What are Prime Numbers?

A prime number is a natural number that has no factors other than 1 and itself among natural numbers greater than 1.

Prime numbers are also called prime numbers. A natural number greater than 1 is a prime number that cannot be divisible by other natural numbers except 1 and itself; otherwise it is called

The number of prime numbers is infinite.

With 36N (N + 1) as the unit, as N increases, the number of prime numbers gradually increases in a wave form.

Although the whole prime number is infinite, some people still ask "How many prime numbers are below 100,000?", "How likely is a random 100-digit prime number?"

Prime numbers have many unique properties:

(1) There are only two divisors of the prime number p: 1 and p.

(2) Basic Theorems of Elementary Mathematics: Any natural number greater than 1 is either itself a prime number or it can be decomposed into the product of several prime numbers, and this decomposition is unique.

(3) The number of prime numbers is

Prime numbers are used in

Prime Goldbach conjecture

The difficulty with Goldbach's conjecture is that any prime number that can be found is not valid in the following formula. 2 * 3 * 5 * 7 *. . . . . . * PN. . . . . . * P = PN + (2 * 3 * 5 * 7 * ... * P-1) * The difference between the even number before PN minus any prime PN must be a composite number.

In his letter to Euler in 1742, Goldbach raised the following conjecture: Any integer greater than 2 can be written as the sum of two prime numbers. Since the mathematical world no longer uses the convention that "1 is also a prime number", the modern statement of the original conjecture was: Any integer greater than 5 can be written as the sum of three prime numbers. In his reply, Euler also proposed another equivalent version, that is, any even number greater than 2 wants to be stated as Euler's version. Let the proposition "any sufficiently large even number can be expressed as the sum of a prime factor not exceeding a number and another prime factor not exceeding b number" as "a + b". In 1966, Chen Jingrun proved that "1 + 2" is true, that is, "any sufficiently large even number can be expressed as the sum of two prime numbers, or the sum of a prime number and a half prime number." The common conjecture stated today is Euler's version, that is, any even number greater than 2 can be written as the sum of two prime numbers, also known as "strong Goldbach conjecture" or "Goldbach conjecture about even numbers".

From the Goldbach conjecture about even numbers, we can conclude that any odd number greater than 7 can be written as the sum of three prime numbers. The latter is called "weak Goldbach conjecture" or "Goldbach conjecture about odd numbers".

If the Goldbach conjecture about even numbers is correct, then the Goldbach conjecture about odd numbers will also be correct. In 1937, the former Soviet mathematician Vinogradov had proved that sufficiently large odd prime numbers can be written as the sum of three prime numbers, also known as "Goldbach-Vinogradov theorem" or "three prime number theorem." In 2013, Peruvian mathematician Harald Helfergot declared at the Paris Normal University that he proved a "weak Goldbach conjecture", that is, "any odd number greater than 7 can be represented as 3 odd Sum of prime numbers. "

Prime Riemann conjecture

The Riemann conjecture is a conjecture about the zero distribution of the Riemann's function (s). It was proposed by the mathematician Bonnhard Riemann (1826 ~ 1866) in 1859. German mathematician Hilbert lists 23 mathematical problems. Among them, the Riemann hypothesis is in question 8. There is no simple rule for the distribution of prime numbers in natural numbers. Riemann found that the frequency of occurrence of primes is closely related to the Riemann zeta function. The Riemann conjecture proposes that the real part of the non-trivial zero point of the Riemann's zeta function z (s) (in this case, s is not a value of -2, -4, -6, etc.) is 1/2. That is, all non-trivial zeros should lie on the line 1/2 + ti ("critical line"). t is a real number and i is the basic unit of an imaginary number. No one gives a convincing rational proof of Riemann's conjecture.

In the study of the Riemann conjecture, mathematicians call the straight line Re (s) = 1/2 on the complex plane a critical line. Using this term, the Riemann conjecture can also be expressed as: all non-trivial zeros of the Riemann zeta function are on the critical line.

The Riemann conjecture was proposed by Riemann in 1859. In the process of proving the prime number theorem, Riemann put forward a conclusion: the zeros of the Zeta function are all on the straight line Res (s) = 1/2. He gave up after trying hard and failed to prove it, because it had little effect on his proof of the prime number theorem. But this problem has not been solved, and even simpler conjectures than this assumption have not been proved. Many problems in function theory and analytic number theory rely on the Riemann hypothesis. The generalized Riemann hypothesis in algebraic number theory is even more profound. If the Riemann hypothesis can be proved, it can lead to the solution of many problems.

Prime twins

Prove that 36N (N + 1) +-1 form twin primes are infinitely many.

The twin primes of the 36N (N + 1) +-1 form are called Yandang twin primes. If this number pair is not a twin prime, it must be divisible by one or both sides by a prime number less than it.

In this pair of numbers 2 and 3 cannot divide them, so 2 and 3 do not participate in the screening; when using a sieve of 5, N is the negative number (6n-1) produced by dividing 5 and 2 and can be divided by 5, N Divide by more than 1, 1, 3, 4, and 0 are not divisible by 5; no matter how N is divided by the remainder 1.2.3.4.0, the number of positives (6n + 1) generated cannot be divided by 5, so that all natural numbers 1/5 was screened out.

When using 7 as a sieve, the number of positives produced by dividing N by 7 more than 2 and 4 can be divided by 7. Even if N is divided by 7 more than 1.2.3.4.5.6.0, the number of negatives produced cannot be divided by 7. In this way, 2/7 is screened out.

When using 11 as a sieve, no matter how N is divided by 11 and 1.2.3.4.5.6.7.8.9.10.0, the number of negatives and positives produced cannot be divisible by 11. 11 is an invalid sieve and does not participate in the screening.
In a word, there are 4 cases when all prime numbers are screened for N. First, they do not participate in the screen. Second, a single participant in the screening, such as 5, (5 is the only one participating in the screening). Third, paired unilateral screening. Fourth, both sides participated in the screening.
N is infinite, sifted out by 1/5, and the rest is infinite. After being sieved by 7 and 2/7, the rest is infinite. No matter whether the sieve is 2 / P or 4 / P, the rest is always infinite.
The number of twin primes in Yandang Mountain is infinite.

In 1849, Polinnack proposed the conjecture of twin primes , which speculates that there are an infinite number of twin primes . In the conjecture, the " twin prime numbers " refer to a pair of prime numbers that differ by two. For example, 3 and 5, 5 and 7, 11 and 13, 10,016,957 and 10,016,959 are all twin primes.

The British mathematicians Godfrey Hardy and John Littlewood have proposed a "strong twin prime conjecture". This conjecture not only proposes infinitely many pairs of twin primes, but also gives its asymptotic distribution. On May 14, 2013, an online report in Nature magazine Zhang Yitang proved that "there are infinitely many prime pairs with a difference of less than 70 million", and this research was immediately considered to be the ultimate number theory problem of the twin prime conjecture. Major breakthroughs have been made, and some even think that its impact on the academic world will exceed Chen Jingrun's "1 + 2" proof. ^[3]

36N (N + 1) + -1 twin primes of the form N, 100000000 have 109128 pairs.

Prime number

Mason prime selection

Mason prime numbers are compiled according to the distribution law of prime numbers in the 2 ^ N-1 sequence.

There is no odd prime number within the square root of 2 ^ N-1. The number of this exponent is the Mersenne prime. All the subsequent exponents that can be divided by this Mersenne prime have the factor of the Mersenne prime. The periodic repetitions are all coded with this factor.

In the 2 ^ N-1 sequence, N is composite. 2 ^ N-1 is composed of the previous prime factor and the newly added factor number. Just remove the previous factor number to get the new factor number, and the new factor. The number also repeats with this index as a cycle, and the factor is added to the repeated number later.

The three exponents are prime numbers. The remaining factor prime numbers can be selected and divided by the exponent according to minus one. After the selection, the factor numbers are repeated in the subsequent cycles. The remaining factor prime numbers are 2 ^ N-1. After the selection is completed within the square root, there is also a Mason number that is empty. This Mason number is a Mason prime, and it is a factor number on the number of repeated cycles.

The operation according to the above method is as follows:

Expand the exponents of 2 ^ n-1, and start from 2.
The value of exponent 2 is 3, and there are no odd prime numbers in the square root of 3, so it is a Mason prime. Any factor that can be divisible by 2 has a factor of 3, followed by a factor of 2 and a factor of 3;
The value of the exponent 3 is 7, and there are no odd prime numbers in the square root of 7, so it is also a Mason prime number; any index that is divisible by 3 has a factor of 7, and the factor 7 is coded with 3 as the period;
The value of exponent 4 is 15, and 4 is divisible by 2 so there is a factor of 3. Dividing 15 by 3 is equal to 5, and 5 is a new factor. Every factor that is divisible by 4 has a factor of 5. 4 is the factor of 5 for the period;
The exponent is 5 and the value is 31. There are only two odd prime numbers 3 and 5 within the square root of 31. 3 and 5 are already factors of exponents 2 and 4. Therefore, this Mason number is a Mason prime. Any exponent that is divisible by 5, All have a factor of 31, followed by a factor of 31 with a period of 5;
The value of exponent 6 is 63. 6 is more special. Any factor that can be divided by 6 will increase by a factor of 3. It is the only number that does not add other factors. Later, a factor of 6 is used to add a factor. Number 3
The value of exponent 7 is 127. Within the square root of 127, only the odd prime number 11 has not been used (actually, it will be used in the future). 11 minus 1 cannot be divided by 7, so 127 is also a Mason prime. Any index that is divisible by 7 has The number of factors of 127, with a factor of 127 followed by a period of 7;
The value of exponent 8 is 255, and the number of factors is increased by 17. Any factor that can be divisible by 8 has a factor of 17, and the factor number is 17 with 8 as the period.
The value of the index 9 is 511, and the number of new factors of 73 is added. Any factor that is divisible by 9 has a factor of 73. The number of factors is 73 after the period is 9;
The value of the index 10 is 1023, and the number of factors of 11 is newly added. Any factor that is divisible by 10 has a factor of 11, and the number of factors is 11 after the period is 10;
The value of exponent 11 is odd prime numbers such as 43, 41, 37, 29, 23, 19, 13 and so on within the square root of 2047, 2047 (as long as you edit it a few more, the remaining number is even less). All numbers are subtracted by 1. That number is divisible by 11. Only 23-1 is divisible by 11. 2047/23 = 89. 23 and 89 are its factors. Both factors are newly added because it It is a Mason composite number, and any index that is divisible by 11 has a factor of 23 and 89, and the factor of 23 and 89 is added in the cycle of 11;

The value of exponent 12 is 4095. The existing factor number is 3 * 7 * 5 * 3. The new factor number is 13. Any number that can be divisible by 12 is added with the factor number 13.

The value of exponent 13 is 83, 79, 73, 71, 67, 61, 59, 53, 47, 43, 41, 29, 19 and other prime numbers in the square root of 8191, 8191. Only 79-1 can be divided by 13. Dividing by 79 is also not possible, so 8191 is also a Mason prime.
As you continue to edit, at least one prime factor will be added to each exponent. Not both the Mason prime and the Mason composite can be selected. The ones that are not selected are the Mason composite and the Mason prime.
When editing the Mason number, according to the theorem that the prime number minus 1 can be divisible by the exponent, choose from the remaining prime numbers.

The prime numbers within the square root of 2 ^ N-1 are all selected by the factor number, and the Mason prime number is left behind.

The approximate calculation formula of Mason prime:

3 * 5 / 3.8 * 7 / 5.8 * 11 / 9.8 * 13 / 11.8 * ...... * P / P-1.2 -1 = M

P is the index of the Mason number, and M is the number of the Mason prime numbers below P.

The following are the calculated and actual numbers:

Index 5, calculation 2.947, actual 3, error 0.053;

Index 7, calculate 3.764, actual 4, error 0.236;

Index 13, calculated 4.891, actual 5, error 0.109;

Index 17, calculated 5.339, actual 6, error 0.661;

Index 19, calculation 5.766, actual 7, error 1.234;

Index 31, calculation 6.746, actual 8, error 1.254;

Index 61, calculated 8.445, actual 9, error 0.555;

Index 89, calculation 9.201, actual 10, error 0.799;

Index 107, calculation 9.697, actual 11, error 1.303;

Index 127, calculation 10.036, actual 12, error 1.964;

Index 521, calculation 13.818, actual 13, error -0.818;

Index 607, calculated 14.259, actual 14, error -0.259;

Index 1279, calculation 16.306, actual 15, error-1.306;

Index 2203, calculation 17.573, actual 16, error-1.573;

Index 2281, calculation 17.941, actual 17, error -0.941;

All odd prime numbers are factors of quasi-Mason numbers (2 ^ N-1), so the factors of Mason composite numbers are only a part of prime numbers.

In the 2 ^ N-1 sequence, a prime appears for the first time in the number of the exponent N as a prime factor. This prime appears as a factor in the 2 ^ N-1 sequence with a period of N. The exponents that are even in this series are equal to three times four times the number of pyramids.

In the 2 ^ N-1 series, if the exponent is greater than 6, in addition to the Mason prime, one or more primes are added as factors, and the newly added factor minus 1 can be divided by this index.

The factors of a Mason composite appear only once in a Mason composite.

One is the prime number of a Mason prime, which is never a factor of a Mason composite.

One is the factor of the previous Mason composite, which will never be the factor of the following Mason composite.

All Mason composite numbers reduced by 1 can be divisible by this Mason composite index, and the quotient is even.

A prime number appears as a factor for the first time in a quasi-Mason number that is not a Mason composite number. This prime number minus 1 can be divided by the index of this quasi-Mason number. The quotient is not necessarily an even number.

Mason prime numbers are all in [4 ^ (1-1) + 4 ^ (2-1) + 4 ^ (3-1) + ... + 4 ^ (n-1)] * 6 + 1 series The number in parentheses is temporarily called a quadruple pyramid number.

Any factor whose prime number is four times the number of the pyramid will not be a factor of the Mason composite number in the future.

The number in the 4 ^ (1-1) + 4 ^ (2-1) + 4 ^ (3-1) + ...... + 4 ^ (n-1) sequence is not equal to 6NM +-( N + M) times 6 plus 1 are Mason primes.

Prime numbers below the square root of 2 ^ P-1 all appear as prime factors in previous quasi-Mason numbers, so this Mason number must be a Mason prime. But its inverse theorem is not valid. If the prime number has not appeared in the previous quasi-Mason number, it may not be a factor of the Mason composite number.

In the 17th century, there was also a French mathematician named Mason, who once made a conjecture: when ^p in 2 ^p -1 is a prime number, 2 ^p -1 is a prime number. He verified that when p = 2, 3, 5, 7, 17, 19, the values of the algebraic formulas obtained were all prime numbers. Later, when Euler proved that p = 31, 2 ^p -1 was a prime number. When p = 2,3,5,7, 2 ^p -1 are all prime numbers, but when p = 11, the resulting 2,047 = 23 × 89 is not a prime number.

250 years after Mason's death, Cole, an American mathematician, proved that 2 ⁶⁷ -1 = 193,707,721 × 761,838,257,287, which is a composite number. This is the ninth Mason number. In the 20th century, it has been proved that the 10th Mason number is a prime number, and the 11th Mason number is a composite number. The chaotic arrangement of prime numbers also makes it difficult for people to find the law of prime numbers.

So far, humans have only found 49 Mason prime numbers. The prime number discovered by the University of Central Missouri on January 7, 2016 is the largest prime number ever discovered, and it is also a Mason prime number. Because of this rare and fascinating prime number, it is known as "the treasure of mathematics". It is worth mentioning that Chinese mathematician and linguist Zhou Haizhong cleverly used the connection observation method and incomplete induction method based on the known Mason prime numbers and their permutations, and formally proposed the conjecture of Mason quality distribution in 1992. An important conjecture is known internationally as "Zhou's conjecture."

(GIMPS) The project found the largest prime number 2 ^74,207,281 -1 known to mankind on January 7, 2016. The prime number has 22,338,618 digits, which is the 49th Mersenne prime number.

On December 26, 2017, Jonathan Pace, a Gimps volunteer in Germantown, Tennessee, USA, discovered the 50th Mersenne prime 277232917-1. This oversized prime has 23,249,425 digits, which once again refreshed the record of the largest known prime. The new record is M82589933, discovered by Patrick Roche in Ocala, Florida, on December 7, 2018.