In other words, when n > 2,
xn+yn=zn( 1)
There is no positive integer. This is the world-famous Fermat's last theorem.
"I have a wonderful proof about this proposition," Fermat said, "but the blank here is too small to write down."
People have never been able to find verma's "proof". Many mathematicians have conquered this castle, but not yet. So Fermat's last theorem is actually Fermat's great conjecture. People only found this equation in Fermat's letters and manuscripts.
x4+y4=z4(2)
Without the proof of positive integer solution, I'm afraid the "great theorem" he really proved is a special case of n = 4.
Because (2) has no positive integer solution, the equation
x4k+y4k=z4k(3)
No solution (if (3) has a solution, there are positive integers x0, y0, z0.
x04k+y04k=z04k(3)
Then (x0k) 4+(y0k) 4 = (z0k) 4.
This is contradictory to the unsolvable (2)!
In the same way, we only need to prove that for odd prime number p, indefinite equation
xp+yp=zp(4)
If there is no positive integer solution, then Fermat's Last Theorem holds (because every integer with n > 2 is either divisible by 4 or has an odd prime number p as a factor).
The proof of (4) is very difficult. More than 90 years after Fermat's death, Euler took the first step. In the letter 1753 to Goldbach on August 4th, he claimed that he had proved that (4) had no solution when p = 3. But he found that the proof of P = 3 is completely different from the proof of N = 4. He thinks that the general proof (that is, (4) there is no positive integer solution for all prime numbers P) is very far away.
Sophie Gilman (1776— 183 1), a female mathematician with an alias of LeBron, took the second step to solve Fermat's last theorem. Her theorem is:
"If the indefinite equation
x5+y5=z5
If there is a solution, then 5 | XYZ. "
It is customary to divide the discussion of Equation (4) into two cases. That is, if this equation
xp+yp=zp
If there is no solution satisfying p | XYZ, it is said that Fermat's Last Theorem holds for p in the first case.
If the equation
xp+yp=zp
If there is no solution satisfying p | XYZ, it is said that Fermat's Last Theorem holds for p in the second case.
So Gilman proved that p = 5, and Fermat's last theorem holds in the first case. She also proved that if p and 2p+ 1 are both odd prime numbers, then Fermat's Last Theorem holds in the first case. She further proved that Fermat's Last Theorem in the first case holds for odd prime numbers P ≤ 100.
In the more than 90 years after Euler solved P = 3, although many mathematicians tried to prove Fermat's Last Theorem, little progress was made. Except Gilman's results, only the cases of p = 5 and p = 7 are solved.
The honor of conquering P = 5 is shared by two mathematicians, one is Dirichlet, who just turned 20, and the other is Lested, who is over 70 and famous. They completed this proof in September 1825 and1October 165438+ respectively.
P = 7 was proved by the French mathematician Lame in 1839.
In this way, it is more and more difficult to deal with each odd prime P one by one, and Fermat's last theorem can't be solved for all P's. Is there any way to prove Fermat's last theorem for all P's or at least for a batch of P's? German mathematician Kumar has created a new method to look at Fermat's last theorem from a new and profound point of view, which brings hope to the solution of the general situation.
Using the ideal theory, Kumar proved that Fermat's Last Theorem holds for P < 100. In recognition of his achievements, the Paris Academy of Sciences awarded him 1857 a prize of 3,000 francs.
Kumar found that Bernoulli number has an important connection with Fermat's Last Theorem, and he introduced the concept of normal prime number: if prime number P does not divide the denominator of B2, B4...BP-3 is called normal prime number, and if p is divided by the denominator of one of B2, B4...BP-3 is called irregular prime number. For example, 5 is a normal number because the denominators of B2 are 6 and 5×6. 7 is also a normal prime number, because B2 has a denominator of 6, B4 has a denominator of 30, and 7×6 and 7×30.
In 1850, Kumar proved that Fermat's Last Theorem holds for normal prime numbers, thus proving that Fermat's Last Theorem holds for a large number of prime numbers. He found that only 37, 59 and 67 in 100 are irregular prime numbers. After special treatment of these three numbers, he proved that Fermat's Last Theorem holds for P < 100.
How many normal prime numbers are there? Kumar has infinite guesses, but this guess has never been confirmed. Interestingly, in 1953, Kalitz proved that the number of irregular prime numbers is infinite.
In recent years, great progress has been made in the study of Fermat's Last Theorem. In 1983, Valtins of West Germany proved that "the (nondegenerate) curve F(x, y) = 0 on algebraic number field k has a limited number of k points at most when it exceeds g > 1."
As a special case, the curve on rational number field q
xn+yn- 1=0(5)
When genus g > 1, there are at most finite rational points.
Here the genus G is a geometric quantity, and G is available for curve (5).
g=(n- 1)(n-2)2
According to (6), when n > 3, the genus of (5) is greater than 1, so there are at most a limited number of rational points (x, y) satisfying (5).
equation
xn+yn=2n
Can become
x2n+y4n- 1=0
If x2 and y2 are changed to (x, y), then (7) becomes (5). Therefore, it is concluded from (5) that there are only finite rational number solutions X, Y, and it is immediately concluded that (1) there are only finite positive integer solutions X, Y, Z, but here X, Y, Z and kx, ky, kz(k is a positive integer) are all counted as the same solution set.
Therefore, even if Fermat's Last Theorem is not true for a certain n, Equation (7) has a positive integer solution, but the solution has a finite number of groups at most.
In 1984, Edelman and Heath Brown proved that Fermat's last theorem in the first case holds for infinite p. Their work makes use of an important result of Fauvre: there are infinite pairs of prime numbers P and Q, which satisfy Q | P- 1 and Q > P2/3. Favrey's result is based on a new estimate of Lustman, which has caused many breakthroughs in number theory.
Despite centuries of efforts, it is uncertain whether Fermat's Last Theorem is correct. In 1977, wagstaff proved that the great theorem holds for p < 125000. Recently, Lohan further proved that the last theorem holds for p < 4 1 million. However, Fermat's last theorem is still a conjecture. If anyone can give a counterexample, the great theorem will be overturned. But counterexamples are hard to cite.