# Quod Erat Demonstrandum

## 2009/05/13

### 數數唸

1961

$f(x)$ is differentiable at $x = a$”does NOT imply ”$f'(x)$ is continuous at $x = a$

Just think about a classic example

$f(x) = x^2\sin \frac{1}{x}$ for $x \ne 0$ and $f(0) = 0$.

Justin had just sent me an interesting question.

Prove by mathematical induction that $x^3 + y^3 + z^3 = 3^n$ has integral solution $(x,y,z)$ for any positive integer $n$.

Since

$1^3 + 1^3 + 1^3 = 3^1$
$0^3 + 1^3 + 2^3 = 3^2$
$0^3 + 0^3 + 3^3 = 3^3$

the statement is true for $n = 1,2,3$.

For any positive integer $k$,
suppose $x_0^3 + y_0^3 + z_0^3 = 3^k$ for some integers $x_0, y_0, z_0$,
then $(3x_0)^3 + (3y_0)^3 + (3z_0)^3 = 3^{k+3}$
i.e. the statement is true for $n = k + 3$

Another problem is, whether, for fixed $n$, $x^3 + y^3 + z^3 = 3^n$ has unique integral solution (up to permutation)? Could you give a proof or counter-example? Thank you in advance.

（\ ___ /）

[SBA]

1.「連續 N 期買同一注」是個買「六合彩」的好策略嗎？
2.「電腦彩票每次都有幾個數字不出現」如何影響中獎的機會？
3. 試比較評論以下的命題：
(a)「連續買 19 次六合彩都唔中頭獎，咁買第 20 次都唔中頭獎的機會便很高！」
(b)「擲一元硬幣 19 次，每次結果都是『公』，咁擲第 20 次都是『公』的機會便很高！」