/kloc-in the 7th century, there was a French lawyer named Fermat (Fermat, 160 1- 1665). He likes mathematics very much and often studies advanced mathematics problems in his spare time. Therefore, he is called the king of amateur mathematicians. When Fermat studied mathematics, he didn't like proof, but he did. With rich imagination and profound insight, he put forward a series of important mathematical conjectures, which deeply influenced the development of mathematics. His Fermat's Last Theorem has attracted countless mathematicians for hundreds of years, and it was not until 1994 that it was proved by wiles of Princeton University.

In 1640, he proposed a formula:' 2+1'. He checked the situation that n is equal to 1 ratio 4, and found that they are all prime numbers (as shown in the following table), and directly guessed that as long as n is a natural number, the formula must be a prime number. "

n

2+ 1

1

2+ 1=5 (prime number)

2

2+ 1= 17 (prime number)

three

2+ 1=257 (prime number)

four

2+ 1=65537 (prime number)

Fermat's favorite branch of mathematics is number theory. He deeply studied the properties of prime numbers and found an interesting phenomenon. Is calculation = a prime number?

How much is that? Is it a prime number?

13. Say, what is this? Is it a prime number?

What's the last one? Is it a prime number?

Answer:

=5; This is a prime number.

= 17; This is a prime number.

=257; This is a prime number.

=65537; This is a prime number.

Fermat did not continue to calculate that year. He guessed that as long as n is a natural number, the number derived from this formula must be a prime number; This is a famous conjecture, because it is very troublesome to calculate after n=5, and few people verify it.

1732, the great mathematician Euler studied this problem seriously. He found that if a natural number was further calculated, Fermat would not be all prime numbers.

When n=5, = = 4294967297,4294967297 can be decomposed into 64 1×67004 17, which is not a prime number. In other words, Fermat conjecture can't be a formula for finding prime numbers. In fact, mathematicians have been looking for such a formula for thousands of years. But until now, no one has found such a formula, and no one has found evidence that such a formula must not exist; Whether such a formula exists has become a famous mathematical problem.

Fermat is a very important figure in the history of mathematics. Although Fermat's formula is wrong, mathematicians look for big prime numbers from another direction, that is, when they finish all the numbers, they mention,' If 2- 1 is a prime number, then N=2(2- 1) must be a perfect number.' Therefore, mathematicians try to check different numbers. There was no computer to help at that time, so many results were wrong. /kloc-in the 7th century, a French Catholic monk Masseni suggested that * * has 1 1 prime numbers when n is not greater than 257. Although there are many mistakes in his results, later generations called this prime number mersenne prime in the form of' 2-1'. "