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21 lines
8.3 KiB
HTML
<!DOCTYPE html>
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<meta http-equiv="Content-Type" content="text/html; charset=UTF-8" />
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<style type="text/css">
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.katex-html {display: none;}
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.katex-mathml {font-size: 0px;}
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.katex-mathml math mrow {font-size: 16px;}
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</style>
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<h1>素数对猜想</h1>
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<div class="ques-view"><p>让我们定义<span class="katex"><span class="katex-mathml"><math><mrow><msub><mi>d</mi><mi>n</mi></msub></mrow>d_n</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit">d</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;"></span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathit mtight">n</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;"></span></span></span></span></span></span></span></span></span>为：<span class="katex"><span class="katex-mathml"><math><mrow><msub><mi>d</mi><mi>n</mi></msub><mo>=</mo><msub><mi>p</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>−</mo><msub><mi>p</mi><mi>n</mi></msub></mrow>d_n = p_{n+1}-p_n</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.902771em;vertical-align:-0.208331em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit">d</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;"></span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathit mtight">n</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;"></span></span></span></span></span></span><span class="mrel">=</span><span class="mord"><span class="mord mathit">p</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;"></span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathit mtight">n</span><span class="mbin mtight">+</span><span class="mord mathrm mtight">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;"></span></span></span></span></span></span><span class="mbin">−</span><span class="mord"><span class="mord mathit">p</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;"></span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathit mtight">n</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;"></span></span></span></span></span></span></span></span></span>，其中<span class="katex"><span class="katex-mathml"><math><mrow><msub><mi>p</mi><mi>i</mi></msub></mrow>p_i</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.43056em;"></span><span class="strut bottom" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit">p</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;"></span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathit mtight">i</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;"></span></span></span></span></span></span></span></span></span>是第<span class="katex"><span class="katex-mathml"><math><mrow><mi>i</mi></mrow>i</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.65952em;"></span><span class="strut bottom" style="height:0.65952em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit">i</span></span></span></span>个素数。显然有<span class="katex"><span class="katex-mathml"><math><mrow><msub><mi>d</mi><mn>1</mn></msub><mo>=</mo><mn>1</mn></mrow>d_1 = 1</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit">d</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;"></span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathrm mtight">1</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;"></span></span></span></span></span></span><span class="mrel">=</span><span class="mord mathrm">1</span></span></span></span>，且对于<span class="katex"><span class="katex-mathml"><math><mrow><mi>n</mi><mo>></mo><mn>1</mn></mrow>n>1</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.64444em;"></span><span class="strut bottom" style="height:0.68354em;vertical-align:-0.0391em;"></span><span class="base textstyle uncramped"><span class="mord mathit">n</span><span class="mrel">></span><span class="mord mathrm">1</span></span></span></span>有<span class="katex"><span class="katex-mathml"><math><mrow><msub><mi>d</mi><mi>n</mi></msub></mrow>d_n</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit">d</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;"></span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathit mtight">n</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;"></span></span></span></span></span></span></span></span></span>是偶数。“素数对猜想”认为“存在无穷多对相邻且差为2的素数”。</p>
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<p>现给定任意正整数<code>N</code>(<span class="katex"><span class="katex-mathml"><math><mrow><mo><</mo><mn>1</mn><msup><mn>0</mn><mn>5</mn></msup></mrow><10^5</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.8141079999999999em;"></span><span class="strut bottom" style="height:0.853208em;vertical-align:-0.0391em;"></span><span class="base textstyle uncramped"><span class="mrel"><</span><span class="mord mathrm">1</span><span class="mord"><span class="mord mathrm">0</span><span class="msupsub"><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;"></span></span><span class="reset-textstyle scriptstyle uncramped mtight"><span class="mord mathrm mtight">5</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;"></span></span></span></span></span></span></span></span></span>)，请计算不超过<code>N</code>的满足猜想的素数对的个数。</p>
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<h3 id="-">输入格式:</h3>
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<p>输入在一行给出正整数<code>N</code>。</p>
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<h3 id="-">输出格式:</h3>
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<p>在一行中输出不超过<code>N</code>的满足猜想的素数对的个数。</p>
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<h3 id="-">输入样例:</h3>
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<pre><code class="lang-in">20
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</code></pre>
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<h3 id="-">输出样例:</h3>
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<pre><code class="lang-out">4
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</code></pre>
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</div> |