When are two prime numbers coprime

Two prime numbers are coprime in any case, unless they are equal

What is prime?
Ten examples?

A prime number is a number in which no integer can be divisible except 1 and itself. For example: 2.3.5.7.11.13.17.19.23.39.31
Prime numbers are prime numbers. The numbers outside prime numbers are called composite numbers. For example, 4.6.8.16.32.64.72
But notice that 1 is neither prime nor composite

What is prime? What is prime?

Prime number is also called prime number. Prime number is usually used in life. Prime number is a number which has no other factors except one and itself. It is called prime number. It refers to a natural number which is greater than 1 and cannot be divided by other natural numbers except one and the integer itself, Natural numbers with only two positive factors (1 and themselves) are prime numbers. Numbers larger than 1 but not prime numbers are called composite numbers. 1 and 0 are neither prime nor composite numbers
A natural number with only 1 and its two positive factors is called prime number. (for example, from 2 / 1 = 2,2 / 2 = 1, we can see that the factor of 2 has only 1 and its two divisors, so 2 is prime number. The opposite is composite number: "in addition to 1 and its two factors, there are other factors, which are called composite number." for example, 4 / 1 = 4,4 / 2 = 2,4 / 4 = 1, In addition to the two factors of 1 and 4 itself, there is also a factor of 2, so 4 is a composite number.)
There are 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97 primes in 100, and there are 25 primes in 100
Note:
(1) 2 and 3 are the only two connected primes
(2) 2 is the only even prime number
The square of a prime has only three factors
What is a prime number? In all integers larger than 1, except 1 and itself, there is no other divisor. This kind of integer is called a prime number, and prime number is also called a prime number. This final rule is only a literal explanation. Can there be an algebraic formula that stipulates that when the number represented by letters is any specified value, the value of the algebraic formula substituted is a prime number?
For example, 101, 401, 601 and 701 are prime numbers, but 301 and 901 are combined numbers
Someone has done such calculations: 1 ^ 2 + 1 + 41 = 43, 2 ^ 2 + 2 + 41 = 47, 3 ^ 2 + 3 + 41 = 53 So we can have such a formula: let a positive number be n, then the value of n ^ 2 + N + 41 must be a prime number. This formula holds until n = 39. But when n = 40, the formula does not hold, because 40 ^ 2 + 40 + 41 = 1681 = 41 * 41
Fermat, known as "the greatest French mathematician in the 17th century", also studied the properties of prime numbers. He found that if FN = 2 ^ (2 ^ n), then when n is equal to 0, 1, 2, 3 and 4 respectively, FN gives 3, 5, 17, 257 and 65537, which are all prime numbers. Because F5 is too large (F5 = 14292967297), he directly guessed that FN is a prime number for all natural numbers, In the 67 years after Fermat's death, Euler, a 25-year-old Swiss mathematician, proved that F5 = 14292967297 = 641 * 6700417 is not a prime number, but a composite number
What's more interesting is that in the future, mathematicians have never found any FN value to be prime, all of them are composite numbers. At present, because the square is large, there are few proofs. Now mathematicians have obtained the maximum value of FN: n = 1495. This is a super astronomical number, with 10 ^ 10584 digits. Of course, although it is very large, But it's not a prime. Prime and Fermat have a big joke!
There was also a French mathematician named Mason in the 17th century. He once made a conjecture: 2 ^ P-1 algebraic formula, when p is prime, 2 ^ P-1 is prime. He checked that when p = 2, 3, 5, 7, 17, 19, the values of the obtained algebraic formula are prime. Later, Euler proved that when p = 31, 2 ^ P-1 is prime
Two hundred and fifty years after Mason's death, the American mathematician Kohler proved that 2 ^ 67-1 = 193707721 * 761838257287 is a composite number. This is the ninth Mason number. In the 20th century, people successively proved that the 10th Mason number is a prime number and the 11th Mason number is a composite number, It also makes it difficult for people to find the law of prime number
Now, the largest Mason number found by mathematicians is a number with 378632 bits: 2 ^ 1257787-1. Although a large number of prime numbers can be found in mathematics, the law of prime numbers still can not be followed
The first 50 million prime numbers
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[Abstract] mathematicians can't deal with him even if they don't play according to the card principle
How to distribute prime numbers? Both professional mathematicians and amateurs have been deeply attracted by this problem
Prime number is a natural number larger than 1. There are no other natural numbers except itself and 1. The distribution of prime number has two contradictory characteristics. I will list some facts to make you believe these two characteristics forever
First, although the definition of prime number is very simple, and it is also the building block of natural number (any natural number can be expressed as the continuous product of the power of prime factors, and the expression is unique), it is the most unruly one of the objects studied by mathematicians. In natural numbers, prime numbers grow like weeds, and do not seem to obey other laws except the law of opportunity, Nobody dares to say that the next one will come out there
The second point is more surprising, because? T 篕 P on the contrary, prime numbers show amazing regularity. That is to say, there are rules restricting the behavior of prime numbers, and they absolutely obey these rules like soldiers
In order to support the first point, I write down prime numbers and composite numbers below 100 (except 2, even numbers are not listed)
[browse original]
Then list 10 million plus or minus 100 prime numbers: prime numbers between 9999000 and 10000000
9,999,901
9,999,907
9,999,929
9,999,931
9,999,937
9,999,943
9,999,971
9,999,973
9,999,991
Prime numbers between 10000000 and 10000100
10,000,019
10,000,079
You see! There is no reason to say that this number is prime and that number is not prime. When you see these numbers, do you think of the mystery of the universe as mysterious and unpredictable as the twinkling stars in the sky? Even mathematicians can't uncover this mystery, if they can, No one wants to find a larger square than the previous one, or a power of 2 - usually a good student only remembers 210 = 1024
In 1876, Lucas proved that 2127-1 was a prime number, which lasted for 75 years
2127-1
=1701411834604469231731687303715884105727
Until 1951, in the new era of electronic computer, more prime numbers were found one after another (see the records in the table below). The current record is 219937-1 with 6002 digits. If you don't believe it, you can check Guiness World record

When are two prime numbers coprime