What is a Mersenne Prime Number?

Mason primes are derived from Mason numbers.

Mason prime

As early as 300 BC, ancient Greek mathematicians

For more than 2300 years, humans have only found 51 Mason primes. Because of their rarity and fascination, they have been hailed as "the sea pearl". Since Mason put forward his assertion, the known maximum primes that people have discovered are almost all Mason primes, so the process of finding a new Mason prime is almost the same as the process of finding a new maximum prime.

The search for Mason primes is extremely difficult. It requires not only advanced theories and sophisticated skills, but also difficult calculations.

As of December 2018, a total of 51 Mason primes were found. Now list their values, digits, discovery time, discoverers, etc. as follows:

Early 1996, American mathematician and programmer

Mason prime numbers have been an important part of number theory research since ancient times. Many great mathematicians in history have studied this special form of prime numbers. From the ancient Greek era until the 17th century, the meaning of people looking for Mason primes seems to be just to find

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;

This formula is based on the distribution of Mason prime numbers. The tens is the first, and the 1 is excluded, so subtract 1. Without considering the overlap problem, P should be reduced by 1. Here, the overlap problem has been considered, so P is reduced by 1.2. The index of the Messen number is gradually increased, and whether 1.2 is suitable or not has to be tested in practice.

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 prime number of the previous Mason composite number, which will never be the factor of the subsequent Mason composite number.

All Mason composite factors minus 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.

The factors of Mason composite numbers are prime numbers of the form 8N + 1 and 8N-1.

Test Mason Prime

In the 2 ^ n-1 series where the index n is infinite, Mason numbers and Mason prime numbers account for only a small proportion of them.

According to Fermat's Little Theorem, every odd prime number will appear in the 2 ^ n-1 sequence with a number factor, but some appear in advance and some appear at the end. Only Mason prime numbers appear first in this sequence. No other prime number appears first, and the latest prime number appears in the number minus one, which is where Fermat's Little Theorem is.

Every odd prime number appears very regularly as a factor in a 2 ^ n-1 sequence. A prime appears for the first time in a 2 ^ n-1 number (including a Mason prime). This prime appears repeatedly with a period of n. In the 2 ^ n-1 sequence, if 3 appears for the first time in n = 2, the factor that can be divisible by 2 has a factor of 3. For the first time, if n appears in n = 3, the index can be divided by 3. Both have a factor of 7; 5 appears for the first time in n = 4, and the exponent is divisible by 4 and has a factor of 5. A prime number appears in the 2 ^ n-1 sequence n. No matter whether n is a prime number or not, as long as all odd prime numbers less than n are used to sieve, the index n is in it. If the composite number overlaps with the previous prime number, there is no need to rescreen.

In order to sieve all the number factors in the 2 ^ n-1 sequence, all prime numbers less than or equal to the square root of 2 ^ n-1 must be used to sieve, so the remaining unscreened is the Mason prime.

The sequence of 2 ^ n-1 is infinite, and an infinite number of natural numbers is a fraction of how many times you sieve. It is always infinite. So Mason primes are infinite.

Sieve method of mason prime

According to Fermat's Little Theorem, every odd prime number will appear in the 2 ^ n-1 sequence as a prime factor, but some appear early and some appear late.

Each odd prime number appears for the first time in the number of the 2 ^ n-1 series index n. This n is the corresponding number of the prime number, and it repeatedly appears with n as a cycle.

It is already known that Mason prime numbers all appear in Mason numbers. As long as the Mason number is filtered out, the rest is the Mason prime.

Expand the Mason series, starting from the corresponding number 3 of 3, 2 points and 1 point; the corresponding number of 5 is 4, and 4 is a composite number, there is no need to operate; the corresponding number of 7 is 3, at 3 points; the corresponding number of 11 is 10 is a composite number without operation; the corresponding number of 13 is 12, and 12 is a composite number without operation; in this way, keep going, and the number before the index of the Mason number can be filtered. Any Mason number that has been counted twice or more is crossed out, and the rest is a Mason number that is only once. These Mason numbers are all Mason primes.

This sieve method is a bit difficult to find its corresponding number when the prime number is large.