# 立方和

${\displaystyle a^{3}\pm b^{3}\equiv (a\pm b)(a^{2}\mp ab+b^{2})}$

## 驗證

### 主驗證

${\displaystyle a^{2}b-a^{2}b+ab^{2}-ab^{2}=0\,\!}$

${\displaystyle =a^{3}-a^{2}b+ab^{2}+a^{2}b-ab^{2}+b^{3}\,\!}$

${\displaystyle =a(a^{2}-ab+b^{2})+b(a^{2}-ab+b^{2})\,\!}$
${\displaystyle =(a+b)(a^{2}-ab+b^{2})\,\!}$

### 和立方驗證

${\displaystyle (x+y)^{3}\,\!}$
${\displaystyle =x^{3}+3x^{2}y+3xy^{2}+y^{3}\,\!}$

${\displaystyle x^{3}+y^{3}=(x+y)^{3}-3x^{2}y-3xy^{2}\,\!}$

${\displaystyle =(x+y)^{3}-3xy(x+y)\,\!}$
${\displaystyle =(x+y)\left[(x+y)^{2}-3xy\right]\,\!}$
${\displaystyle =(x+y)(x^{2}+2xy+y^{2}-3xy)\,\!}$
${\displaystyle =(x+y)(x^{2}-xy+y^{2})\,\!}$

### 幾何驗證

${\displaystyle x^{3}+y^{3}\,\!}$

${\displaystyle (x+y)^{3}\,\!}$

• ${\displaystyle x\times y\times (x+y)}$
• ${\displaystyle x\times (x+y)\times y}$
• ${\displaystyle (x+y)\times y\times x}$

${\displaystyle =xy(x+y)+xy(x+y)+xy(x+y)\,\!}$
${\displaystyle =3xy(x+y)\,\!}$

${\displaystyle =(x+y)\left[(x+y)^{2}-3xy\right]\,\!}$

${\displaystyle (x+y)^{2}}$可透過和平方公式，得到：

${\displaystyle =(x+y)(x^{2}+2xy+y^{2}-3xy)\,\!}$
${\displaystyle =(x+y)(x^{2}-xy+y^{2})\,\!}$

### 反驗證

${\displaystyle (a+b)(a^{2}-ab+b^{2})\,\!}$
${\displaystyle =a(a^{2}-ab+b^{2})+b(a^{2}-ab+b^{2})\,\!}$
${\displaystyle =a^{3}-a^{2}b+ab^{2}+a^{2}b-ab^{2}+b^{3}\,\!}$
${\displaystyle =a^{3}+b^{3}\,\!}$

${\displaystyle a^{3}+b^{3}=(a+b)(a^{2}-ab+b^{2})}$
x) ${\displaystyle +a^{2}}$ ${\displaystyle -ab}$ ${\displaystyle +b^{2}}$
${\displaystyle +a}$ ${\displaystyle +a^{3}}$ ${\displaystyle -a^{2}b}$ ${\displaystyle +ab^{2}}$
${\displaystyle +b}$ ${\displaystyle +a^{2}b}$ ${\displaystyle -ab^{2}}$ ${\displaystyle +b^{3}}$

## 立方差

${\displaystyle 125u^{3}-343v^{3}\,\!}$

${\displaystyle =(5u)^{3}-(7v)^{3}\,\!}$

${\displaystyle =(5u)^{3}+(-7v)^{3}\,\!}$

${\displaystyle =\left[5u+(-7v)\right]\left[25u^{2}-(5u)(-7v)+(-7v)^{2}\right]}$
${\displaystyle =(5u-7v)(25u^{2}+35uv+49v^{2})\,\!}$

## 兩組立方和的數

${\displaystyle 1729=1^{3}+12^{3}\,\!}$
${\displaystyle 1729=9^{3}+10^{3}\,\!}$

${\displaystyle 4104=9^{3}+15^{3}\,\!}$
${\displaystyle 4104=2^{3}+16^{3}\,\!}$

## 參考文獻

1. ^ Factorization of x^3 + y^3. Queen's College On The Web （英语）.
2. ^ History Dudeney Canterbury puzzles elliptic curves. [1994-11-12] （英语）.
3. ^ The Hardy-Ramanujan Number. Jimloy.com （英语）.