=3*5, so 15 is not a prime number; another example is, 12=6*2=4*3, so 12 is not a prime number either.

On the other hand, 13 cannot be expressed as the product of any other two integers other than 13*1, so 13 is a prime number.

Some numbers, if you try to figure them out based on impressions alone, you can't be sure whether they are prime or not.

For some numbers you can immediately tell that they are not prime.

A number, no matter how big it is, cannot be prime as long as its single digit is 2, 4, 5, 6, 8 or 0.

In addition, if the sum of the digits of a number is divisible by 3, it cannot be a prime number.

But if its single digits are 1, 3, 7, or 9, and the sum of its digits is not divisible by 3, then it may be a prime number (but it may not be prime).

There is no ready-made formula that can tell you whether a number is prime or not.

You can only try to see if you can express this number as the product of two smaller numbers.

One way to find prime numbers is to start with 2 and list them all on a "keep it if it is, drop it if it's not" method (until you don't want to go any further, say, 10,000).

The first number is 2, which is a prime number, so you should keep it, and then continue counting, deleting every other number, so that you have all the numbers that are divisible by 2 and therefore not prime.

Remove.

Among the smallest numbers left, the one ranked after 2 is 3. This is the second prime number, so it should be kept, and then count backward from it, deleting every two numbers, so that

Remove all numbers that are divisible by 3.

The next number to be removed is 5, and then every fourth number is removed to remove all numbers divisible by 5.

The next number is 7, delete every 6 numbers from now on; the next number is 11, delete every 10 numbers from now on; the next number is 13, delete every 12 numbers from now on.

.

... Just do it in accordance with the law.

You may think that if you continue deleting like this, as more and more numbers are deleted, a situation like this will eventually occur; all the numbers after a certain number will be deleted, right after the largest prime number.

There will never be any more prime numbers.

But in reality, such a situation will not occur.

No matter how big the number you take is, whether it is millions or tens of millions, there will always be prime numbers that are larger than it and have not been deleted.

In fact, as early as 300 BC, the Greek mathematician Euclid had proved that no matter how big the number you take, there will definitely be a larger prime number. Suppose you take the first 6 prime numbers and put them

Multiply them together: 2*3*5*7*11*13=30030, and then add 1 to get 3003 1.

This number cannot be divisible by 2, 3, 5, 7, 11, or 13, because the result of division will always be 1.

If 30031 is not divisible by any number except itself, it is a prime number.

If it can be divided by other numbers, then the numbers that 30031 is decomposed into must be greater than 13.

In fact, 3 0031 = 59 * 509.

This can be done for the first one hundred, the first one hundred million, or the first any number of prime numbers.

If you calculate their product and then add 1, the resulting number is either a prime number or the product of several prime numbers that are greater than the listed prime numbers.

No matter how big the number is, there is always a prime number larger than it. Therefore, the number of prime numbers is infinite.

As the number increases, we will encounter pairs of adjacent odd numbers that are both prime numbers again and again, such as 5, 7; 11, 13; 17, 19; 29, 31; 41, 43; and so on.

For as many numbers as mathematicians have access to, they can always find pairs of prime numbers like this.

Are there infinite pairs of such prime numbers?

No one knows.

Mathematicians thought it was infinite, but they have never been able to prove it.

This is why mathematicians are interested in prime numbers.

Prime numbers provide mathematicians with seemingly easy but in fact very difficult problems to solve, a challenge they have not yet been able to meet.