Charming special prime

Recently, an amateur mathematician from Florida, USA, discovered the largest prime number known to humans, 2^82589933-1; the number has 24,620,048, and if it is printed with a common font size, its length will exceed 100 kilometers! Relevant experts believe that this is a major breakthrough in the field of mathematics.

As we all know, the starting point of mathematics is natural numbers, and the basis of natural numbers is prime numbers. Prime numbers, also known as prime numbers or irreducible numbers, refer to numbers that cannot be divisible by other natural numbers except for 1 and the integer itself in a natural number greater than 1, in other words, the number is no longer except 1 and itself. There are other numbers of divisors. A number larger than 1 but not a prime number is called a composite number, and 1 and 0 are neither prime nor non-combined, and the composite number is obtained by multiplying several prime numbers.

More than 2,300 years ago, the ancient Greek mathematician Euclid has proved that there are many prime numbers, such as 2, 3, 5, 7, 11, 13 and so on. Because prime numbers have many unique properties, it has attracted many mathematicians and countless amateur math enthusiasts; people have a process of understanding and researching it, which is gradually deepened. In mathematics, there are several special primes that are very fascinating.

Mersenne prime number
A number of the form 2^P-1 (i.e., the P-th power of 2 minus 1, wherein the index P is a prime number) is called a Mason number. It is named after the 17th century French mathematician M. Mersenne; if the Mason number is prime, it is called the Mason prime, such as 2^2-1=3, 2^3-1=7, 2^ 5-1=31, 2^7-1=127, etc.

Based on the research of Euclid, Fermat and others, Mason did a lot of calculations and verifications on 2^P-1z, and in 1644 he interrupted in his book “Physical Mathematical Sense”: No Among the prime numbers greater than 257, when P = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, 257, 2^P-1 is a prime number, and the others are all composite numbers. The first seven numbers (ie 2, 3, 5, 7, 13, 17, 19) have been confirmed by the predecessors, while the next four numbers (ie 31, 67, 127, 257) are Mason’s own inferences. . At that time, some people believed that Mason’s assertion (also known as “Mason’s conjecture”) was correct, but later people realized that his assertion actually contained a number of errors and omissions.

In the era of handwriting, people have gone through countless hardships, and only found a total of 12 Mersenne prime numbers. The advent of electronic computers, especially the era of grid computing, has greatly accelerated the pace of Mason’s prime search. So far, 51 Mersenne primes have been found, the latest and largest being 2^82589933-1; this number is also the largest prime number known to man. Nearly 700,000 people around the world are participating in an international cooperation project called “Internet Mersen Prime Search”, which uses more than 1.85 million nuclear CPUs to connect to find new Mason prime numbers.

In the basic research of the Mason prime, the French mathematician F. Lucas and the American mathematician D. Lehmer made important contributions; the Lucas-Remer method named after them is The best known method for detecting Mason digitality is currently known. In addition, Chinese mathematician and linguist Zhou Haizhong gave an accurate expression of the Meisen prime number distribution; this research result was internationally hailed as “Zhou’s guess” and was highly evaluated.

17th century French mathematician Mason

Fermat prime number
The number of 2^(2^N)+1 (where the exponent N is a non-negative integer) is called the Fermat number, which is named after the 17th century French mathematician P. Fermat; The horse number is a prime number, which is called the Fermat prime number, such as 2^(2^0)+1=3, 2^(2^1)+1=5, 2^(2^2)+1=17, 2 ^(2^3)+1=257 and so on.

Fermat proposed a conjecture in 1640: the number of 2^(2^N)+1 must be prime, but he did not give a complete proof. In 1732, Swiss mathematician and physicist L. Euler discovered that 2^(2^5)+1=641×6700417 is not a prime number; this means that it is a composite number, so Fermat’s guess is wrong. Later, people have found many counterexamples, such as 2^(2^6)+1=274177×67280421310721, 2^(2^7)+1=59649589127497217×5704689200685129054721 and so on. So far, the Fermat prime number has not found one more than the five confirmed by Fermat himself!

At present, many mathematical problems are studied using the high-speed computing functions of computers, but even so, the results are limited. For example, at present, only five Fermat numbers are known as prime numbers, and 226 Fermat numbers are found to be composite numbers, but this can be said to be very small for an unlimited number of Fermats.

Incidentally, Fermat’s small theorem is the basis of the modern prime number determination method. If the inverse proposition of the theorem is established, then it is easy to discriminate the prime number. Unfortunately, the inverse theorem of Fermat’s small theorem does not hold, and the composite number that makes it unsuccessful is called pseudo-prime number. It can be seen that the pseudo-prime number plays an important role in the study of mathematics.

In fact, for thousands of years, people have been looking for such a formula that can find all the prime numbers. But until now, no one has found such a formula, and no one can find evidence, saying that such a formula must not exist. Although Fermat’s conjecture failed, it is interesting to note that in 1801, the German mathematician and physicist J. Gauss proved that the reference to the Fermat prime can be equally divided by a ruler and a compass; he himself This made a positive heptagon.

Twin prime number
The twin prime number, also known as the twin prime number, refers to a pair of prime numbers, which differ by 2, such as (3,5), (5,7), (11,13), (17,19), and so on. The pair of maximum twins currently known is 2996863034895 × 2 ^ 1290000 ± 1; it has 388,342 digits, which was discovered in September 2016.

In the era of handwriting, people have gone through countless hardships, and only found a total of 12 Mersenne prime numbers.

In addition to the five that Fermat himself confirmed, Fermat’s prime number did not find one more!

Euclidean was the first person to notice the interesting phenomenon of twin prime numbers. He had boldly guessed that there were infinite pairs of twin prime numbers. This conjecture is called the “twin prime number conjecture.”

The French mathematician A. Polignac proposed a more general conjecture in 1849 (the “Polygnac conjecture”): for all positive integers K, there are infinite prime pairs (P, P). +2K). When K is equal to 1, it is a twin prime guess, and when K is equal to other positive integers, it is called a weak twin prime guess (ie, a weakened version of the twin prime guess). Therefore, some mathematicians have used Polignac as the proponent of the twin prime speculation.

British mathematicians G. Hardy and J. Littlewood proposed a conjecture similar to the Polignac conjecture in 1921. It is now known as the Hardy-Rittelwood conjecture or strong. Twin prime speculation” (ie an enhanced version of the twin prime guess). This conjecture not only proposes that there are infinite pairs of twin prime numbers, but also gives its asymptotic distribution. Since the distribution of twin prime numbers is extremely uneven and becomes more and more sparse as the number increases, it is very difficult to study the distribution pattern of twin prime numbers.

It is worth mentioning that in the basic research of twin prime numbers, the American Chinese number scientist Zhang Yitang has made a major breakthrough; he proved in 2013 that there are infinite numbers of infinite differences of less than 70 million. Although 70 million looks like a very large number, no matter how big the number is, the existence of a limited range means that the difference between connected prime numbers does not always increase; moreover, from 2 to 70 million, and 70 million to infinity Same day.

Palin prime
Palindromic primes are integers that are both prime numbers and palindromes, such as 11, 101, 131, and 151 in decimal. Except for the smallest palindrome prime number 11, there is no palindrome prime number for the even number of digits. It is known that there are two palindrome prime numbers, three palindrome prime numbers, five five palindrome prime numbers, seven palindrome prime numbers 668, and nine palindrome prime numbers 5172. The largest known palindrome number is currently 320,237 digits, which was discovered in March 2014.

It should be pointed out that the palindrome prime number is related to the carry system of the counting system. At present, people do not know whether there are an infinite number of palindromes in decimal. In binary, the palindromes include the Mason prime and the Fermat prime.

Some of these special primes have practical value (such as Mersenne prime numbers), and some do not see any practical value (such as twin prime numbers). In any case, they are famous mathematical problems. Exploring these problems and uncovering their mysteries is precisely the long-term scientific pursuit of people. As the German mathematician D. Hilbert said: “We must know that we will know.”