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在自然数范围内,我们把满足方程a~2+b~2=c~2的三个数(a,b,c)称为一组勾股数。如何编制勾股数组呢?下面介绍四种方法。一、任取两个互质数m和n,即(m,n)=1,其中一个是偶数,另一个是奇效,当m>n时,则编制成的勾股数组为(m~2-n~2,2mn,m~2+n~2)(1)公式(1)不是方程a~2+b~2=c~2的一般解,因为不是每一组勾股数a、b、c都能满足公式(1)。例如,公式(1)就没有给出勾股数组(9,12,15)。但是,如果这组勾股数约去公因数,得到的勾股数组(3,4,5)则可由公式(1)在m=2,n=1时直接给出。
In the range of natural numbers, we call the three numbers (a, b, c) that satisfy the equations a ~ 2 + b ~ 2 = c ~ 2 as a set of Pythagorean numbers. How to prepare the Pythagorean array? Here are four ways. First, take two mutual prime m and n, that is, (m, n) = 1, one is even, the other is a miracle, when m> n, then compiled into the Pythagan array is (m ~ 2- n ~ 2, 2mn, m ~ 2 + n ~ 2) (1) Equation (1) is not a general solution to the equations a ~ 2 + b ~ 2 = c ~ 2 because it is not a, b, c can satisfy formula (1). For example, equation (1) does not give the Pythagorean array (9,12,15). However, if the group of gypsies is about to go to the common divisor, the resulting Pythagorean arrays (3,4,5) can be given directly by equation (1) when m = 2 and n = 1.