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Mathematical handwritten newspaper lace design picture

Math handwritten newspaper lace design picture 1 Math handwritten newspaper content April 6, 2000, living in Yang Na, Plymouth, Michigan, USA? Mr. Yan Na Khairat Valla won the $50,000 prize in mathematics because he discovered the largest prime number known so far, which is a mersenne prime:

26972593- 1。

This is also the first prime number with more than one million digits that we know. To be precise, if this prime number is written in the familiar decimal form, it has two million digits. If you write in this form, you will probably need 150 to 200 articles.

But Mr. Haji Latvala is not a mathematician. He may even know nothing about the mathematical theory of finding prime numbers-even though it won him this prize. All he did was download a program from the Internet. When he is not using his Pentium II350 computer, this program runs quietly. After the calculation of 1 1 1 day, the prime number mentioned above was found.

Second, mersenne prime.

We call a natural number greater than 1 a prime number if only 1 and itself can divide it exactly. If a natural number greater than 1 is not a prime number, we call it a composite number. 1 is neither prime nor composite.

For example, you can easily verify that 7 is a prime number; And 15 is a composite number, because except 1 and 15, 3 and 5 can be divisible by 15. By definition, 2 is a prime number and the only even prime number. As early as 300 BC in ancient Greece, the great mathematician Euclid proved that there were infinitely many prime numbers.

There are many simple and beautiful but extremely difficult questions about prime numbers that have not been answered so far. There is a famous Goldbach conjecture, which means that any even number greater than 6 can be expressed as the sum of two odd prime numbers. There is also the problem of double prime numbers. Prime pairs with a difference of 2, such as 5 and 7,465,438+0 and 43, are called twin prime numbers. The question of twin prime numbers is: Do twin prime numbers have infinity? By the way, the solutions of these seemingly simple mathematical problems will be extremely complicated and need the most advanced mathematical tools. If you are not arrogant enough to think that all mathematicians (many of them are excellent) and math lovers who have spent countless talents on these problems for hundreds or even thousands of years are not as smart as you, then don't try to solve these problems by elementary methods, it will only take time and energy.

The ancient Greeks were also interested in another number. They call it a perfect number. A natural number greater than 1 is called a perfect number if the sum of all its factors (including 1 but excluding itself) is equal to itself. For example, 6= 1+2+3 is the smallest perfect number. The ancient Greeks regarded it as Venus, a symbol of love. 28= 1+2+4+7+ 14 is another perfect number. Euclid proved that an even number is a perfect number if and only if it has the following form:

Mathematical handwritten newspaper lace design picture 2

2p- 1

Where 2p- 1 is a prime number. The above 6 and 28 correspond to the case of p=2 and 3. As long as we find a prime number in the form of 2p- 1, we will know an even perfect number. As long as we find all prime numbers in the form of 2p- 1, we will find all even perfect numbers. So Mr. Haji Latvala not only discovered the largest known prime number in the world, but also discovered the largest even perfect number in the world. Well, you have to ask, what about odd perfect numbers? The answer is: we haven't even found an odd perfect number, and we don't even know whether there is an odd perfect number. We only know that if there is an odd perfect number, it must be very, very big! Whether odd perfect numbers exist or not is also a simple and beautiful but extremely difficult famous mathematical problem.

For a long time, people thought that for all prime numbers p,

M_p=2p- 1

Are prime numbers (note that in order for 2p- 1 to be prime, p itself must be prime. Why? However, in 1536, Hudalricus Regius points out that m _11= 21-1= 2047 = 23 * 89 is not a prime number.

Pietro? Cataldi first made a systematic study of this kind of figures. In the results published in 1603, he said that p= 17, 19, 23, 29, 3 1 and 37, 2p- 1 is a prime number. But in 1640, Fermat proved that Caldi's results about p=23 and 37 were wrong with the famous Fermat's little theorem (not to be confused with Fermat's last theorem), and Euler proved that the results about p=29 were wrong in 1738, and later he proved that the conclusion about p=3 1 was correct. It is worth pointing out that cataldi reached his conclusion by checking one by one by hand. Fermat and Euler used the most advanced mathematical knowledge at that time to avoid many complicated calculations and possible mistakes.

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French priest Mei Sen published his achievements in 1644. He declared that p=2, 3, 5, 7, 13, 17, 19, 3 1 67, 127 and 257, 2p- 1 are all prime numbers.

It is quite difficult to judge whether a large number is a prime number by hand. Father Mei Sen himself admits that his calculation is not necessarily accurate. It was not until 1750 a century later that Euler announced that he had discovered Father Mei Sen's mistake: M_4 1 and M_47 were also prime numbers. But as great as Euler, he also made calculation mistakes-in fact, neither M_4 1 nor M_47 is a prime number. But this does not mean that Father Mei Sen's results are correct. It was not until 1883, that is, more than 200 years after Father Mei Sen's results were published, that the first error was discovered: M_6 1 is a prime number. Then four other errors are found: M_67 and M_257 are not prime numbers, while M_89 and M_ 107 are prime numbers. Until 1947, for P.

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