X+y=z has infinite integer solutions, which are called triples; X 2+Y 2 = Z 2 also has an infinite integer solution. This conclusion was proved by his students in the Pythagorean era. It is called Pythagorean triple, and we in China call it Pythagorean number. But x 3+y 3 = z 3 has never found an integer solution.
The closest is: 6 3+8 3 = 9- 1, or the difference 1. So Fermat, the greatest amateur mathematician so far, put forward a conjecture: generally speaking, it is impossible to write a power higher than twice as the sum of two powers of the same power. Therefore, there are:
Known: A 2+B 2 = C 2
Let c=b+k, k = 1.2.3 ..., then a 2+b 2 = (b+k) 2.
Because the integer c must be greater than A and B and at least greater than 1, k = 1.2.3. ...
Let: a = d (n/2), b = h (n/2) and c = p (n/2);
Then a 2+b 2 = c 2 can be written as d n+h n = p n, n = 1.2.3. ...
When n= 1, d+h=p, and d, h and p can be any integer.
When n=2, a=d, b=h and c=p, then d 2+h 2 = p 2 = > a 2+b 2 = c 2.
When n≥3, a 2 = d n, b 2 = h n, c 2 = p n.
Because, a = d (n/2), b = h (n/2) and c = p (n/2); To ensure that D, H and P are integers, we must ensure that A, B and C are complete squares.
A, b and c must be the squares of integers, so that d, h and p can be integers in the formula of d n+h n = p n.
If d, h and p cannot exist in the formula as integers at the same time, Fermat's last theorem holds.
Extended data:
1In June, 993, Newton College of Cambridge held an academic conference called "L Function and Arithmetic". One of the organizers was Coates, a doctoral supervisor in wiles, so from June 1993 to June/23rd12003, wiles was privileged to give three lectures on the topic of "Modular Form, Elliptic Curve and Galois Representation".
1994 65438+1October 25th1:041sec, wiles sent a complete proof email of Fermat's last theorem to the world mathematics community through his former student, Professor Karl Rubin of Ohio State University, including a long article. Richard taylor and andrew wiles are the authors of another article, "The Ring Theory Properties of Some Heck Algebras". So far Fermat's last theorem has been proved.
Wiles and his former doctoral student, richard taylor, spent nearly a year fixing this loophole in a way that wiles had given up before. This part of the proof is related to Iwasawa's theory. This proves the Gushan-Zhicun conjecture and finally proves Fermat's last theorem.
References:
Baidu encyclopedia-Fermat's last theorem