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xw_y_am@rmbp 2017-09-30 11:03:18 +08:00
parent 6fe1881e82
commit 772f1f8d15
4 changed files with 22 additions and 19 deletions
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4
python/131.md
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@ -2,12 +2,12 @@ $$
m^3=n^3+n^2p m^3=n^3+n^2p
$$ $$
let $m=n+k$ , then 设 $m=n+k$ ，则
$$ $$
3n^2k+3nk^2+k^3=n^2p\\ 3n^2k+3nk^2+k^3=n^2p\\
(p-3k)n^2-3k^2n-k^3=0 (p-3k)n^2-3k^2n-k^3=0
$$ $$
witch $p>3k$ . solving this function, we get 其中有 $p>3k$ 。解此方程，得
$$ $$
n = \frac{3k^2 \pm k\sqrt{4kp-3k^2}}{p-3k} n = \frac{3k^2 \pm k\sqrt{4kp-3k^2}}{p-3k}
$$ $$

18
python/137.md
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@ -6,33 +6,33 @@ $$
xA_F(x)= \; x^2F_1+x^3F_2+x^4F_3+\cdots xA_F(x)= \; x^2F_1+x^3F_2+x^4F_3+\cdots
$$ $$
$(1)+(2)$, and consider that $F_1=F_2$, then we have (1)+(2)，且考虑到 $F_1=F_2$ ，有
$$ $$
(1+x)A_F(x)=xF_2+x^2F_2+x^3F_3+\cdots (1+x)A_F(x)=xF_2+x^2F_2+x^3F_3+\cdots
$$ $$
so 所以
$$ $$
x(1+x)A_F(x)=A_F(x)-xF_1 x(1+x)A_F(x)=A_F(x)-xF_1
$$ $$
then 即
$$ $$
A_F(x)=\frac{x}{1-x-x^2} A_F(x)=\frac{x}{1-x-x^2}
$$ $$
we want that $A_F(x)$ could be a nature number, so let 想要 $A_F(x)$ 是自然数，只需设
$$ $$
\frac{x}{1-x-x^2}=n \frac{x}{1-x-x^2}=n
$$ $$
then 即
$$ $$
nx^2+(n+1)x-n=0 nx^2+(n+1)x-n=0
$$ $$
solve this, get 解此方程，得
$$ $$
x=\frac{\sqrt{5n^2+2n+1}-(n+1)}{2n} x=\frac{\sqrt{5n^2+2n+1}-(n+1)}{2n}
$$ $$
only $5n^2+2n+1$ is square number that can make x to be a rational. 只有 $5n^2+2n+1$ 是平方数时， $x$ 才可能是有理数。
let's try to analyse how $n$ increasing when $5n^2+2n+1$ is square number. 尝试分析 $N=5n^2+2n+1$ 是平方数时， $n$ 的变化规律
| n | increasing rate of n | | n | increasing rate of n |
| ---------- | -------------------- | | ---------- | -------------------- |
@ -49,4 +49,4 @@ let's try to analyse how $n$ increasing when $5n^2+2n+1$ is square number.
| 507544127 | 6.854101982 | | 507544127 | 6.854101982 |
| 3478759200 | 6.854101969 | | 3478759200 | 6.854101969 |
it is obvious that the rate of $n$ is converge to $6.8541\cdots$ . with using search engein we know $\varphi^4=6.8541\cdots$ and $\varphi=\frac{\sqrt{5}-1}{2}$. 显然 $n$ 的变化率收敛于 $6.8541\cdots$ 。通过搜索引擎可知 $\varphi^4=6.8541\cdots$ 其中 $\varphi=\frac{\sqrt{5}-1}{2}$ 。

6
python/tools/algebra.py
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@ -4,3 +4,9 @@ def factorial(n, multi=1):
return multi return multi
else: else:
return factorial(n - 1, multi * n) return factorial(n - 1, multi * n)
def is_root(n, exp=2):
root = int(n ** (1 / exp))
if root ** exp == n:
return root
return 0

13
python/tools/number_theory.py
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@ -20,12 +20,11 @@ def pow_mod(base, exp, mod):
def make_prime(limit): def make_prime(limit):
if limit < 5: if limit < 2: return
if limit < 2: return yield 2
yield 2 if limit < 3: return
if limit < 3: return yield 3
yield 3 if limit < 5: return
return
n = (limit + 1) // 6 n = (limit + 1) // 6
@ -46,8 +45,6 @@ def make_prime(limit):
a[f::p] = [False] * ((n - f - 1) // p + 1) a[f::p] = [False] * ((n - f - 1) // p + 1)
b[g::p] = [False] * ((n - g - 1) // p + 1) b[g::p] = [False] * ((n - g - 1) // p + 1)
yield 2
yield 3
for i in range(n): for i in range(n):
if b[i]: if b[i]:
yield 6 * i + 5 yield 6 * i + 5