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-indicator - 版本历史
2026-04-18T05:30:10Z
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物灵：已还原Visaa11（讨论）的编辑至最后由荆哲修订的版本
2025-04-06T06:19:37Z
<p>已还原<a href="/wiki/Special:%E7%94%A8%E6%88%B7%E8%B4%A1%E7%8C%AE/Visaa11" title="Special:用户贡献/Visaa11">Visaa11</a>（<a href="/index.php?title=User_talk:Visaa11&action=edit&redlink=1" class="new" title="User talk:Visaa11（页面不存在）">讨论</a>）的编辑至最后由<a href="/wiki/User:%E8%8D%86%E5%93%B2" class="mw-redirect" title="User:荆哲">荆哲</a>修订的版本</p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">2025年4月6日 (日) 14:19的版本</td>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>'''-indicator''' (pronunciation: <del style="font-weight: bold; text-decoration: none;">'</del>''Eye-indicator<del style="font-weight: bold; text-decoration: none;">'</del>'', Shidinn language: - <del style="font-weight: bold; text-decoration: none;">, pronunciation: /æe-ẽɪ̃nzʷɐonzʷɐ̃nzɹ̩ xæ̃nzɹ̩sʷəʊnzɹ̩/</del>) is a mathematical function mapping [https://mathworld.wolfram.com/PrimeNumber.html prime numbers] to [https://mathworld.wolfram.com/p-adicNumber.html ''p''-adic numbers], indicating how "similar" two primes are, in the sense of [https://mathworld.wolfram.com/QuadraticResidue.html quadratic residues].</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>'''-indicator''' (pronunciation: ''Eye-indicator'', Shidinn language: - ) is a mathematical function mapping [https://mathworld.wolfram.com/PrimeNumber.html prime numbers] to [https://mathworld.wolfram.com/p-adicNumber.html ''p''-adic numbers], indicating how "similar" two primes are, in the sense of [https://mathworld.wolfram.com/QuadraticResidue.html quadratic residues].</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Definition==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Definition==</div></td></tr>
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物灵
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2025年4月3日 (四) 16:22 Visaa11
2025-04-03T16:22:35Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">2025年4月4日 (五) 00:22的版本</td>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>'''-indicator''' (pronunciation: ''Eye-indicator'', Shidinn language: - ) is a mathematical function mapping [https://mathworld.wolfram.com/PrimeNumber.html prime numbers] to [https://mathworld.wolfram.com/p-adicNumber.html ''p''-adic numbers], indicating how "similar" two primes are, in the sense of [https://mathworld.wolfram.com/QuadraticResidue.html quadratic residues].</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>'''-indicator''' (pronunciation: <ins style="font-weight: bold; text-decoration: none;">'</ins>''Eye-indicator<ins style="font-weight: bold; text-decoration: none;">'</ins>'', Shidinn language: - <ins style="font-weight: bold; text-decoration: none;">, pronunciation: /æe-ẽɪ̃nzʷɐonzʷɐ̃nzɹ̩ xæ̃nzɹ̩sʷəʊnzɹ̩/</ins>) is a mathematical function mapping [https://mathworld.wolfram.com/PrimeNumber.html prime numbers] to [https://mathworld.wolfram.com/p-adicNumber.html ''p''-adic numbers], indicating how "similar" two primes are, in the sense of [https://mathworld.wolfram.com/QuadraticResidue.html quadratic residues].</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Definition==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Definition==</div></td></tr>
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Visaa11
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2025年3月15日 (六) 08:57 荆哲
2025-03-15T08:57:47Z
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category:English]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category:English]]</div></td></tr>
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荆哲
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2025年3月15日 (六) 08:57 荆哲
2025-03-15T08:57:00Z
<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">2025年3月15日 (六) 16:57的版本</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== See also ==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== See also ==</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{译文|zh = -指示函数}}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{译文|zh = -指示函数}}</div></td></tr>
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荆哲
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2025年3月15日 (六) 08:56 荆哲
2025-03-15T08:56:17Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">2025年3月15日 (六) 16:56的版本</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Usage==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Usage==</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>For any two "primes" ''q''<sub>1</sub> and ''q''<sub>2</sub> (no matter finite or infinite), the [https://mathworld.wolfram.com/p-adicNorm.html ''p''-adic distance] |<sub>''p''</sub>(''q''<sub>1</sub>)-<sub>''p''</sub>(''q''<sub>2</sub>)|<sub>''p''</sub> measures how similar ''q''<sub>1</sub> and ''q''<sub>2</sub> are. If |<sub>''p''</sub>(''q''<sub>1</sub>)-<sub>''p''</sub>(''q''<sub>2</sub>)|<sub>''p''</sub> < ''p''<sup>-''n''</sup> for a positive integer ''n'', then we can obtain that both negative integers -1..-''n'' and positive integers 1..''n'' (think why?) "performs" consistently on ''q''<sub>1</sub>-adic and ''q''<sub>2</sub>-adic (about whether the integer is a square).</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>For any two "primes" ''q''<sub>1</sub> and ''q''<sub>2</sub> (no matter finite or infinite), the [https://mathworld.wolfram.com/p-adicNorm.html ''p''-adic distance] |<sub>''p''</sub>(''q''<sub>1</sub>)-<sub>''p''</sub>(''q''<sub>2</sub>)|<sub>''p''</sub> measures how similar ''q''<sub>1</sub> and ''q''<sub>2</sub> are. If |<sub>''p''</sub>(''q''<sub>1</sub>)-<sub>''p''</sub>(''q''<sub>2</sub>)|<sub>''p''</sub> < ''p''<sup>-''n''</sup> for a positive integer ''n'', then we can obtain that both negative integers -1..-''n'' and positive integers 1..''n'' (think why?) "performs" consistently on ''q''<sub>1</sub>-adic and ''q''<sub>2</sub>-adic (about whether the integer is a square).</div></td></tr>
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<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">== See also ==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">{{译文|zh = -指示函数}}</ins></div></td></tr>
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荆哲
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2025年3月15日 (六) 08:52 荆哲
2025-03-15T08:52:41Z
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><sub>''p''</sub>(''q'') := ∑<sub>''n''∈'''N'''<sup>+</sup></sub> Kronecker(-''n''|''q'') ''p''<sup>''n''</sup></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><sub>''p''</sub>(''q'') := ∑<sub>''n''∈'''N'''<sup>+</sup></sub> Kronecker(-''n''|''q'') ''p''<sup>''n''</sup></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>where Kronecker means the [https://mathworld.wolfram.com/KroneckerSymbol.html Kronecker symbol]; for ''q''=∞, since the local field on the "infinity prime", i.e. '''Q'''<sub>∞</sub>, stands for the real number field '''R''', and all negative numbers <del style="font-weight: bold; text-decoration: none;">are </del>not square roots in '''R''', we define:</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>where Kronecker means the [https://mathworld.wolfram.com/KroneckerSymbol.html Kronecker symbol]; for ''q''=∞, since the local field on the <ins style="font-weight: bold; text-decoration: none;">so-called </ins>"infinity prime", i.e. '''Q'''<sub>∞</sub>, stands for the real number field '''R''', and all negative numbers <ins style="font-weight: bold; text-decoration: none;">do </ins>not <ins style="font-weight: bold; text-decoration: none;">have </ins>square roots in '''R''', we define:</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><sub>''p''</sub>(∞) := ∑<sub>''n''∈'''N'''<sup>+</sup></sub> -''p''<sup>''n''</sup> = ''p''/(''p''-1)</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><sub>''p''</sub>(∞) := ∑<sub>''n''∈'''N'''<sup>+</sup></sub> -''p''<sup>''n''</sup> = ''p''/(''p''-1)</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Obviously, for any prime ''q'' (including "infinity prime"), the <del style="font-weight: bold; text-decoration: none;">terms </del>of the infinity sum are periodic, thus the value of -indicator is definitely a [https://mathworld.wolfram.com/RationalNumber.html rational number]. Furthermore, <sub>''p''</sub>(''q'') times (''p''<sup>''q''</sup>-1) is an integer if ''q'' is a finite odd prime, and so is <sub>''p''</sub>(2) times (''p''<sup>8</sup>-1).</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Obviously, for any prime ''q'' (including "infinity prime"), the <ins style="font-weight: bold; text-decoration: none;">coefficients </ins>of the infinity sum <ins style="font-weight: bold; text-decoration: none;">above </ins>are periodic, thus the value of -indicator is definitely a [https://mathworld.wolfram.com/RationalNumber.html rational number]. Furthermore, <sub>''p''</sub>(''q'') times (''p''<sup>''q''</sup>-1) is an integer if ''q'' is a finite odd prime, and so is <sub>''p''</sub>(2) times (''p''<sup>8</sup>-1).</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Examples==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Examples==</div></td></tr>
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荆哲
https://wiki.xdi8.top/index.php?title=%EE%82%84-indicator&diff=38133&oldid=prev
2025年3月15日 (六) 03:04 荆哲
2025-03-15T03:04:40Z
<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="zh">
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">←上一版本</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">2025年3月15日 (六) 11:04的版本</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1">第1行：</td>
<td colspan="2" class="diff-lineno">第1行：</td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>'''-indicator''' (pronunciation: ''Eye-indicator'', Shidinn language: - ) is a mathematical function mapping prime numbers to ''p''-adic numbers, indicating how "similar" two primes are, in the sense of quadratic residues.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>'''-indicator''' (pronunciation: ''Eye-indicator'', Shidinn language: - ) is a mathematical function mapping <ins style="font-weight: bold; text-decoration: none;">[https://mathworld.wolfram.com/PrimeNumber.html </ins>prime numbers<ins style="font-weight: bold; text-decoration: none;">] </ins>to <ins style="font-weight: bold; text-decoration: none;">[https://mathworld.wolfram.com/p-adicNumber.html </ins>''p''-adic numbers<ins style="font-weight: bold; text-decoration: none;">]</ins>, indicating how "similar" two primes are, in the sense of <ins style="font-weight: bold; text-decoration: none;">[https://mathworld.wolfram.com/QuadraticResidue.html </ins>quadratic residues<ins style="font-weight: bold; text-decoration: none;">]</ins>.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Definition==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Definition==</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l6">第6行：</td>
<td colspan="2" class="diff-lineno">第6行：</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><sub>''p''</sub>(''q'') := ∑<sub>''n''∈'''N'''<sup>+</sup></sub> Kronecker(-''n''|''q'') ''p''<sup>''n''</sup></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><sub>''p''</sub>(''q'') := ∑<sub>''n''∈'''N'''<sup>+</sup></sub> Kronecker(-''n''|''q'') ''p''<sup>''n''</sup></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>where Kronecker means the Kronecker symbol; for ''q''=∞, since the local field on the "infinity prime", i.e. '''Q'''<sub>∞</sub>, stands for the real number field '''R''', and all negative numbers are not square roots in '''R''', we define:</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>where Kronecker means the <ins style="font-weight: bold; text-decoration: none;">[https://mathworld.wolfram.com/KroneckerSymbol.html </ins>Kronecker symbol<ins style="font-weight: bold; text-decoration: none;">]</ins>; for ''q''=∞, since the local field on the "infinity prime", i.e. '''Q'''<sub>∞</sub>, stands for the real number field '''R''', and all negative numbers are not square roots in '''R''', we define:</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><sub>''p''</sub>(∞) := ∑<sub>''n''∈'''N'''<sup>+</sup></sub> -''p''<sup>''n''</sup> = ''p''/(''p''-1)</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><sub>''p''</sub>(∞) := ∑<sub>''n''∈'''N'''<sup>+</sup></sub> -''p''<sup>''n''</sup> = ''p''/(''p''-1)</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Obviously, for any prime ''q'' (including "infinity prime"), the terms of the infinity sum are periodic, thus the value of -indicator is definitely a rational number. Furthermore, <sub>''p''</sub>(''q'') times (''p''<sup>''q''</sup>-1) is an integer if ''q'' is a finite odd prime, and so is <sub>''p''</sub>(2) times (''p''<sup>8</sup>-1).</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Obviously, for any prime ''q'' (including "infinity prime"), the terms of the infinity sum are periodic, thus the value of -indicator is definitely a <ins style="font-weight: bold; text-decoration: none;">[https://mathworld.wolfram.com/RationalNumber.html </ins>rational number<ins style="font-weight: bold; text-decoration: none;">]</ins>. Furthermore, <sub>''p''</sub>(''q'') times (''p''<sup>''q''</sup>-1) is an integer if ''q'' is a finite odd prime, and so is <sub>''p''</sub>(2) times (''p''<sup>8</sup>-1).</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Examples==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Examples==</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l22">第22行：</td>
<td colspan="2" class="diff-lineno">第22行：</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Usage==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Usage==</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>For any two "primes" ''q''<sub>1</sub> and ''q''<sub>2</sub> (no matter finite or infinite), the ''p''-adic distance |<sub>''p''</sub>(''q''<sub>1</sub>)-<sub>''p''</sub>(''q''<sub>2</sub>)|<sub>''p''</sub> measures how similar ''q''<sub>1</sub> and ''q''<sub>2</sub> are. If |<sub>''p''</sub>(''q''<sub>1</sub>)-<sub>''p''</sub>(''q''<sub>2</sub>)|<sub>''p''</sub> < ''p''<sup>-''n''</sup> for a positive integer ''n'', then we can obtain that both negative integers -1..-''n'' and positive integers 1..''n'' (think why?) "performs" consistently on ''q''<sub>1</sub>-adic and ''q''<sub>2</sub>-adic (about whether the integer is a square).</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>For any two "primes" ''q''<sub>1</sub> and ''q''<sub>2</sub> (no matter finite or infinite), the <ins style="font-weight: bold; text-decoration: none;">[https://mathworld.wolfram.com/p-adicNorm.html </ins>''p''-adic distance<ins style="font-weight: bold; text-decoration: none;">] </ins>|<sub>''p''</sub>(''q''<sub>1</sub>)-<sub>''p''</sub>(''q''<sub>2</sub>)|<sub>''p''</sub> measures how similar ''q''<sub>1</sub> and ''q''<sub>2</sub> are. If |<sub>''p''</sub>(''q''<sub>1</sub>)-<sub>''p''</sub>(''q''<sub>2</sub>)|<sub>''p''</sub> < ''p''<sup>-''n''</sup> for a positive integer ''n'', then we can obtain that both negative integers -1..-''n'' and positive integers 1..''n'' (think why?) "performs" consistently on ''q''<sub>1</sub>-adic and ''q''<sub>2</sub>-adic (about whether the integer is a square).</div></td></tr>
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荆哲
https://wiki.xdi8.top/index.php?title=%EE%82%84-indicator&diff=38132&oldid=prev
2025年3月15日 (六) 02:58 荆哲
2025-03-15T02:58:42Z
<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="zh">
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">←上一版本</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">2025年3月15日 (六) 10:58的版本</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1">第1行：</td>
<td colspan="2" class="diff-lineno">第1行：</td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>'''-indicator''' (pronunciation: Eye-indicator, Shidinn language: - ) is a mathematical function mapping prime numbers to ''p''-adic numbers, indicating how "similar" two primes are, in the sense of quadratic residues.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>'''-indicator''' (pronunciation: <ins style="font-weight: bold; text-decoration: none;">''</ins>Eye-indicator<ins style="font-weight: bold; text-decoration: none;">''</ins>, Shidinn language: - ) is a mathematical function mapping prime numbers to ''p''-adic numbers, indicating how "similar" two primes are, in the sense of quadratic residues.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Definition==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Definition==</div></td></tr>
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荆哲
https://wiki.xdi8.top/index.php?title=%EE%82%84-indicator&diff=38131&oldid=prev
荆哲：/* Examples */
2025-03-15T02:58:22Z
<p><span class="autocomment">Examples</span></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="zh">
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">←上一版本</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">2025年3月15日 (六) 10:58的版本</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l19">第19行：</td>
<td colspan="2" class="diff-lineno">第19行：</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* <sub>3</sub>(7)=-453/1093</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* <sub>3</sub>(7)=-453/1093</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* <sub>3</sub>(11)=-24249/88573</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* <sub>3</sub>(11)=-24249/88573</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>...</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* </ins>...</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Usage==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Usage==</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>For any two "primes" ''q''<sub>1</sub> and ''q''<sub>2</sub> (no matter finite or infinite), the ''p''-adic distance |<sub>''p''</sub>(''q''<sub>1</sub>)-<sub>''p''</sub>(''q''<sub>2</sub>)|<sub>''p''</sub> measures how similar ''q''<sub>1</sub> and ''q''<sub>2</sub> are. If |<sub>''p''</sub>(''q''<sub>1</sub>)-<sub>''p''</sub>(''q''<sub>2</sub>)|<sub>''p''</sub> < ''p''<sup>-''n''</sup> for a positive integer ''n'', then we can obtain that both negative integers -1..-''n'' and positive integers 1..''n'' (think why?) "performs" consistently on ''q''<sub>1</sub>-adic and ''q''<sub>2</sub>-adic (about whether the integer is a square).</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>For any two "primes" ''q''<sub>1</sub> and ''q''<sub>2</sub> (no matter finite or infinite), the ''p''-adic distance |<sub>''p''</sub>(''q''<sub>1</sub>)-<sub>''p''</sub>(''q''<sub>2</sub>)|<sub>''p''</sub> measures how similar ''q''<sub>1</sub> and ''q''<sub>2</sub> are. If |<sub>''p''</sub>(''q''<sub>1</sub>)-<sub>''p''</sub>(''q''<sub>2</sub>)|<sub>''p''</sub> < ''p''<sup>-''n''</sup> for a positive integer ''n'', then we can obtain that both negative integers -1..-''n'' and positive integers 1..''n'' (think why?) "performs" consistently on ''q''<sub>1</sub>-adic and ''q''<sub>2</sub>-adic (about whether the integer is a square).</div></td></tr>
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荆哲
https://wiki.xdi8.top/index.php?title=%EE%82%84-indicator&diff=38130&oldid=prev
2025年3月15日 (六) 02:58 荆哲
2025-03-15T02:58:07Z
<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">←上一版本</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">2025年3月15日 (六) 10:58的版本</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1">第1行：</td>
<td colspan="2" class="diff-lineno">第1行：</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>'''-indicator''' (pronunciation: Eye-indicator, Shidinn language: - ) is a mathematical function mapping prime numbers to ''p''-adic numbers, indicating how "similar" two primes are, in the sense of quadratic residues.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>'''-indicator''' (pronunciation: Eye-indicator, Shidinn language: - ) is a mathematical function mapping prime numbers to ''p''-adic numbers, indicating how "similar" two primes are, in the sense of quadratic residues.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==Definition==</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>For any odd prime ''p'', define the -indicator <sub>''p''</sub> : {''q'':''q'' is a prime}∪{∞}→'''Q'''<sub>''p''</sub> , where '''Q'''<sub>''p''</sub> is the ''p''-adic number field:</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>For any odd prime ''p'', define the -indicator <sub>''p''</sub> : {''q'':''q'' is a prime}∪{∞}→'''Q'''<sub>''p''</sub> , where '''Q'''<sub>''p''</sub> is the ''p''-adic number field:</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l11">第11行：</td>
<td colspan="2" class="diff-lineno">第12行：</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Obviously, for any prime ''q'' (including "infinity prime"), the terms of the infinity sum are periodic, thus the value of -indicator is definitely a rational number. Furthermore, <sub>''p''</sub>(''q'') times (''p''<sup>''q''</sup>-1) is an integer if ''q'' is a finite odd prime, and so is <sub>''p''</sub>(2) times (''p''<sup>8</sup>-1).</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Obviously, for any prime ''q'' (including "infinity prime"), the terms of the infinity sum are periodic, thus the value of -indicator is definitely a rational number. Furthermore, <sub>''p''</sub>(''q'') times (''p''<sup>''q''</sup>-1) is an integer if ''q'' is a finite odd prime, and so is <sub>''p''</sub>(2) times (''p''<sup>8</sup>-1).</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Here are some examples:</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==Examples==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Here are some examples <ins style="font-weight: bold; text-decoration: none;">for ''p''=3</ins>:</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* <sub>3</sub>(2)=-12/41</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* <sub>3</sub>(2)=-12/41</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* <sub>3</sub>(3)=-3/13</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* <sub>3</sub>(3)=-3/13</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l19">第19行：</td>
<td colspan="2" class="diff-lineno">第21行：</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>...</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>...</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==Usage==</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>For any two "primes" ''q''<sub>1</sub> and ''q''<sub>2</sub> (no matter finite or infinite), the ''p''-adic distance |<sub>''p''</sub>(''q''<sub>1</sub>)-<sub>''p''</sub>(''q''<sub>2</sub>)|<sub>''p''</sub> measures how similar ''q''<sub>1</sub> and ''q''<sub>2</sub> are. If |<sub>''p''</sub>(''q''<sub>1</sub>)-<sub>''p''</sub>(''q''<sub>2</sub>)|<sub>''p''</sub> < ''p''<sup>-''n''</sup> for a positive integer ''n'', then we can obtain that both negative integers -1..-''n'' and positive integers 1..''n'' (think why?) "performs" consistently on ''q''<sub>1</sub>-adic and ''q''<sub>2</sub>-adic (about whether the integer is a square).</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>For any two "primes" ''q''<sub>1</sub> and ''q''<sub>2</sub> (no matter finite or infinite), the ''p''-adic distance |<sub>''p''</sub>(''q''<sub>1</sub>)-<sub>''p''</sub>(''q''<sub>2</sub>)|<sub>''p''</sub> measures how similar ''q''<sub>1</sub> and ''q''<sub>2</sub> are. If |<sub>''p''</sub>(''q''<sub>1</sub>)-<sub>''p''</sub>(''q''<sub>2</sub>)|<sub>''p''</sub> < ''p''<sup>-''n''</sup> for a positive integer ''n'', then we can obtain that both negative integers -1..-''n'' and positive integers 1..''n'' (think why?) "performs" consistently on ''q''<sub>1</sub>-adic and ''q''<sub>2</sub>-adic (about whether the integer is a square).</div></td></tr>
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荆哲