Conjunction, 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.
If expressed by indefinite equation, Fermat's last theorem is: when n
& gt
2. Indefinite equation xn
+
y
n
=
z
n
Xyz≠0 has no integer solution. In order to prove this result, only equation x4 needs to be proved.
+
y
four
=
z
four
，(x
,
y)
=
1 and equation xp
+
Foaming strip
=
Zero power
，(x
,
y)
=
(x)
,
z)
=
(y
,
z)
=
1 (p is an odd prime number) has no integer solution of xyz≠0.
n
=
The situation of 4 has been solved by Leibniz and Euler. Fermat himself proved p.
=
3, but the proof is incomplete. Legendre [1823] and Dirichlet [1825] proved p.
=
The situation of 5. In 1839, Lame proved P.
=
The situation of 7. 1847, the German mathematician Cuomo made a breakthrough in Fermat's conjecture. He founded the ideal number theory, which made him prove that when P
& lt
100, except p.
=
Fermat's conjecture holds except for 37, 59 and 67. Later, he conducted in-depth research and proved that Fermat's conjecture about the above three numbers was also established. Among modern mathematicians, Van Dhivert made an important contribution to Fermat's conjecture. He began to study Fermat's conjecture in the 1920s, and first discovered and corrected the defects in Cuomo's proof. In the next 30 years, he did a lot of work and got some sufficient conditions for the establishment of Fermat's conjecture. He and two other mathematicians proved that when P.
& lt
In 4002, the Fermat conjecture was established.
Modern mathematicians also use large electronic calculators to explore Fermat's conjecture and make P.
The figures have been greatly improved. By 1977, wagstaff proved p.
& lt
At 125000, Fermat's conjecture holds. Chinese mathematical society Communication 1987 No.2 According to foreign reports, Fermat conjecture has made amazing research achievements in recent years: Grandville and Heath-Braun proved that "Fermat's Last Theorem holds for almost all indexes". That is, if N(x) represents the number of exponents in an integer not exceeding x, then
The result of [falting] is used to prove. Another important result is that if there is a counterexample of Fermat's conjecture, there is X.
& gt
0，y
& gt
0，z
& gt
0，n
& gt
2, making xn
+
y
n
=
z
n
, and then x
& gt
10 1,800,000。