# 毕氏数

## 找出毕氏数

${\displaystyle a=m^{2}-n^{2))$
${\displaystyle b=2mn}$
${\displaystyle c=m^{2}+n^{2))$

${\displaystyle m}$${\displaystyle n}$互质，而且${\displaystyle m}$${\displaystyle n}$为一奇一偶，计算出来的${\displaystyle (a,b,c)}$就是素毕氏数。（若${\displaystyle m}$${\displaystyle n}$都是奇数${\displaystyle (a,b,c)}$就会全是偶数，不符合互质。）

## 例子

${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle c}$
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

${\displaystyle a^{2}=(c-b)(c+b)}$

${\displaystyle 1229779565176982820}$
${\displaystyle 1230126649417435981}$
${\displaystyle 1739416382736996181}$

${\displaystyle 1229779565176982820}$
${\displaystyle 378089444731722233953867379643788099}$
${\displaystyle 378089444731722233953867379643788101}$

${\displaystyle 1229779565176982820=2^{2}\times 3\times 5\times 7\times 11\times 13\times 17\times 19\times 23\times 29\times 31\times 37\times 41\times 43\times 47}$

## 性质

• ${\displaystyle \gcd(a,b)=\gcd(b,c)=\gcd(c,a)=\gcd(a,b,c)}$
• ${\displaystyle (a,b)}$至少一个是3的倍数
• ${\displaystyle (a,b)}$至少一个是4的倍数
• ${\displaystyle (a,b,c)}$至少一个是5的倍数