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:

Fermat's Last Theorem was put forward by French mathematician Pierre de Fermat in the 7th century.

He asserted that when the integer n > 2, the equation x n+y n = z n about x, y and z has no positive integer solution.

Wolfsk of Germany once announced that he would award 65,438+ten thousand marks to the first person who proved the theorem within 100 years after his death, which attracted many people to try and submit their "proofs".

After it was put forward, it was proved by British mathematician andrew wiles in 1995 after more than 300 years of history.