Fermat's conjecture, also known as Fermat's last theorem or Fermat's problem, is one of the most famous world problems in number theory. 1637, the French mathematician Fermat wrote next to Proposition No.8 in Volume II of Arithmetic edited by the Greek mathematician Diophantu: "It is impossible to divide a cubic into two cubic powers, a quartic power into two quartic powers, and a power higher than a quadratic power into two powers of the same power in general. I'm sure I found a wonderful proof about this, but the space here is too small to write down. After Fermat's death, people could not find the proof of this conjecture, which aroused the interest of many mathematicians. Euler, Legendre, Gauss, Abel, Dirichlet, Cauchy and other mathematicians tried to prove it, but no one got universal proof. Over the past 300 years, countless outstanding scholars have made great efforts to prove this conjecture, and at the same time, many important mathematical concepts and branches have emerged.

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Fermat guessed around * 1637, Fermat read the Latin translation of Diophantine's book Arithmetic, and read the eighth proposition in Volume 2, "Divide a sum of squares into two squares", and wrote a paragraph in the margin of the book, which means "Divide a cubic number into two cubic numbers, divide a quartic power into two quartic powers, or generally divide a cubic number into two quartic powers. Described in modern mathematical language, Fermat conjecture means that when n > 2, the equation

xn+yn=zn

There is no positive integer solution.

Fermat conjecture is usually called Fermat's last theorem. To prove Fermat's conjecture is right, prove it.

x4+y4=z4

When p is an odd prime number, XP+YP = ZP has no positive integer solution. Fermat said that he proved the former by infinite descent. In 1676, Bessie also proved n = 4, Euler proved n = 3 and 4, and Jean-Claude and Dirichlet proved n = 5. /kloc-0 in the middle of the 9th century, Cuomo proved that n <100 (except 37, 59 and 67) is an odd prime number. 1908, German mathematician Wolfskeil awarded the prize of 65,438+million marks to the first person who proved Fermat's last theorem. Three and a half centuries have passed since Fermat put forward this conjecture, and the problem remains unsolved. The main achievements in recent years are:

(1) 1977 wagstaff proved that Fermat's last theorem is correct for every prime number p < 125000.

(2) 1983, Valdings proved the conjecture put forward by the British mathematician Mo Deer in 1922: What if? E(x, y) is a rational polynomial, algebraic curve? If the genus of E(x, y) = 0 ≥2, then? E(x，y) = 0。 Only a limited number of people can understand at most. This ensures that when n≥4, there are at most finite N's that make Xn+Yn = Zn have integer solutions.

(3) In 1985, Edelman and Hayes Brown proved that there are infinitely many prime numbers P, so that there are no integers x, y, z, {p that satisfy XP+YP = ZP and cannot be divisible by xyz}.

(4)1On June 23rd, 993, British mathematician K.WILER gave an academic report entitled "Modular Form, Elliptic Curve and Galois Representation" at Newton Institute of Mathematics, Cambridge University. Finally, I announced that "I proved Fermat's conjecture". The initial reactions of experts and authorities are mostly positive.

A positive integer in the form of * 22n+ 1 is called fermat number and dEnoted as en, where E0 = 3,? E 1=5，？ E2= 17，？ E3 = 257 and E4 = 65537 are both prime numbers. Fermat once guessed that all fermat number in 1640 are prime numbers, but in 1732 Euler pointed out that 641E5: E5 = 641× 6700417, thus denying Fermat. But how many fermat number are prime numbers, finite or infinite? Is there an infinite number of fermat number? These problems are unsolved. As we all know, 48 fermat number are not prime numbers. Whether E 17 is a prime number or a composite number is still unknown. Fermat number has something to do with drawing with a ruler. Gauss proved that if En is a prime number, a positive En polygon can be made with a ruler.