Which Numbers are the Sum of Two Squares?

最新推荐文章于 2025-07-22 11:31:02 发布
最新推荐文章于 2025-07-22 11:31:02 发布 · 961 阅读 · 0

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The main goal of today's lecture is to prove the following theorem.

Theorem 1.1   A number  $ n$ is a sum of two squares if and only if all prime factors of  $ n$ of the form $ 4m+3$ have even exponent in the prime factorization of  $ n$.

Before tackling a proof, we consider a few examples.

Example 1.2  

  • $ 5 = 1^2 + 2^2$.
  • $ 7$ is not a sum of two squares.
  • $ 2001$ is divisible by $ 3$ because $ 2+1$ is, but not by $ 9$ since $ 2+1$ is not, so $ 2001$ is not a sum of two squares.
  • $ 2\cdot 3^4\cdot 5\cdot 7^2\cdot 13$ is a sum of two squares.
  • $ 389$ is a sum of two squares, since $ 389\equiv 1\pmod{4}$ and $ 389$ is prime.
  • $ 21=3\cdot 7$ is not a sum of two squares even though $ 21\equiv 1\pmod{4}$.

In preparation for the proof of Theorem 1.1, we recall a result that emerged when we analyzed how partial convergents of a continued fraction converge.

Lemma 1.3   If $ x\in\mathbb{R}$ and $ n\in\mathbb{N}$, then there is a fraction $ \displaystyle \frac{a}{b}$ in lowest terms such that $ 0<b\leq n$ and

$\displaystyle \left\vert x - \frac{a}{b} \right\vert \leq \frac{1}{b(n+1)}.$

Proof. Let $ [a_0,a_1,\ldots]$ be the continued fraction expansion of  $ x$. As we saw in the proof of Theorem 2.3 in Lecture 18, for each  $ m$

$\displaystyle \left\vert x - \frac{p_m}{q_m}\right\vert < \frac{1}{q_m \cdot q_{m+1}}. $

Since $ q_{m+1}$ is always at least  $ 1$ bigger than $ q_m$ and $ q_0=1$, either there exists an  $ m$ such that $ q_m\leq n < q_{m+1}$, or the continued fraction expansion of  $ x$ is finite and $ n$ is larger than the denominator of the rational number  $ x$. In the first case,

$\displaystyle \left\vert x - \frac{p_m}{q_m}\right\vert < \frac{1}{q_m \cdot q_{m+1}} \leq \frac{1}{q_m \cdot (n+1)},$

so $ \displaystyle \frac{a}{b} = \frac{p_m}{q_m}$ satisfies the conclusion of the lemma. In the second case, just let $ \displaystyle \frac{a}{b} = x$.

$ \qedsymbol$

Definition 1.4   A representation $ n=x^2 + y^2$ is primitive if $ \gcd(x,y)=1$.

Lemma 1.5   If  $ n$ is divisible by a prime  $ p$ of the form $ 4m+3$, then  $ n$ has no primitive representations.

Proof. If  $ n$ has a primitive representation, $ n=x^2 + y^2$, then

$\displaystyle p \mid x^2 + y^2$    and $\displaystyle \quad \gcd(x,y)=1, $

so $ p\nmid x$ and $ p\nmid y$. Thus $ x^2 + y^2 \equiv 0\pmod{p}$ so, since $ \mathbb{Z}/p\mathbb{Z}$ is a field we can divide by $ y^2$ and see that

$\displaystyle (x/y)^2 \equiv -1\pmod{p}. $

Thus the quadratic residue symbol $ \left(\frac{-1}{p}\right)$ equals $ +1$. However,

$\displaystyle \left(\frac{-1}{p}\right) = (-1)^{\frac{p-1}{2}} = (-1)^\frac{4m+3-1}{2} = (-1)^{2m+1} = -1. $

$ \qedsymbol$

Proof. [Proof of Theorem  1.1] $ \left(\Longrightarrow\right)$ Suppose that  $ p$ is of the form $ 4m+3$, that $ p^r\mid\mid n$ (exactly divides) with  $ r$ odd, and that $ n=x^2 + y^2$. Letting $ d=\gcd(x,y)$, we have

$\displaystyle x = dx', \quad y = dy', \quad n = d^2 n' $

with $ \gcd(x',y')=1$ and

$\displaystyle (x')^2 + (y')^2 = n'. $

Because $ r$ is odd, $ p\mid n'$, so Lemma 1.5 implies that $ \gcd(x',y')>1$, a contradiction.

$ \left(\Longleftarrow\right)$ Write $ n=n_1^2 n_2$ where $ n_2$ has no prime factors of the form $ 4m+3$. It suffices to show that $ n_2$ is a sum of two squares. Also note that

$\displaystyle (x_1^2 + y_1^2)(x_2^2+y_2^2) = (x_1x_2+y_1y_2)^2 + (x_1y_2-x_2y_1)^2, $

so a product of two numbers that are sums of two squares is also a sum of two squares. 1Also, the prime  $ 2$ is a sum of two squares. It thus suffices to show that if  $ p$ is a prime of the form $ 4m+1$, then  $ p$ is a sum of two squares.

Since

$\displaystyle (-1)^{\frac{p-1}{2}} = (-1)^{\frac{4m+1-1}{2}} = +1, $

$ -1$ is a square modulo  $ p$; i.e., there exists  $ r$ such that $ r^2\equiv -1\pmod{p}$. Taking $ n=\lfloor \sqrt{p}\rfloor$ in Lemma  1.3 we see that there are integers $ a, b$ such that $ 0<b<\sqrt{p}$ and

$\displaystyle \left\vert -\frac{r}{p} - \frac{a}{b}\right\vert \leq\frac{1}{b(n+1)} < \frac{1}{b\sqrt{p}}. $

If we write

$\displaystyle c = rb + pa $

then

$\displaystyle \vert c\vert < \frac{pb}{b\sqrt{p}} = \frac{p}{\sqrt{p}} = \sqrt{p} $

and

$\displaystyle 0 < b^2 + c^2 < 2p. $

But $ c \equiv rb\pmod{p}$, so

$\displaystyle b^2 + c^2 \equiv b^2 + r^2 b^2 \equiv b^2(1+r^2) \equiv 0\pmod{p}. $

Thus $ b^2 + c^2 = p$.

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import numpy as np import pandas as pd import matplotlib.pyplot as plt import seaborn as sns import plotly.graph_objects as go import plotly.express as px from scipy.stats import gaussian_kde import matplotlib.font_manager as fm from matplotlib.colors import LinearSegmentedColormap # 设置中文字体支持 plt.rcParams['font.sans-serif'] = ['SimHei', 'Arial Unicode MS', 'Microsoft YaHei', 'WenQuanYi Micro Hei'] plt.rcParams['axes.unicode_minus'] = False # 设置随机种子，确保结果可复现 np.random.seed(42) # 根据92.02%的高准确率生成模拟数据 # 总样本数 n_samples = 1000 # 正确样本比例 (92.02%) correct_ratio = 0.9202 n_correct = int(n_samples * correct_ratio) n_incorrect = n_samples - n_correct # 生成预测不确定性数据 # 正确样本的不确定性较低，分布更集中 correct_uncertainty = np.random.normal(0.3, 0.15, n_correct) # 错误样本的不确定性较高，分布更分散 incorrect_uncertainty = np.random.normal(1.2, 0.4, n_incorrect) # 合并数据 uncertainty = np.concatenate([correct_uncertainty, incorrect_uncertainty]) correctness = np.concatenate([np.ones(n_correct), np.zeros(n_incorrect)]) # 添加峰高变异系数作为第三维度特征 cv_correct = np.random.normal(0.2, 0.1, n_correct) # 正确样本峰高变异系数较低 cv_incorrect = np.random.normal(0.6, 0.2, n_incorrect) # 错误样本峰高变异系数较高 cv = np.concatenate([cv_correct, cv_incorrect]) # 创建数据框 df = pd.DataFrame({ 'uncertainty': uncertainty, 'correctness': correctness, 'result': ['正确' if c == 1 else '错误' for c in correctness], 'cv': cv }) # 确保不确定性值为非负数 df['uncertainty'] = df['uncertainty'].clip(lower=0) # 计算整体准确率 overall_accuracy = df['correctness'].mean() print(f"模拟数据准确率: {overall_accuracy:.4f}") # 创建自定义颜色映射 def create_green_cmap(): colors = ["#f0f9e8", "#bae4bc", "#7bccc4", "#2b8cbe"] return LinearSegmentedColormap.from_list("green_cmap", colors) # 保存所有图像的函数 def save_all_figures(): # 方案1：核密度估计（KDE）+ 统计摘要图 plt.figure(figsize=(12, 8)) kde = gaussian_kde(df['uncertainty']) x_range = np.linspace(0, df['uncertainty'].max(), 200) y_kde = kde(x_range) # 计算统计指标 mean_uncert = df['uncertainty'].mean() median_uncert = df['uncertainty'].median() q25, q75 = np.percentile(df['uncertainty'], [25, 75]) std_uncert = df['uncertainty'].std() plt.plot(x_range, y_kde, 'b-', linewidth=2, label='KDE分布') plt.fill_between(x_range, y_kde, color='royalblue', alpha=0.2, label='分布区域') # 标注统计指标 plt.axvline(mean_uncert, color='r', linestyle='--', label=f'均值: {mean_uncert:.2f}') plt.axvline(median_uncert, color='g', linestyle=':', label=f'中位数: {median_uncert:.2f}') plt.axvline(q25, color='purple', linestyle='-.', label=f'25%分位数: {q25:.2f}') plt.axvline(q75, color='orange', linestyle='-.', label=f'75%分位数: {q75:.2f}') plt.title(f'预测不确定性分布 (准确率: {overall_accuracy * 100:.2f}%)', fontsize=16, pad=20) plt.xlabel('预测方差', fontsize=14) plt.ylabel('概率密度', fontsize=14) plt.legend(loc='upper right', fontsize=12) plt.grid(alpha=0.2, linestyle='--') # 添加统计信息框 stats_text = f'统计摘要:\n样本数: {n_samples}\n标准差: {std_uncert:.2f}\n最小值: {df["uncertainty"].min():.2f}\n最大值: {df["uncertainty"].max():.2f}' plt.text(0.95, 0.95, stats_text, transform=plt.gca().transAxes, fontsize=12, verticalalignment='top', horizontalalignment='right', bbox=dict(boxstyle='round', facecolor='white', alpha=0.8)) plt.tight_layout() plt.savefig('1_kde_distribution.png', dpi=300, bbox_inches='tight') plt.close() # 方案2：分组小提琴图 + 抖动散点图 plt.figure(figsize=(12, 8)) sns.set_style("whitegrid") # 创建自定义调色板 palette = {"正确": "#4caf50", "错误": "#f44336"} # 绘制小提琴图 sns.violinplot(x='result', y='uncertainty', data=df, palette=palette, inner='quartile', linewidth=2, saturation=0.8) # 绘制散点图（带透明度） sns.stripplot(x='result', y='uncertainty', data=df, palette=palette, alpha=0.4, size=4, jitter=0.2) # 添加中位数线 medians = df.groupby('result')['uncertainty'].median() for i, category in enumerate(medians.index): plt.hlines(medians[category], i - 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Assign the `x` variable to `hue` and set `legend=False` for the same effect. C:\python\py\3.py:117: FutureWarning: Passing `palette` without assigning `hue` is deprecated and will be removed in v0.14.0. Assign the `x` variable to `hue` and set `legend=False` for the same effect. C:\python\py\3.py:140: UserWarning: Glyph 27491 (\N{CJK UNIFIED IDEOGRAPH-6B63}) missing from font(s) Arial. C:\python\py\3.py:140: UserWarning: Glyph 30830 (\N{CJK UNIFIED IDEOGRAPH-786E}) missing from font(s) Arial. C:\python\py\3.py:140: UserWarning: Glyph 38169 (\N{CJK UNIFIED IDEOGRAPH-9519}) missing from font(s) Arial. C:\python\py\3.py:140: UserWarning: Glyph 35823 (\N{CJK UNIFIED IDEOGRAPH-8BEF}) missing from font(s) Arial. C:\python\py\3.py:140: UserWarning: Glyph 39044 (\N{CJK UNIFIED IDEOGRAPH-9884}) missing from font(s) Arial. C:\python\py\3.py:140: UserWarning: Glyph 27979 (\N{CJK UNIFIED IDEOGRAPH-6D4B}) missing from font(s) Arial. 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C:\python\PyCharm 2024.3.5\plugins\python-ce\helpers\pycharm_matplotlib_backend\backend_interagg.py:124: UserWarning: Glyph 27491 (\N{CJK UNIFIED IDEOGRAPH-6B63}) missing from font(s) Arial. C:\python\PyCharm 2024.3.5\plugins\python-ce\helpers\pycharm_matplotlib_backend\backend_interagg.py:124: UserWarning: Glyph 30830 (\N{CJK UNIFIED IDEOGRAPH-786E}) missing from font(s) Arial. C:\python\PyCharm 2024.3.5\plugins\python-ce\helpers\pycharm_matplotlib_backend\backend_interagg.py:124: UserWarning: Glyph 38169 (\N{CJK UNIFIED IDEOGRAPH-9519}) missing from font(s) Arial. C:\python\PyCharm 2024.3.5\plugins\python-ce\helpers\pycharm_matplotlib_backend\backend_interagg.py:124: UserWarning: Glyph 35823 (\N{CJK UNIFIED IDEOGRAPH-8BEF}) missing from font(s) Arial. C:\python\PyCharm 2024.3.5\plugins\python-ce\helpers\pycharm_matplotlib_backend\backend_interagg.py:124: UserWarning: Glyph 39044 (\N{CJK UNIFIED IDEOGRAPH-9884}) missing from font(s) Arial. C:\python\PyCharm 2024.3.5\plugins\python-ce\helpers\pycharm_matplotlib_backend\backend_interagg.py:124: UserWarning: Glyph 27979 (\N{CJK UNIFIED IDEOGRAPH-6D4B}) missing from font(s) Arial. C:\python\PyCharm 2024.3.5\plugins\python-ce\helpers\pycharm_matplotlib_backend\backend_interagg.py:124: UserWarning: Glyph 32467 (\N{CJK UNIFIED IDEOGRAPH-7ED3}) missing from font(s) Arial. C:\python\PyCharm 2024.3.5\plugins\python-ce\helpers\pycharm_matplotlib_backend\backend_interagg.py:124: UserWarning: Glyph 26524 (\N{CJK UNIFIED IDEOGRAPH-679C}) missing from font(s) Arial. C:\python\PyCharm 2024.3.5\plugins\python-ce\helpers\pycharm_matplotlib_backend\backend_interagg.py:124: UserWarning: Glyph 26041 (\N{CJK UNIFIED IDEOGRAPH-65B9}) missing from font(s) Arial. C:\python\PyCharm 2024.3.5\plugins\python-ce\helpers\pycharm_matplotlib_backend\backend_interagg.py:124: UserWarning: Glyph 24046 (\N{CJK UNIFIED IDEOGRAPH-5DEE}) missing from font(s) Arial. C:\python\PyCharm 2024.3.5\plugins\python-ce\helpers\pycharm_matplotlib_backend\backend_interagg.py:124: UserWarning: Glyph 19981 (\N{CJK UNIFIED IDEOGRAPH-4E0D}) missing from font(s) Arial. C:\python\PyCharm 2024.3.5\plugins\python-ce\helpers\pycharm_matplotlib_backend\backend_interagg.py:124: UserWarning: Glyph 23450 (\N{CJK UNIFIED IDEOGRAPH-5B9A}) missing from font(s) Arial. C:\python\PyCharm 2024.3.5\plugins\python-ce\helpers\pycharm_matplotlib_backend\backend_interagg.py:124: UserWarning: Glyph 24615 (\N{CJK UNIFIED IDEOGRAPH-6027}) missing from font(s) Arial. C:\python\PyCharm 2024.3.5\plugins\python-ce\helpers\pycharm_matplotlib_backend\backend_interagg.py:124: UserWarning: Glyph 19982 (\N{CJK UNIFIED IDEOGRAPH-4E0E}) missing from font(s) Arial. C:\python\PyCharm 2024.3.5\plugins\python-ce\helpers\pycharm_matplotlib_backend\backend_interagg.py:124: UserWarning: Glyph 20998 (\N{CJK UNIFIED IDEOGRAPH-5206}) missing from font(s) Arial. C:\python\PyCharm 2024.3.5\plugins\python-ce\helpers\pycharm_matplotlib_backend\backend_interagg.py:124: UserWarning: Glyph 31867 (\N{CJK UNIFIED IDEOGRAPH-7C7B}) missing from font(s) Arial. C:\python\PyCharm 2024.3.5\plugins\python-ce\helpers\pycharm_matplotlib_backend\backend_interagg.py:124: UserWarning: Glyph 20934 (\N{CJK UNIFIED IDEOGRAPH-51C6}) missing from font(s) Arial. C:\python\PyCharm 2024.3.5\plugins\python-ce\helpers\pycharm_matplotlib_backend\backend_interagg.py:124: UserWarning: Glyph 29575 (\N{CJK UNIFIED IDEOGRAPH-7387}) missing from font(s) Arial. C:\python\PyCharm 2024.3.5\plugins\python-ce\helpers\pycharm_matplotlib_backend\backend_interagg.py:124: UserWarning: Glyph 20010 (\N{CJK UNIFIED IDEOGRAPH-4E2A}) missing from font(s) Arial. C:\python\PyCharm 2024.3.5\plugins\python-ce\helpers\pycharm_matplotlib_backend\backend_interagg.py:124: UserWarning: Glyph 26679 (\N{CJK UNIFIED IDEOGRAPH-6837}) missing from font(s) Arial. 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C:\python\PyCharm 2024.3.5\plugins\python-ce\helpers\pycharm_matplotlib_backend\backend_interagg.py:124: UserWarning: Glyph 26041 (\N{CJK UNIFIED IDEOGRAPH-65B9}) missing from font(s) Arial. C:\python\PyCharm 2024.3.5\plugins\python-ce\helpers\pycharm_matplotlib_backend\backend_interagg.py:124: UserWarning: Glyph 24046 (\N{CJK UNIFIED IDEOGRAPH-5DEE}) missing from font(s) Arial. C:\python\PyCharm 2024.3.5\plugins\python-ce\helpers\pycharm_matplotlib_backend\backend_interagg.py:124: UserWarning: Glyph 21306 (\N{CJK UNIFIED IDEOGRAPH-533A}) missing from font(s) Arial. C:\python\PyCharm 2024.3.5\plugins\python-ce\helpers\pycharm_matplotlib_backend\backend_interagg.py:124: UserWarning: Glyph 38388 (\N{CJK UNIFIED IDEOGRAPH-95F4}) missing from font(s) Arial. C:\python\PyCharm 2024.3.5\plugins\python-ce\helpers\pycharm_matplotlib_backend\backend_interagg.py:124: UserWarning: Glyph 19981 (\N{CJK UNIFIED IDEOGRAPH-4E0D}) missing from font(s) Arial. C:\python\PyCharm 2024.3.5\plugins\python-ce\helpers\pycharm_matplotlib_backend\backend_interagg.py:124: UserWarning: Glyph 21516 (\N{CJK UNIFIED IDEOGRAPH-540C}) missing from font(s) Arial. C:\python\PyCharm 2024.3.5\plugins\python-ce\helpers\pycharm_matplotlib_backend\backend_interagg.py:124: UserWarning: Glyph 30340 (\N{CJK UNIFIED IDEOGRAPH-7684}) missing from font(s) Arial. C:\python\PyCharm 2024.3.5\plugins\python-ce\helpers\pycharm_matplotlib_backend\backend_interagg.py:124: UserWarning: Glyph 39044 (\N{CJK UNIFIED IDEOGRAPH-9884}) missing from font(s) Arial. C:\python\PyCharm 2024.3.5\plugins\python-ce\helpers\pycharm_matplotlib_backend\backend_interagg.py:124: UserWarning: Glyph 27979 (\N{CJK UNIFIED IDEOGRAPH-6D4B}) missing from font(s) Arial. C:\python\PyCharm 2024.3.5\plugins\python-ce\helpers\pycharm_matplotlib_backend\backend_interagg.py:124: UserWarning: Glyph 27491 (\N{CJK UNIFIED IDEOGRAPH-6B63}) missing from font(s) Arial. C:\python\PyCharm 2024.3.5\plugins\python-ce\helpers\pycharm_matplotlib_backend\backend_interagg.py:124: UserWarning: Glyph 30830 (\N{CJK UNIFIED IDEOGRAPH-786E}) missing from font(s) Arial. C:\python\PyCharm 2024.3.5\plugins\python-ce\helpers\pycharm_matplotlib_backend\backend_interagg.py:124: UserWarning: Glyph 29575 (\N{CJK UNIFIED IDEOGRAPH-7387}) missing from font(s) Arial. C:\python\PyCharm 2024.3.5\plugins\python-ce\helpers\pycharm_matplotlib_backend\backend_interagg.py:124: UserWarning: Glyph 24635 (\N{CJK UNIFIED IDEOGRAPH-603B}) missing from font(s) Arial. C:\python\PyCharm 2024.3.5\plugins\python-ce\helpers\pycharm_matplotlib_backend\backend_interagg.py:124: UserWarning: Glyph 20307 (\N{CJK UNIFIED IDEOGRAPH-4F53}) missing from font(s) Arial. C:\python\PyCharm 2024.3.5\plugins\python-ce\helpers\pycharm_matplotlib_backend\backend_interagg.py:124: UserWarning: Glyph 20934 (\N{CJK UNIFIED IDEOGRAPH-51C6}) missing from font(s) Arial. C:\python\PyCharm 2024.3.5\plugins\python-ce\helpers\pycharm_matplotlib_backend\backend_interagg.py:124: UserWarning: Glyph 26679 (\N{CJK UNIFIED IDEOGRAPH-6837}) missing from font(s) Arial. C:\python\PyCharm 2024.3.5\plugins\python-ce\helpers\pycharm_matplotlib_backend\backend_interagg.py:124: UserWarning: Glyph 26412 (\N{CJK UNIFIED IDEOGRAPH-672C}) missing from font(s) Arial. Traceback (most recent call last): File "C:\python\py\3.py", line 339, in <module> save_all_figures() ~~~~~~~~~~~~~~~~^^ File "C:\python\py\3.py", line 218, in save_all_figures fig.update_layout( ~~~~~~~~~~~~~~~~~^ title=dict( ^^^^^^^^^^^ ...<29 lines>... hovermode="x unified" ^^^^^^^^^^^^^^^^^^^^^ ) ^ File "C:\python\py\.venv\Lib\site-packages\plotly\graph_objs\_figure.py", line 218, in update_layout return super().update_layout(dict1, overwrite, **kwargs) ~~~~~~~~~~~~~~~~~~~~~^^^^^^^^^^^^^^^^^^^^^^^^^^^^ File "C:\python\py\.venv\Lib\site-packages\plotly\basedatatypes.py", line 1415, in update_layout self.layout.update(dict1, overwrite=overwrite, **kwargs) ~~~~~~~~~~~~~~~~~~^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ File "C:\python\py\.venv\Lib\site-packages\plotly\basedatatypes.py", line 5195, in update BaseFigure._perform_update(self, kwargs, overwrite=overwrite) ~~~~~~~~~~~~~~~~~~~~~~~~~~^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ File "C:\python\py\.venv\Lib\site-packages\plotly\basedatatypes.py", line 3971, in _perform_update BaseFigure._perform_update(plotly_obj[key], val) ~~~~~~~~~~~~~~~~~~~~~~~~~~^^^^^^^^^^^^^^^^^^^^^^ File "C:\python\py\.venv\Lib\site-packages\plotly\basedatatypes.py", line 3949, in _perform_update raise err ValueError: Invalid property specified for object of type plotly.graph_objs.layout.YAxis: 'titlefont' Did you mean "tickfont"? Valid properties: anchor If set to an opposite-letter axis id (e.g. `x2`, `y`), this axis is bound to the corresponding opposite-letter axis. If set to "free", this axis' position is determined by `position`. automargin Determines whether long tick labels automatically grow the figure margins. autorange Determines whether or not the range of this axis is computed in relation to the input data. See `rangemode` for more info. If `range` is provided and it has a value for both the lower and upper bound, `autorange` is set to False. Using "min" applies autorange only to set the minimum. Using "max" applies autorange only to set the maximum. Using *min reversed* applies autorange only to set the minimum on a reversed axis. Using *max reversed* applies autorange only to set the maximum on a reversed axis. Using "reversed" applies autorange on both ends and reverses the axis direction. autorangeoptions :class:`plotly.graph_objects.layout.yaxis.Autorangeopti ons` instance or dict with compatible properties autoshift Automatically reposition the axis to avoid overlap with other axes with the same `overlaying` value. This repositioning will account for any `shift` amount applied to other axes on the same side with `autoshift` is set to true. Only has an effect if `anchor` is set to "free". autotickangles When `tickangle` is set to "auto", it will be set to the first angle in this array that is large enough to prevent label overlap. autotypenumbers Using "strict" a numeric string in trace data is not converted to a number. Using *convert types* a numeric string in trace data may be treated as a number during automatic axis `type` detection. Defaults to layout.autotypenumbers. calendar Sets the calendar system to use for `range` and `tick0` if this is a date axis. This does not set the calendar for interpreting data on this axis, that's specified in the trace or via the global `layout.calendar` categoryarray Sets the order in which categories on this axis appear. Only has an effect if `categoryorder` is set to "array". Used with `categoryorder`. categoryarraysrc Sets the source reference on Chart Studio Cloud for `categoryarray`. categoryorder Specifies the ordering logic for the case of categorical variables. By default, plotly uses "trace", which specifies the order that is present in the data supplied. Set `categoryorder` to *category ascending* or *category descending* if order should be determined by the alphanumerical order of the category names. Set `categoryorder` to "array" to derive the ordering from the attribute `categoryarray`. If a category is not found in the `categoryarray` array, the sorting behavior for that attribute will be identical to the "trace" mode. The unspecified categories will follow the categories in `categoryarray`. Set `categoryorder` to *total ascending* or *total descending* if order should be determined by the numerical order of the values. Similarly, the order can be determined by the min, max, sum, mean, geometric mean or median of all the values. color Sets default for all colors associated with this axis all at once: line, font, tick, and grid colors. Grid color is lightened by blending this with the plot background Individual pieces can override this. constrain If this axis needs to be compressed (either due to its own `scaleanchor` and `scaleratio` or those of the other axis), determines how that happens: by increasing the "range", or by decreasing the "domain". Default is "domain" for axes containing image traces, "range" otherwise. constraintoward If this axis needs to be compressed (either due to its own `scaleanchor` and `scaleratio` or those of the other axis), determines which direction we push the originally specified plot area. Options are "left", "center" (default), and "right" for x axes, and "top", "middle" (default), and "bottom" for y axes. dividercolor Sets the color of the dividers Only has an effect on "multicategory" axes. dividerwidth Sets the width (in px) of the dividers Only has an effect on "multicategory" axes. domain Sets the domain of this axis (in plot fraction). dtick Sets the step in-between ticks on this axis. Use with `tick0`. Must be a positive number, or special strings available to "log" and "date" axes. If the axis `type` is "log", then ticks are set every 10^(n*dtick) where n is the tick number. For example, to set a tick mark at 1, 10, 100, 1000, ... set dtick to 1. To set tick marks at 1, 100, 10000, ... set dtick to 2. To set tick marks at 1, 5, 25, 125, 625, 3125, ... set dtick to log_10(5), or 0.69897000433. "log" has several special values; "L<f>", where `f` is a positive number, gives ticks linearly spaced in value (but not position). For example `tick0` = 0.1, `dtick` = "L0.5" will put ticks at 0.1, 0.6, 1.1, 1.6 etc. To show powers of 10 plus small digits between, use "D1" (all digits) or "D2" (only 2 and 5). `tick0` is ignored for "D1" and "D2". If the axis `type` is "date", then you must convert the time to milliseconds. For example, to set the interval between ticks to one day, set `dtick` to 86400000.0. "date" also has special values "M<n>" gives ticks spaced by a number of months. `n` must be a positive integer. To set ticks on the 15th of every third month, set `tick0` to "2000-01-15" and `dtick` to "M3". To set ticks every 4 years, set `dtick` to "M48" exponentformat Determines a formatting rule for the tick exponents. For example, consider the number 1,000,000,000. If "none", it appears as 1,000,000,000. If "e", 1e+9. If "E", 1E+9. If "power", 1x10^9 (with 9 in a super script). If "SI", 1G. If "B", 1B. fixedrange Determines whether or not this axis is zoom-able. If true, then zoom is disabled. gridcolor Sets the color of the grid lines. griddash Sets the dash style of lines. Set to a dash type string ("solid", "dot", "dash", "longdash", "dashdot", or "longdashdot") or a dash length list in px (eg "5px,10px,2px,2px"). gridwidth Sets the width (in px) of the grid lines. hoverformat Sets the hover text formatting rule using d3 formatting mini-languages which are very similar to those in Python. For numbers, see: https://github.com/d3/d3-format/tree/v1.4.5#d3-format. And for dates see: https://github.com/d3/d3-time- format/tree/v2.2.3#locale_format. We add two items to d3's date formatter: "%h" for half of the year as a decimal number as well as "%{n}f" for fractional seconds with n digits. For example, *2016-10-13 09:15:23.456* with tickformat "%H~%M~%S.%2f" would display "09~15~23.46" insiderange Could be used to set the desired inside range of this axis (excluding the labels) when `ticklabelposition` of the anchored axis has "inside". Not implemented for axes with `type` "log". This would be ignored when `range` is provided. labelalias Replacement text for specific tick or hover labels. For example using {US: 'USA', CA: 'Canada'} changes US to USA and CA to Canada. The labels we would have shown must match the keys exactly, after adding any tickprefix or ticksuffix. For negative numbers the minus sign symbol used (U+2212) is wider than the regular ascii dash. That means you need to use −1 instead of -1. labelalias can be used with any axis type, and both keys (if needed) and values (if desired) can include html-like tags or MathJax. layer Sets the layer on which this axis is displayed. If *above traces*, this axis is displayed above all the subplot's traces If *below traces*, this axis is displayed below all the subplot's traces, but above the grid lines. Useful when used together with scatter-like traces with `cliponaxis` set to False to show markers and/or text nodes above this axis. linecolor Sets the axis line color. linewidth Sets the width (in px) of the axis line. matches If set to another axis id (e.g. `x2`, `y`), the range of this axis will match the range of the corresponding axis in data-coordinates space. Moreover, matching axes share auto-range values, category lists and histogram auto-bins. Note that setting axes simultaneously in both a `scaleanchor` and a `matches` constraint is currently forbidden. Moreover, note that matching axes must have the same `type`. maxallowed Determines the maximum range of this axis. minallowed Determines the minimum range of this axis. minexponent Hide SI prefix for 10^n if |n| is below this number. This only has an effect when `tickformat` is "SI" or "B". minor :class:`plotly.graph_objects.layout.yaxis.Minor` instance or dict with compatible properties mirror Determines if the axis lines or/and ticks are mirrored to the opposite side of the plotting area. If True, the axis lines are mirrored. If "ticks", the axis lines and ticks are mirrored. If False, mirroring is disable. If "all", axis lines are mirrored on all shared-axes subplots. If "allticks", axis lines and ticks are mirrored on all shared-axes subplots. nticks Specifies the maximum number of ticks for the particular axis. The actual number of ticks will be chosen automatically to be less than or equal to `nticks`. Has an effect only if `tickmode` is set to "auto". overlaying If set a same-letter axis id, this axis is overlaid on top of the corresponding same-letter axis, with traces and axes visible for both axes. If False, this axis does not overlay any same-letter axes. In this case, for axes with overlapping domains only the highest- numbered axis will be visible. position Sets the position of this axis in the plotting space (in normalized coordinates). Only has an effect if `anchor` is set to "free". range Sets the range of this axis. If the axis `type` is "log", then you must take the log of your desired range (e.g. to set the range from 1 to 100, set the range from 0 to 2). If the axis `type` is "date", it should be date strings, like date data, though Date objects and unix milliseconds will be accepted and converted to strings. If the axis `type` is "category", it should be numbers, using the scale where each category is assigned a serial number from zero in the order it appears. Leaving either or both elements `null` impacts the default `autorange`. rangebreaks A tuple of :class:`plotly.graph_objects.layout.yaxis.Rangebreak` instances or dicts with compatible properties rangebreakdefaults When used in a template (as layout.template.layout.yaxis.rangebreakdefaults), sets the default property values to use for elements of layout.yaxis.rangebreaks rangemode If "normal", the range is computed in relation to the extrema of the input data. If "tozero", the range extends to 0, regardless of the input data If "nonnegative", the range is non-negative, regardless of the input data. Applies only to linear axes. scaleanchor If set to another axis id (e.g. `x2`, `y`), the range of this axis changes together with the range of the corresponding axis such that the scale of pixels per unit is in a constant ratio. Both axes are still zoomable, but when you zoom one, the other will zoom the same amount, keeping a fixed midpoint. `constrain` and `constraintoward` determine how we enforce the constraint. You can chain these, ie `yaxis: {scaleanchor: *x*}, xaxis2: {scaleanchor: *y*}` but you can only link axes of the same `type`. The linked axis can have the opposite letter (to constrain the aspect ratio) or the same letter (to match scales across subplots). Loops (`yaxis: {scaleanchor: *x*}, xaxis: {scaleanchor: *y*}` or longer) are redundant and the last constraint encountered will be ignored to avoid possible inconsistent constraints via `scaleratio`. Note that setting axes simultaneously in both a `scaleanchor` and a `matches` constraint is currently forbidden. Setting `false` allows to remove a default constraint (occasionally, you may need to prevent a default `scaleanchor` constraint from being applied, eg. when having an image trace `yaxis: {scaleanchor: "x"}` is set automatically in order for pixels to be rendered as squares, setting `yaxis: {scaleanchor: false}` allows to remove the constraint). scaleratio If this axis is linked to another by `scaleanchor`, this determines the pixel to unit scale ratio. For example, if this value is 10, then every unit on this axis spans 10 times the number of pixels as a unit on the linked axis. Use this for example to create an elevation profile where the vertical scale is exaggerated a fixed amount with respect to the horizontal. separatethousands If "true", even 4-digit integers are separated shift Moves the axis a given number of pixels from where it would have been otherwise. Accepts both positive and negative values, which will shift the axis either right or left, respectively. If `autoshift` is set to true, then this defaults to a padding of -3 if `side` is set to "left". and defaults to +3 if `side` is set to "right". Defaults to 0 if `autoshift` is set to false. Only has an effect if `anchor` is set to "free". showdividers Determines whether or not a dividers are drawn between the category levels of this axis. Only has an effect on "multicategory" axes. showexponent If "all", all exponents are shown besides their significands. If "first", only the exponent of the first tick is shown. If "last", only the exponent of the last tick is shown. If "none", no exponents appear. showgrid Determines whether or not grid lines are drawn. If True, the grid lines are drawn at every tick mark. showline Determines whether or not a line bounding this axis is drawn. showspikes Determines whether or not spikes (aka droplines) are drawn for this axis. Note: This only takes affect when hovermode = closest showticklabels Determines whether or not the tick labels are drawn. showtickprefix If "all", all tick labels are displayed with a prefix. If "first", only the first tick is displayed with a prefix. If "last", only the last tick is displayed with a suffix. If "none", tick prefixes are hidden. showticksuffix Same as `showtickprefix` but for tick suffixes. side Determines whether a x (y) axis is positioned at the "bottom" ("left") or "top" ("right") of the plotting area. spikecolor Sets the spike color. If undefined, will use the series color spikedash Sets the dash style of lines. Set to a dash type string ("solid", "dot", "dash", "longdash", "dashdot", or "longdashdot") or a dash length list in px (eg "5px,10px,2px,2px"). spikemode Determines the drawing mode for the spike line If "toaxis", the line is drawn from the data point to the axis the series is plotted on. If "across", the line is drawn across the entire plot area, and supercedes "toaxis". If "marker", then a marker dot is drawn on the axis the series is plotted on spikesnap Determines whether spikelines are stuck to the cursor or to the closest datapoints. spikethickness Sets the width (in px) of the zero line. tick0 Sets the placement of the first tick on this axis. Use with `dtick`. If the axis `type` is "log", then you must take the log of your starting tick (e.g. to set the starting tick to 100, set the `tick0` to 2) except when `dtick`=*L<f>* (see `dtick` for more info). If the axis `type` is "date", it should be a date string, like date data. If the axis `type` is "category", it should be a number, using the scale where each category is assigned a serial number from zero in the order it appears. tickangle Sets the angle of the tick labels with respect to the horizontal. For example, a `tickangle` of -90 draws the tick labels vertically. tickcolor Sets the tick color. tickfont Sets the tick font. tickformat Sets the tick label formatting rule using d3 formatting mini-languages which are very similar to those in Python. For numbers, see: https://github.com/d3/d3-format/tree/v1.4.5#d3-format. And for dates see: https://github.com/d3/d3-time- format/tree/v2.2.3#locale_format. We add two items to d3's date formatter: "%h" for half of the year as a decimal number as well as "%{n}f" for fractional seconds with n digits. For example, *2016-10-13 09:15:23.456* with tickformat "%H~%M~%S.%2f" would display "09~15~23.46" tickformatstops A tuple of :class:`plotly.graph_objects.layout.yaxis.Ti ckformatstop` instances or dicts with compatible properties tickformatstopdefaults When used in a template (as layout.template.layout.yaxis.tickformatstopdefaults), sets the default property values to use for elements of layout.yaxis.tickformatstops ticklabelindex Only for axes with `type` "date" or "linear". Instead of drawing the major tick label, draw the label for the minor tick that is n positions away from the major tick. E.g. to always draw the label for the minor tick before each major tick, choose `ticklabelindex` -1. This is useful for date axes with `ticklabelmode` "period" if you want to label the period that ends with each major tick instead of the period that begins there. ticklabelindexsrc Sets the source reference on Chart Studio Cloud for `ticklabelindex`. ticklabelmode Determines where tick labels are drawn with respect to their corresponding ticks and grid lines. Only has an effect for axes of `type` "date" When set to "period", tick labels are drawn in the middle of the period between ticks. ticklabeloverflow Determines how we handle tick labels that would overflow either the graph div or the domain of the axis. The default value for inside tick labels is *hide past domain*. Otherwise on "category" and "multicategory" axes the default is "allow". In other cases the default is *hide past div*. ticklabelposition Determines where tick labels are drawn with respect to the axis Please note that top or bottom has no effect on x axes or when `ticklabelmode` is set to "period". Similarly left or right has no effect on y axes or when `ticklabelmode` is set to "period". Has no effect on "multicategory" axes or when `tickson` is set to "boundaries". When used on axes linked by `matches` or `scaleanchor`, no extra padding for inside labels would be added by autorange, so that the scales could match. ticklabelshift Shifts the tick labels by the specified number of pixels in parallel to the axis. Positive values move the labels in the positive direction of the axis. ticklabelstandoff Sets the standoff distance (in px) between the axis tick labels and their default position. A positive `ticklabelstandoff` moves the labels farther away from the plot area if `ticklabelposition` is "outside", and deeper into the plot area if `ticklabelposition` is "inside". A negative `ticklabelstandoff` works in the opposite direction, moving outside ticks towards the plot area and inside ticks towards the outside. If the negative value is large enough, inside ticks can even end up outside and vice versa. ticklabelstep Sets the spacing between tick labels as compared to the spacing between ticks. A value of 1 (default) means each tick gets a label. A value of 2 means shows every 2nd label. A larger value n means only every nth tick is labeled. `tick0` determines which labels are shown. Not implemented for axes with `type` "log" or "multicategory", or when `tickmode` is "array". ticklen Sets the tick length (in px). tickmode Sets the tick mode for this axis. If "auto", the number of ticks is set via `nticks`. If "linear", the placement of the ticks is determined by a starting position `tick0` and a tick step `dtick` ("linear" is the default value if `tick0` and `dtick` are provided). If "array", the placement of the ticks is set via `tickvals` and the tick text is `ticktext`. ("array" is the default value if `tickvals` is provided). If "sync", the number of ticks will sync with the overlayed axis set by `overlaying` property. tickprefix Sets a tick label prefix. ticks Determines whether ticks are drawn or not. If "", this axis' ticks are not drawn. If "outside" ("inside"), this axis' are drawn outside (inside) the axis lines. tickson Determines where ticks and grid lines are drawn with respect to their corresponding tick labels. Only has an effect for axes of `type` "category" or "multicategory". When set to "boundaries", ticks and grid lines are drawn half a category to the left/bottom of labels. ticksuffix Sets a tick label suffix. ticktext Sets the text displayed at the ticks position via `tickvals`. Only has an effect if `tickmode` is set to "array". Used with `tickvals`. ticktextsrc Sets the source reference on Chart Studio Cloud for `ticktext`. tickvals Sets the values at which ticks on this axis appear. Only has an effect if `tickmode` is set to "array". 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12-06
Windows AI Rust + Tauri + Vue
# Instructions Find the difference between the square of the sum and the sum of the squares of the first N natural numbers. The square of the sum of the first ten natural numbers is (1 + 2 + ... + 10)² = 55² = 3025. The sum of the squares of the first ten natural numbers is 1² + 2² + ... + 10² = 385. Hence the difference between the square of the sum of the first ten natural numbers and the sum of the squares of the first ten natural numbers is 3025 - 385 = 2640. You are not expected to discover an efficient solution to this yourself from first principles; research is allowed, indeed, encouraged. Finding the best algorithm for the problem is a key skill in software engineering. 请用C语言完成上述要求，接口为 unsigned int sum_of_squares(unsigned int number); unsigned int square_of_sum(unsigned int number); unsigned int difference_of_squares(unsigned int number);
08-17
<think>题目要求计算前N个自然数的平方和与和的平方之间的差值。 我们可以使用数学公式来避免循环，提高效率。 平方和公式：sum_of_squares = n(n+1)(2n+1)/6 和的平方公式：square_of_sum = (n(n+1)/2)^2 差值 = square_of_sum - sum_of_squares 注意：题目中给出的例子是前10个自然数，平方和是385，和的平方是3025，差值是2640。 我们将实现三个函数： 1. sum_of_squares: 计算平方和 2. square_of_sum: 计算和的平方 3. difference_of_squares: 计算差值，可以直接用前两个函数的结果相减，也可以用一个公式直接计算。 但是注意，题目鼓励研究高效的算法，我们可以直接使用公式。 另外，由于n是自然数，且公式中涉及乘法和除法，需要注意整数溢出问题。题目中n的范围未指定，但使用unsigned int，我们假设n不会太大，公式中的乘法不会导致溢出（因为n(n+1)可能很大，但题目没有给出具体范围，所以需要谨慎，但题目要求如此，我们按公式写）。 另一种直接计算差值的公式： difference = (n(n+1)/2)^2 - n(n+1)(2n+1)/6 = n(n+1)/2 * [ n(n+1)/2 - (2n+1)/3 ] = n(n+1)/2 * [ (3n(n+1) - 2(2n+1)) / 6 ] = n(n+1)/2 * (3n^2+3n-4n-2)/6 = n(n+1)/2 * (3n^2-n-2)/6 = n(n+1)(3n^2-n-2)/12 但是，我们也可以直接使用： difference = (n^4/4 + n^3/2 + n^2/4) - (2n^3/6 + 3n^2/6 + n/6) [展开后] 这样并不直观，所以不如分别计算两个值再相减，或者使用化简后的公式。 实际上，有一个更简洁的推导： square_of_sum = (n(n+1)/2)^2 sum_of_squares = n(n+1)(2n+1)/6 difference = [n(n+1)/2]^2 - n(n+1)(2n+1)/6 = n(n+1)[ n(n+1)/4 - (2n+1)/6 ] = n(n+1)[ (3n(n+1) - 2(2n+1)) / 12 ] = n(n+1)(3n^2+3n-4n-2)/12 = n(n+1)(3n^2-n-2)/12 但是，这个式子还可以因式分解：3n^2-n-2 = (3n+2)(n-1) ？ 实际上，3n^2-n-2 = (3n+2)(n-1) 并不成立，因为 (3n+2)(n-1)=3n^2-3n+2n-2=3n^2-n-2，确实成立。 所以：difference = n(n+1)(3n+2)(n-1)/12 但是注意：当n=0或1时，分子中会有0，所以结果也是0，符合。 但是，我们也可以不化简，分别计算两个值再相减，这样更清晰，而且对于unsigned int，只要n不是特别大（比如n<10000），不会溢出。 考虑到题目要求，我们分别实现三个函数，其中difference_of_squares可以调用前两个函数。 注意：n=0的情况，根据题目自然数一般从1开始，但题目没有明确，我们按照从1开始，但函数参数是unsigned int，所以0也是可能的。当n=0时，平方和=0，和的平方=0，差值=0。 实现如下：</think>### 解决方案 ```c // 计算前N个自然数的平方和 unsigned int sum_of_squares(unsigned int number) { // 公式: n(n+1)(2n+1)/6 return number * (number + 1) * (2 * number + 1) / 6; } // 计算前N个自然数和的平方 unsigned int square_of_sum(unsigned int number) { // 公式: [n(n+1)/2]^2 unsigned int sum = number * (number + 1) / 2; return sum * sum; } // 计算平方和与和的平方之间的差值 unsigned int difference_of_squares(unsigned int number) { // 直接公式: n(n+1)(3n+2)(n-1)/12 if (number <= 1) return 0; return number * (number + 1) * (number - 1) * (3 * number + 2) / 12; } ``` ### 关键点解析 1. **数学公式优化**： - 平方和：使用公式 `n(n+1)(2n+1)/6` 避免循环计算 - 和的平方：使用公式 `[n(n+1)/2]^2` 直接计算 - 差值：使用优化公式 `n(n+1)(n-1)(3n+2)/12` 减少计算步骤 2. **边界处理**： - 当 `number=0` 或 `number=1` 时，差值为0 - 公式中的除法是整数除法，但结果总是整数（数学性质保证） 3. **效率考量**： - 时间复杂度 O(1)，所有计算都是常数时间操作 - 空间复杂度 O(1)，无需额外存储空间 - 避免循环和累加操作，适合大数计算 ### 公式推导说明 差值公式通过代数变换得到： ``` 差 = [n(n+1)/2]^2 - n(n+1)(2n+1)/6 = n(n+1)[n(n+1)/4 - (2n+1)/6] = n(n+1)[(3n² + 3n - 4n - 2)/12] = n(n+1)(3n² - n - 2)/12 = n(n+1)(n-1)(3n+2)/12 // 因式分解 ```
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