SidneyZhang/SolutionEuler
1
0
Fork 0
Code Issues Pull Requests Actions Packages Projects Releases Wiki Activity

✨ feat(project)：添加欧拉项目第6题解决方案及相关依赖

Browse Source 
📝 docs(project)：添加Faulhaber公式详细文档说明
⬆️ chore(project)：添加numpy依赖以支持数学计算
This commit is contained in:
Sidney Zhang
2025-12-15 14:55:09 +08:00
parent 65999c8456
commit be3f920e72
4 changed files with 298 additions and 1 deletions
Download Patch File Download Diff File Expand all files Collapse all files

4
pyproject.toml
View File

@@ -4,4 +4,6 @@ version = "0.1.0"
description = "euler 项目的解题。主要为python。"
readme = "README.md"
requires-python = ">=3.12"
dependencies = []
dependencies = [
"numpy>=2.3.5",
]

43
solutions/0006.SumSquareDifference/eular_6.py Normal file
View File

@@ -0,0 +1,43 @@
"""
The sum of the squares of the first ten natural numbers is 1^2 + 2^2 + ... + 10^2 = 385,
The square of the sum of the first ten natural numbers is (1+2+...+10)^2 = 55^2 = 3025,
Hence the difference between the sum of the squares of the first ten natural numbers and the square of the sum is 3025 - 385 = 2640.
Find the difference between the sum of the squares of the first one hundred natural numbers and the square of the sum.
"""
import time
import numpy as np
def timer(func):
def wrapper(*args, **kwargs):
start = time.time()
result = func(*args, **kwargs)
end = time.time()
print(f"{func.__name__} took {end - start:.6f} seconds")
return result
return wrapper
@timer
def sum_square_difference(n: int) -> int:
res = 0
for i in range(1, n + 1):
res += sum(np.array(list(range(1, i))) * i)
return 2 * res
@timer
def better_sum_square(n: int) -> int:
"""https://en.wikipedia.org/wiki/Faulhaber%27s_formula"""
su = ((1 + n) * n // 2) ** 2
sq = (2 * n + 1) * (n + 1) * n // 6
return su - sq
if __name__ == "__main__":
print(sum_square_difference(100))
print(better_sum_square(100))

183
solutions/0006.SumSquareDifference/readme.md Normal file
View File

@@ -0,0 +1,183 @@
## 1. Faulhaber's Formula 概述
Faulhaber's formula 用于计算 **前 n 个正整数的 p 次幂之和**：
$$S_p(n) = \sum_{k=1}^n k^p = 1^p + 2^p + 3^p + \cdots + n^p$$
公式的一般形式为：
$$S_p(n) = \frac{1}{p+1} \sum_{j=0}^p \binom{p+1}{j} B_j \cdot n^{p+1-j}$$
其中 $B_j$ 是伯努利数（Bernoulli numbers）。
### 特例
**p = 0**: $S_0(n) = n$
**p = 1**: $S_1(n) = \frac{n(n+1)}{2}$（三角形数）
**p = 2**: $S_2(n) = \frac{n(n+1)(2n+1)}{6}$（平方金字塔数）
**p = 3**: $S_3(n) = \left[\frac{n(n+1)}{2}\right]^2$（有趣的事实：等于 $S_1(n)^2$）
## 2. 历史背景
- **Johann Faulhaber**（1580-1635）：德国数学家，计算了 $p=1$ 到 $p=17$ 的公式，但未发现通式
- **Jacob Bernoulli**（1654-1705）：独立研究并发现了通用公式，引入了伯努利数
- 公式因此以 Faulhaber 命名，但现代形式主要归功于 Bernoulli
## 3. 伯努利数（Bernoulli Numbers）详解
### 3.1 定义
伯努利数由生成函数定义：
$$\frac{x}{e^x - 1} = \sum_{k=0}^\infty B_k \frac{x^k}{k!}$$
### 3.2 递推关系
更实用的计算方法是**递推公式**：
$$\sum_{k=0}^{m} \binom{m+1}{k} B_k = 0 \quad \text{对于} \ m \ge 1$$
由此可得：
$$B_m = -\frac{1}{m+1} \sum_{k=0}^{m-1} \binom{m+1}{k} B_k$$
### 3.3 前几个伯努利数
| n | $B_n$ | 说明 |
|---|-------|------|
| 0 | 1 | $B_0 = 1$ |
| 1 | -1/2 | |
| 2 | 1/6 | |
| 3 | 0 | （所有奇数大于1的都是0）|
| 4 | -1/30 | |
| 5 | 0 | |
| 6 | 1/42 | |
| 7 | 0 | |
| 8 | -1/30 | |
| 9 | 0 | |
| 10 | 5/66 | |
| 12 | -691/2730 | 这是第一个分子大于1的|
| 14 | 7/6 | |
**重要性质**：
- 对于所有奇数 $n > 1$，有 $B_n = 0$
- $B_1 = -1/2$ 是唯一的非零奇数项
- 符号交替变化：$B_2 > 0, B_4 < 0, B_6 > 0, \ldots$
### 3.4 计算示例
**计算 $B_2$：**
使用递推式（m=2）：
$$\binom{3}{0}B_0 + \binom{3}{1}B_1 + \binom{3}{2}B_2 = 0$$
$$1 \cdot 1 + 3 \cdot \left(-\frac{1}{2}\right) + 3 \cdot B_2 = 0$$
$$1 - \frac{3}{2} + 3B_2 = 0$$
$$3B_2 = \frac{1}{2}$$
$$B_2 = \frac{1}{6}$$
**计算 $B_4$：**
使用递推式（m=4）：
$$\sum_{k=0}^4 \binom{5}{k} B_k = 0$$
$$1 \cdot 1 + 5 \cdot \left(-\frac{1}{2}\right) + 10 \cdot \frac{1}{6} + 10 \cdot 0 + 5 \cdot B_4 = 0$$
$$1 - \frac{5}{2} + \frac{10}{6} + 5B_4 = 0$$
$$-\frac{3}{2} + \frac{5}{3} + 5B_4 = 0$$
$$-\frac{9}{6} + \frac{10}{6} + 5B_4 = 0$$
$$\frac{1}{6} + 5B_4 = 0$$
$$B_4 = -\frac{1}{30}$$
## 4. 使用 Faulhaber's Formula 计算具体例子
### 示例：计算 $S_4(n) = \sum_{k=1}^n k^4$
使用公式：
$$S_4(n) = \frac{1}{5} \sum_{j=0}^4 \binom{5}{j} B_j \cdot n^{5-j}$$
展开：
$$S_4(n) = \frac{1}{5}\left[\binom{5}{0}B_0 n^5 + \binom{5}{1}B_1 n^4 + \binom{5}{2}B_2 n^3 + \binom{5}{3}B_3 n^2 + \binom{5}{4}B_4 n\right]$$
代入伯努利数：
$$S_4(n) = \frac{1}{5}\left[1 \cdot 1 \cdot n^5 + 5 \cdot \left(-\frac{1}{2}\right) n^4 + 10 \cdot \frac{1}{6} n^3 + 10 \cdot 0 \cdot n^2 + 5 \cdot \left(-\frac{1}{30}\right) n\right]$$
化简：
$$S_4(n) = \frac{1}{5}\left[n^5 - \frac{5}{2}n^4 + \frac{10}{6}n^3 - \frac{1}{6}n\right]$$
$$= \frac{1}{5}\left[n^5 - \frac{5}{2}n^4 + \frac{5}{3}n^3 - \frac{1}{6}n\right]$$
$$= \frac{n^5}{5} - \frac{n^4}{2} + \frac{n^3}{3} - \frac{n}{30}$$
通分后：
$$S_4(n) = \frac{6n^5 - 15n^4 + 10n^3 - n}{30}$$
$$= \frac{n(n+1)(2n+1)(3n^2+3n-1)}{30}$$
## 5. 算法实现思路
### 计算伯努利数的 Python 代码框架：
```python
def bernoulli_numbers(n):
"""
计算前n个伯努利数 B_0 到 B_n
"""
B = [0] * (n + 1)
B[0] = 1 # B_0 = 1
for m in range(1, n + 1):
sum_val = 0
for k in range(m):
# 二项式系数 binomial(m+1, k)
binom = binomial_coefficient(m + 1, k)
sum_val += binom * B[k]
B[m] = -sum_val / (m + 1)
return B
```
### 计算幂和：
```python
def faulhaber_sum(p, n):
"""
使用Faulhaber公式计算 sum_{k=1}^n k^p
"""
B = bernoulli_numbers(p)
result = 0
for j in range(p + 1):
binom = binomial_coefficient(p + 1, j)
term = binom * B[j] * (n ** (p + 1 - j))
result += term
return result / (p + 1)
```
## 6. 重要关系与推广
### 6.1 与黎曼ζ函数的关系
对于偶数伯努利数：
$$B_{2k} = (-1)^{k+1} \frac{2(2k)!}{(2\pi)^{2k}} \zeta(2k)$$
例如：
- $\zeta(2) = \sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$
- $B_2 = \frac{1}{6}$
### 6.2 与多项式的关系
$S_p(n)$ 是关于 $n$ 的 $p+1$ 次多项式，且：
- 常数项为 0
- 系数为有理数
- 首项系数为 $\frac{1}{p+1}$
### 6.3 生成函数视角
幂和也有生成函数表示：
$$\sum_{p=0}^\infty S_p(n) \frac{t^p}{p!} = \frac{t}{e^t - 1} \cdot \frac{e^{(n+1)t} - 1}{t}$$
这正是伯努利数生成函数与几何级数的乘积。
## 7. 计算复杂度与优化
- 直接计算：$O(n \cdot p)$（暴力求和）
- Faulhaber公式：$O(p^2)$（计算伯努利数）+ $O(p)$（求值）
- 对于**固定p，多次查询不同n**：预处理伯努利数后每次查询 $O(p)$
- 对于**大p**（如 $p > 10^6$）：需要更高级算法和模运算

69
uv.lock generated
View File

@@ -2,7 +2,76 @@ version = 1
revision = 3
requires-python = ">=3.12"
[[package]]
name = "numpy"
version = "2.3.5"
source = { registry = "https://pypi.org/simple" }
sdist = { url = "https://files.pythonhosted.org/packages/76/65/21b3bc86aac7b8f2862db1e808f1ea22b028e30a225a34a5ede9bf8678f2/numpy-2.3.5.tar.gz", hash = "sha256:784db1dcdab56bf0517743e746dfb0f885fc68d948aba86eeec2cba234bdf1c0", size = 20584950, upload-time = "2025-11-16T22:52:42.067Z" }
wheels = [
{ url = "https://files.pythonhosted.org/packages/44/37/e669fe6cbb2b96c62f6bbedc6a81c0f3b7362f6a59230b23caa673a85721/numpy-2.3.5-cp312-cp312-macosx_10_13_x86_64.whl", hash = "sha256:74ae7b798248fe62021dbf3c914245ad45d1a6b0cb4a29ecb4b31d0bfbc4cc3e", size = 16733873, upload-time = "2025-11-16T22:49:49.84Z" },
{ url = "https://files.pythonhosted.org/packages/c5/65/df0db6c097892c9380851ab9e44b52d4f7ba576b833996e0080181c0c439/numpy-2.3.5-cp312-cp312-macosx_11_0_arm64.whl", hash = "sha256:ee3888d9ff7c14604052b2ca5535a30216aa0a58e948cdd3eeb8d3415f638769", size = 12259838, upload-time = "2025-11-16T22:49:52.863Z" },
{ url = "https://files.pythonhosted.org/packages/5b/e1/1ee06e70eb2136797abe847d386e7c0e830b67ad1d43f364dd04fa50d338/numpy-2.3.5-cp312-cp312-macosx_14_0_arm64.whl", hash = "sha256:612a95a17655e213502f60cfb9bf9408efdc9eb1d5f50535cc6eb365d11b42b5", size = 5088378, upload-time = "2025-11-16T22:49:55.055Z" },
{ url = "https://files.pythonhosted.org/packages/6d/9c/1ca85fb86708724275103b81ec4cf1ac1d08f465368acfc8da7ab545bdae/numpy-2.3.5-cp312-cp312-macosx_14_0_x86_64.whl", hash = "sha256:3101e5177d114a593d79dd79658650fe28b5a0d8abeb8ce6f437c0e6df5be1a4", size = 6628559, upload-time = "2025-11-16T22:49:57.371Z" },
{ url = "https://files.pythonhosted.org/packages/74/78/fcd41e5a0ce4f3f7b003da85825acddae6d7ecb60cf25194741b036ca7d6/numpy-2.3.5-cp312-cp312-manylinux_2_27_aarch64.manylinux_2_28_aarch64.whl", hash = "sha256:8b973c57ff8e184109db042c842423ff4f60446239bd585a5131cc47f06f789d", size = 14250702, upload-time = "2025-11-16T22:49:59.632Z" },
{ url = "https://files.pythonhosted.org/packages/b6/23/2a1b231b8ff672b4c450dac27164a8b2ca7d9b7144f9c02d2396518352eb/numpy-2.3.5-cp312-cp312-manylinux_2_27_x86_64.manylinux_2_28_x86_64.whl", hash = "sha256:0d8163f43acde9a73c2a33605353a4f1bc4798745a8b1d73183b28e5b435ae28", size = 16606086, upload-time = "2025-11-16T22:50:02.127Z" },
{ url = "https://files.pythonhosted.org/packages/a0/c5/5ad26fbfbe2012e190cc7d5003e4d874b88bb18861d0829edc140a713021/numpy-2.3.5-cp312-cp312-musllinux_1_2_aarch64.whl", hash = "sha256:51c1e14eb1e154ebd80e860722f9e6ed6ec89714ad2db2d3aa33c31d7c12179b", size = 16025985, upload-time = "2025-11-16T22:50:04.536Z" },
{ url = "https://files.pythonhosted.org/packages/d2/fa/dd48e225c46c819288148d9d060b047fd2a6fb1eb37eae25112ee4cb4453/numpy-2.3.5-cp312-cp312-musllinux_1_2_x86_64.whl", hash = "sha256:b46b4ec24f7293f23adcd2d146960559aaf8020213de8ad1909dba6c013bf89c", size = 18542976, upload-time = "2025-11-16T22:50:07.557Z" },
{ url = "https://files.pythonhosted.org/packages/05/79/ccbd23a75862d95af03d28b5c6901a1b7da4803181513d52f3b86ed9446e/numpy-2.3.5-cp312-cp312-win32.whl", hash = "sha256:3997b5b3c9a771e157f9aae01dd579ee35ad7109be18db0e85dbdbe1de06e952", size = 6285274, upload-time = "2025-11-16T22:50:10.746Z" },
{ url = "https://files.pythonhosted.org/packages/2d/57/8aeaf160312f7f489dea47ab61e430b5cb051f59a98ae68b7133ce8fa06a/numpy-2.3.5-cp312-cp312-win_amd64.whl", hash = "sha256:86945f2ee6d10cdfd67bcb4069c1662dd711f7e2a4343db5cecec06b87cf31aa", size = 12782922, upload-time = "2025-11-16T22:50:12.811Z" },
{ url = "https://files.pythonhosted.org/packages/78/a6/aae5cc2ca78c45e64b9ef22f089141d661516856cf7c8a54ba434576900d/numpy-2.3.5-cp312-cp312-win_arm64.whl", hash = "sha256:f28620fe26bee16243be2b7b874da327312240a7cdc38b769a697578d2100013", size = 10194667, upload-time = "2025-11-16T22:50:16.16Z" },
{ url = "https://files.pythonhosted.org/packages/db/69/9cde09f36da4b5a505341180a3f2e6fadc352fd4d2b7096ce9778db83f1a/numpy-2.3.5-cp313-cp313-macosx_10_13_x86_64.whl", hash = "sha256:d0f23b44f57077c1ede8c5f26b30f706498b4862d3ff0a7298b8411dd2f043ff", size = 16728251, upload-time = "2025-11-16T22:50:19.013Z" },
{ url = "https://files.pythonhosted.org/packages/79/fb/f505c95ceddd7027347b067689db71ca80bd5ecc926f913f1a23e65cf09b/numpy-2.3.5-cp313-cp313-macosx_11_0_arm64.whl", hash = "sha256:aa5bc7c5d59d831d9773d1170acac7893ce3a5e130540605770ade83280e7188", size = 12254652, upload-time = "2025-11-16T22:50:21.487Z" },
{ url = "https://files.pythonhosted.org/packages/78/da/8c7738060ca9c31b30e9301ee0cf6c5ffdbf889d9593285a1cead337f9a5/numpy-2.3.5-cp313-cp313-macosx_14_0_arm64.whl", hash = "sha256:ccc933afd4d20aad3c00bcef049cb40049f7f196e0397f1109dba6fed63267b0", size = 5083172, upload-time = "2025-11-16T22:50:24.562Z" },
{ url = "https://files.pythonhosted.org/packages/a4/b4/ee5bb2537fb9430fd2ef30a616c3672b991a4129bb1c7dcc42aa0abbe5d7/numpy-2.3.5-cp313-cp313-macosx_14_0_x86_64.whl", hash = "sha256:afaffc4393205524af9dfa400fa250143a6c3bc646c08c9f5e25a9f4b4d6a903", size = 6622990, upload-time = "2025-11-16T22:50:26.47Z" },
{ url = "https://files.pythonhosted.org/packages/95/03/dc0723a013c7d7c19de5ef29e932c3081df1c14ba582b8b86b5de9db7f0f/numpy-2.3.5-cp313-cp313-manylinux_2_27_aarch64.manylinux_2_28_aarch64.whl", hash = "sha256:9c75442b2209b8470d6d5d8b1c25714270686f14c749028d2199c54e29f20b4d", size = 14248902, upload-time = "2025-11-16T22:50:28.861Z" },
{ url = "https://files.pythonhosted.org/packages/f5/10/ca162f45a102738958dcec8023062dad0cbc17d1ab99d68c4e4a6c45fb2b/numpy-2.3.5-cp313-cp313-manylinux_2_27_x86_64.manylinux_2_28_x86_64.whl", hash = "sha256:11e06aa0af8c0f05104d56450d6093ee639e15f24ecf62d417329d06e522e017", size = 16597430, upload-time = "2025-11-16T22:50:31.56Z" },
{ url = "https://files.pythonhosted.org/packages/2a/51/c1e29be863588db58175175f057286900b4b3327a1351e706d5e0f8dd679/numpy-2.3.5-cp313-cp313-musllinux_1_2_aarch64.whl", hash = "sha256:ed89927b86296067b4f81f108a2271d8926467a8868e554eaf370fc27fa3ccaf", size = 16024551, upload-time = "2025-11-16T22:50:34.242Z" },
{ url = "https://files.pythonhosted.org/packages/83/68/8236589d4dbb87253d28259d04d9b814ec0ecce7cb1c7fed29729f4c3a78/numpy-2.3.5-cp313-cp313-musllinux_1_2_x86_64.whl", hash = "sha256:51c55fe3451421f3a6ef9a9c1439e82101c57a2c9eab9feb196a62b1a10b58ce", size = 18533275, upload-time = "2025-11-16T22:50:37.651Z" },
{ url = "https://files.pythonhosted.org/packages/40/56/2932d75b6f13465239e3b7b7e511be27f1b8161ca2510854f0b6e521c395/numpy-2.3.5-cp313-cp313-win32.whl", hash = "sha256:1978155dd49972084bd6ef388d66ab70f0c323ddee6f693d539376498720fb7e", size = 6277637, upload-time = "2025-11-16T22:50:40.11Z" },
{ url = "https://files.pythonhosted.org/packages/0c/88/e2eaa6cffb115b85ed7c7c87775cb8bcf0816816bc98ca8dbfa2ee33fe6e/numpy-2.3.5-cp313-cp313-win_amd64.whl", hash = "sha256:00dc4e846108a382c5869e77c6ed514394bdeb3403461d25a829711041217d5b", size = 12779090, upload-time = "2025-11-16T22:50:42.503Z" },
{ url = "https://files.pythonhosted.org/packages/8f/88/3f41e13a44ebd4034ee17baa384acac29ba6a4fcc2aca95f6f08ca0447d1/numpy-2.3.5-cp313-cp313-win_arm64.whl", hash = "sha256:0472f11f6ec23a74a906a00b48a4dcf3849209696dff7c189714511268d103ae", size = 10194710, upload-time = "2025-11-16T22:50:44.971Z" },
{ url = "https://files.pythonhosted.org/packages/13/cb/71744144e13389d577f867f745b7df2d8489463654a918eea2eeb166dfc9/numpy-2.3.5-cp313-cp313t-macosx_10_13_x86_64.whl", hash = "sha256:414802f3b97f3c1eef41e530aaba3b3c1620649871d8cb38c6eaff034c2e16bd", size = 16827292, upload-time = "2025-11-16T22:50:47.715Z" },
{ url = "https://files.pythonhosted.org/packages/71/80/ba9dc6f2a4398e7f42b708a7fdc841bb638d353be255655498edbf9a15a8/numpy-2.3.5-cp313-cp313t-macosx_11_0_arm64.whl", hash = "sha256:5ee6609ac3604fa7780e30a03e5e241a7956f8e2fcfe547d51e3afa5247ac47f", size = 12378897, upload-time = "2025-11-16T22:50:51.327Z" },
{ url = "https://files.pythonhosted.org/packages/2e/6d/db2151b9f64264bcceccd51741aa39b50150de9b602d98ecfe7e0c4bff39/numpy-2.3.5-cp313-cp313t-macosx_14_0_arm64.whl", hash = "sha256:86d835afea1eaa143012a2d7a3f45a3adce2d7adc8b4961f0b362214d800846a", size = 5207391, upload-time = "2025-11-16T22:50:54.542Z" },
{ url = "https://files.pythonhosted.org/packages/80/ae/429bacace5ccad48a14c4ae5332f6aa8ab9f69524193511d60ccdfdc65fa/numpy-2.3.5-cp313-cp313t-macosx_14_0_x86_64.whl", hash = "sha256:30bc11310e8153ca664b14c5f1b73e94bd0503681fcf136a163de856f3a50139", size = 6721275, upload-time = "2025-11-16T22:50:56.794Z" },
{ url = "https://files.pythonhosted.org/packages/74/5b/1919abf32d8722646a38cd527bc3771eb229a32724ee6ba340ead9b92249/numpy-2.3.5-cp313-cp313t-manylinux_2_27_aarch64.manylinux_2_28_aarch64.whl", hash = "sha256:1062fde1dcf469571705945b0f221b73928f34a20c904ffb45db101907c3454e", size = 14306855, upload-time = "2025-11-16T22:50:59.208Z" },
{ url = "https://files.pythonhosted.org/packages/a5/87/6831980559434973bebc30cd9c1f21e541a0f2b0c280d43d3afd909b66d0/numpy-2.3.5-cp313-cp313t-manylinux_2_27_x86_64.manylinux_2_28_x86_64.whl", hash = "sha256:ce581db493ea1a96c0556360ede6607496e8bf9b3a8efa66e06477267bc831e9", size = 16657359, upload-time = "2025-11-16T22:51:01.991Z" },
{ url = "https://files.pythonhosted.org/packages/dd/91/c797f544491ee99fd00495f12ebb7802c440c1915811d72ac5b4479a3356/numpy-2.3.5-cp313-cp313t-musllinux_1_2_aarch64.whl", hash = "sha256:cc8920d2ec5fa99875b670bb86ddeb21e295cb07aa331810d9e486e0b969d946", size = 16093374, upload-time = "2025-11-16T22:51:05.291Z" },
{ url = "https://files.pythonhosted.org/packages/74/a6/54da03253afcbe7a72785ec4da9c69fb7a17710141ff9ac5fcb2e32dbe64/numpy-2.3.5-cp313-cp313t-musllinux_1_2_x86_64.whl", hash = "sha256:9ee2197ef8c4f0dfe405d835f3b6a14f5fee7782b5de51ba06fb65fc9b36e9f1", size = 18594587, upload-time = "2025-11-16T22:51:08.585Z" },
{ url = "https://files.pythonhosted.org/packages/80/e9/aff53abbdd41b0ecca94285f325aff42357c6b5abc482a3fcb4994290b18/numpy-2.3.5-cp313-cp313t-win32.whl", hash = "sha256:70b37199913c1bd300ff6e2693316c6f869c7ee16378faf10e4f5e3275b299c3", size = 6405940, upload-time = "2025-11-16T22:51:11.541Z" },
{ url = "https://files.pythonhosted.org/packages/d5/81/50613fec9d4de5480de18d4f8ef59ad7e344d497edbef3cfd80f24f98461/numpy-2.3.5-cp313-cp313t-win_amd64.whl", hash = "sha256:b501b5fa195cc9e24fe102f21ec0a44dffc231d2af79950b451e0d99cea02234", size = 12920341, upload-time = "2025-11-16T22:51:14.312Z" },
{ url = "https://files.pythonhosted.org/packages/bb/ab/08fd63b9a74303947f34f0bd7c5903b9c5532c2d287bead5bdf4c556c486/numpy-2.3.5-cp313-cp313t-win_arm64.whl", hash = "sha256:a80afd79f45f3c4a7d341f13acbe058d1ca8ac017c165d3fa0d3de6bc1a079d7", size = 10262507, upload-time = "2025-11-16T22:51:16.846Z" },
{ url = "https://files.pythonhosted.org/packages/ba/97/1a914559c19e32d6b2e233cf9a6a114e67c856d35b1d6babca571a3e880f/numpy-2.3.5-cp314-cp314-macosx_10_15_x86_64.whl", hash = "sha256:bf06bc2af43fa8d32d30fae16ad965663e966b1a3202ed407b84c989c3221e82", size = 16735706, upload-time = "2025-11-16T22:51:19.558Z" },
{ url = "https://files.pythonhosted.org/packages/57/d4/51233b1c1b13ecd796311216ae417796b88b0616cfd8a33ae4536330748a/numpy-2.3.5-cp314-cp314-macosx_11_0_arm64.whl", hash = "sha256:052e8c42e0c49d2575621c158934920524f6c5da05a1d3b9bab5d8e259e045f0", size = 12264507, upload-time = "2025-11-16T22:51:22.492Z" },
{ url = "https://files.pythonhosted.org/packages/45/98/2fe46c5c2675b8306d0b4a3ec3494273e93e1226a490f766e84298576956/numpy-2.3.5-cp314-cp314-macosx_14_0_arm64.whl", hash = "sha256:1ed1ec893cff7040a02c8aa1c8611b94d395590d553f6b53629a4461dc7f7b63", size = 5093049, upload-time = "2025-11-16T22:51:25.171Z" },
{ url = "https://files.pythonhosted.org/packages/ce/0e/0698378989bb0ac5f1660c81c78ab1fe5476c1a521ca9ee9d0710ce54099/numpy-2.3.5-cp314-cp314-macosx_14_0_x86_64.whl", hash = "sha256:2dcd0808a421a482a080f89859a18beb0b3d1e905b81e617a188bd80422d62e9", size = 6626603, upload-time = "2025-11-16T22:51:27Z" },
{ url = "https://files.pythonhosted.org/packages/5e/a6/9ca0eecc489640615642a6cbc0ca9e10df70df38c4d43f5a928ff18d8827/numpy-2.3.5-cp314-cp314-manylinux_2_27_aarch64.manylinux_2_28_aarch64.whl", hash = "sha256:727fd05b57df37dc0bcf1a27767a3d9a78cbbc92822445f32cc3436ba797337b", size = 14262696, upload-time = "2025-11-16T22:51:29.402Z" },
{ url = "https://files.pythonhosted.org/packages/c8/f6/07ec185b90ec9d7217a00eeeed7383b73d7e709dae2a9a021b051542a708/numpy-2.3.5-cp314-cp314-manylinux_2_27_x86_64.manylinux_2_28_x86_64.whl", hash = "sha256:fffe29a1ef00883599d1dc2c51aa2e5d80afe49523c261a74933df395c15c520", size = 16597350, upload-time = "2025-11-16T22:51:32.167Z" },
{ url = "https://files.pythonhosted.org/packages/75/37/164071d1dde6a1a84c9b8e5b414fa127981bad47adf3a6b7e23917e52190/numpy-2.3.5-cp314-cp314-musllinux_1_2_aarch64.whl", hash = "sha256:8f7f0e05112916223d3f438f293abf0727e1181b5983f413dfa2fefc4098245c", size = 16040190, upload-time = "2025-11-16T22:51:35.403Z" },
{ url = "https://files.pythonhosted.org/packages/08/3c/f18b82a406b04859eb026d204e4e1773eb41c5be58410f41ffa511d114ae/numpy-2.3.5-cp314-cp314-musllinux_1_2_x86_64.whl", hash = "sha256:2e2eb32ddb9ccb817d620ac1d8dae7c3f641c1e5f55f531a33e8ab97960a75b8", size = 18536749, upload-time = "2025-11-16T22:51:39.698Z" },
{ url = "https://files.pythonhosted.org/packages/40/79/f82f572bf44cf0023a2fe8588768e23e1592585020d638999f15158609e1/numpy-2.3.5-cp314-cp314-win32.whl", hash = "sha256:66f85ce62c70b843bab1fb14a05d5737741e74e28c7b8b5a064de10142fad248", size = 6335432, upload-time = "2025-11-16T22:51:42.476Z" },
{ url = "https://files.pythonhosted.org/packages/a3/2e/235b4d96619931192c91660805e5e49242389742a7a82c27665021db690c/numpy-2.3.5-cp314-cp314-win_amd64.whl", hash = "sha256:e6a0bc88393d65807d751a614207b7129a310ca4fe76a74e5c7da5fa5671417e", size = 12919388, upload-time = "2025-11-16T22:51:45.275Z" },
{ url = "https://files.pythonhosted.org/packages/07/2b/29fd75ce45d22a39c61aad74f3d718e7ab67ccf839ca8b60866054eb15f8/numpy-2.3.5-cp314-cp314-win_arm64.whl", hash = "sha256:aeffcab3d4b43712bb7a60b65f6044d444e75e563ff6180af8f98dd4b905dfd2", size = 10476651, upload-time = "2025-11-16T22:51:47.749Z" },
{ url = "https://files.pythonhosted.org/packages/17/e1/f6a721234ebd4d87084cfa68d081bcba2f5cfe1974f7de4e0e8b9b2a2ba1/numpy-2.3.5-cp314-cp314t-macosx_10_15_x86_64.whl", hash = "sha256:17531366a2e3a9e30762c000f2c43a9aaa05728712e25c11ce1dbe700c53ad41", size = 16834503, upload-time = "2025-11-16T22:51:50.443Z" },
{ url = "https://files.pythonhosted.org/packages/5c/1c/baf7ffdc3af9c356e1c135e57ab7cf8d247931b9554f55c467efe2c69eff/numpy-2.3.5-cp314-cp314t-macosx_11_0_arm64.whl", hash = "sha256:d21644de1b609825ede2f48be98dfde4656aefc713654eeee280e37cadc4e0ad", size = 12381612, upload-time = "2025-11-16T22:51:53.609Z" },
{ url = "https://files.pythonhosted.org/packages/74/91/f7f0295151407ddc9ba34e699013c32c3c91944f9b35fcf9281163dc1468/numpy-2.3.5-cp314-cp314t-macosx_14_0_arm64.whl", hash = "sha256:c804e3a5aba5460c73955c955bdbd5c08c354954e9270a2c1565f62e866bdc39", size = 5210042, upload-time = "2025-11-16T22:51:56.213Z" },
{ url = "https://files.pythonhosted.org/packages/2e/3b/78aebf345104ec50dd50a4d06ddeb46a9ff5261c33bcc58b1c4f12f85ec2/numpy-2.3.5-cp314-cp314t-macosx_14_0_x86_64.whl", hash = "sha256:cc0a57f895b96ec78969c34f682c602bf8da1a0270b09bc65673df2e7638ec20", size = 6724502, upload-time = "2025-11-16T22:51:58.584Z" },
{ url = "https://files.pythonhosted.org/packages/02/c6/7c34b528740512e57ef1b7c8337ab0b4f0bddf34c723b8996c675bc2bc91/numpy-2.3.5-cp314-cp314t-manylinux_2_27_aarch64.manylinux_2_28_aarch64.whl", hash = "sha256:900218e456384ea676e24ea6a0417f030a3b07306d29d7ad843957b40a9d8d52", size = 14308962, upload-time = "2025-11-16T22:52:01.698Z" },
{ url = "https://files.pythonhosted.org/packages/80/35/09d433c5262bc32d725bafc619e095b6a6651caf94027a03da624146f655/numpy-2.3.5-cp314-cp314t-manylinux_2_27_x86_64.manylinux_2_28_x86_64.whl", hash = "sha256:09a1bea522b25109bf8e6f3027bd810f7c1085c64a0c7ce050c1676ad0ba010b", size = 16655054, upload-time = "2025-11-16T22:52:04.267Z" },
{ url = "https://files.pythonhosted.org/packages/7a/ab/6a7b259703c09a88804fa2430b43d6457b692378f6b74b356155283566ac/numpy-2.3.5-cp314-cp314t-musllinux_1_2_aarch64.whl", hash = "sha256:04822c00b5fd0323c8166d66c701dc31b7fbd252c100acd708c48f763968d6a3", size = 16091613, upload-time = "2025-11-16T22:52:08.651Z" },
{ url = "https://files.pythonhosted.org/packages/c2/88/330da2071e8771e60d1038166ff9d73f29da37b01ec3eb43cb1427464e10/numpy-2.3.5-cp314-cp314t-musllinux_1_2_x86_64.whl", hash = "sha256:d6889ec4ec662a1a37eb4b4fb26b6100841804dac55bd9df579e326cdc146227", size = 18591147, upload-time = "2025-11-16T22:52:11.453Z" },
{ url = "https://files.pythonhosted.org/packages/51/41/851c4b4082402d9ea860c3626db5d5df47164a712cb23b54be028b184c1c/numpy-2.3.5-cp314-cp314t-win32.whl", hash = "sha256:93eebbcf1aafdf7e2ddd44c2923e2672e1010bddc014138b229e49725b4d6be5", size = 6479806, upload-time = "2025-11-16T22:52:14.641Z" },
{ url = "https://files.pythonhosted.org/packages/90/30/d48bde1dfd93332fa557cff1972fbc039e055a52021fbef4c2c4b1eefd17/numpy-2.3.5-cp314-cp314t-win_amd64.whl", hash = "sha256:c8a9958e88b65c3b27e22ca2a076311636850b612d6bbfb76e8d156aacde2aaf", size = 13105760, upload-time = "2025-11-16T22:52:17.975Z" },
{ url = "https://files.pythonhosted.org/packages/2d/fd/4b5eb0b3e888d86aee4d198c23acec7d214baaf17ea93c1adec94c9518b9/numpy-2.3.5-cp314-cp314t-win_arm64.whl", hash = "sha256:6203fdf9f3dc5bdaed7319ad8698e685c7a3be10819f41d32a0723e611733b42", size = 10545459, upload-time = "2025-11-16T22:52:20.55Z" },
]
[[package]]
name = "projecteuler"
version = "0.1.0"
source = { virtual = "." }
dependencies = [
{ name = "numpy" },
]
[package.metadata]
requires-dist = [{ name = "numpy", specifier = ">=2.3.5" }]