Question

solve the equation
x2 + y2 = z2
​

Answer 1

Answer:

The Equation x^2 + y^2 = z^2

The equation x^2 + y^2 = z^2 is associated with the Pythagorean theorem: In a right triangle the sum of the squares on the sides is equal to the square on the hypotenuse.

We all learn that (3,4,5) is a “Pythagorean triple”: 3^2 + 4^2 = 5^2. A Pythagorean triple is a “triple” of three positive integers (x, y, z) so that x^2 + y^2 = z^2. The triple is said to be primitive if x and y have no common factors other than ±1.

In other words

(x, y, z) is primitive if x^2 + y^2 = z^2 and gcd(x, y) =1.

Examples: (3, 4, 5), (5, 12, 13), (7, 24, 25), (8, 15, 17), (9, 40, 41), (11, 60, 61), (12, 35, 37), (13, 84, 85), (16, 63, 65), (20, 21, 29), (28, 45, 53), (33, 56, 65), (36, 77, 85), (39, 80, 89), (48, 55, 73), (65, 72, 97)

Answer 2

Answer:

The equation x2 + y2 = z2 is associated with the Pythagorean theorem: In a right

triangle the sum of the squares on the sides is equal to the square on the hypotenuse.

We all learn that (3,4,5) is a “Pythagorean triple”: 32 + 42 = 52. A Pythagorean

triple is a “triple” of three positive integers (x, y, z) so that x2 + y2 = z2. The triple is

said to be primitive if x and y have no common factors other than ±1. In other words

(x, y, z) is primitive if x2 + y2 = z2 and gcd(x, y) =1.

Examples: (3, 4, 5), (5, 12, 13), (7, 24, 25), (8, 15, 17), (9, 40, 41), (11, 60, 61), (12,

35, 37), (13, 84, 85), (16, 63, 65), (20, 21, 29), (28, 45, 53), (33, 56, 65), (36, 77, 85),

(39, 80, 89), (48, 55, 73),

we are interested in positive integer solutions. Certainly for any

positive x and y there is a z defined by z = �x2 + y2 but “usually” z is irrational.

Interest in Pythagorean triples precedes Pythagoras by more than 1000 years. The

Babylonian tablet “Plimpton 322” is dated to 1900-1600 BCE contains a table which is

15 rows by 3 columns of integers. In the first column are values of y and in the second

are values of z and the third column is simply an enumeration 1-15. The largest triple

on Plimpton 322 is (13500,12709,18541):

13, 5002 + 12, 7092 = 18, 5412