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'''<font color="#ff8000"> 埃尔德什数Erdős number</font>'''（匈牙利语：[ˈɛrdøːʃ]）根据数学论文的著作权来来对数学家保罗·埃尔德什与其他作者之间的“协作距离”进行描述。同样的原则也应用于很多当特定某个人与众多同行之间保持合作关系的其他领域。
 
'''<font color="#ff8000"> 埃尔德什数Erdős number</font>'''（匈牙利语：[ˈɛrdøːʃ]）根据数学论文的著作权来来对数学家保罗·埃尔德什与其他作者之间的“协作距离”进行描述。同样的原则也应用于很多当特定某个人与众多同行之间保持合作关系的其他领域。
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== Overview 概况==
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== 概况==
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Paul Erdős (1913–1996) was an influential [[Hungarian people|Hungarian]] mathematician who in the latter part of his life spent a great deal of time writing papers with a large number of colleagues, working on solutions to outstanding mathematical problems. He published more papers during his lifetime (at least 1,525) than any other mathematician in history. ([[Leonhard Euler]] published more total pages of mathematics but fewer separate papers: about 800.) Erdős spent a large portion of his later life living out of a suitcase, visiting his over 500 collaborators around the world.
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保罗·埃尔德什Paul Erdős(1913年至1996年）是一位在业界产生有影响力的匈牙利数学家，其一生中大量的时间都在与很多同事合作撰写论文，致力于解决困扰已久的疑难数学问题。<ref name="newman2001">{{cite journal|last=Newman|first=Mark E. J.|author-link=Mark Newman|title=The structure of scientific collaboration networks|journal=[[Proceedings of the National Academy of Sciences of the United States of America]]| year=2001| doi=10.1073/pnas.021544898| volume=98|issue=2|pages=404–409|pmid=11149952|pmc=14598}}</ref> 他一生中所发表的论文（至少1,525篇<ref>{{cite web |url=http://www.oakland.edu/enp/pubinfo/ |title=Publications of Paul Erdős | first=Jerry | last=Grossman |access-date=1 Feb 2011}}</ref>）比历史上其他任何数学家都多<ref name="newman2001"/>。莱昂哈德·欧拉Leonhard Euler发表过的数学论文页数更多，但单独的论文却较少（大约800篇）。<ref>{{cite web| url=https://www.math.dartmouth.edu/~euler/FAQ.html| work=The Euler Archive| title=Frequently Asked Questions| publisher=Dartmouth College}}</ref>而埃尔德什的大部分时间都在旅居中，其拜访过全球500多个合作者。
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Paul Erdős (1913–1996) was an influential [[Hungarian people|Hungarian]] mathematician who in the latter part of his life spent a great deal of time writing papers with a large number of colleagues, working on solutions to outstanding mathematical problems.<ref name="newman2001">{{cite journal|last=Newman|first=Mark E. J.|author-link=Mark Newman|title=The structure of scientific collaboration networks|journal=[[Proceedings of the National Academy of Sciences of the United States of America]]| year=2001| doi=10.1073/pnas.021544898| volume=98|issue=2|pages=404–409|pmid=11149952|pmc=14598}}</ref> He published more papers during his lifetime (at least 1,525<ref>{{cite web |url=http://www.oakland.edu/enp/pubinfo/ |title=Publications of Paul Erdős | first=Jerry | last=Grossman |access-date=1 Feb 2011}}</ref>) than any other mathematician in history.<ref name="newman2001"/> ([[Leonhard Euler]] published more total pages of mathematics but fewer separate papers: about 800.)<ref>{{cite web| url=https://www.math.dartmouth.edu/~euler/FAQ.html| work=The Euler Archive| title=Frequently Asked Questions| publisher=Dartmouth College}}</ref> Erdős spent a large portion of his later life living out of a suitcase, visiting his over 500 collaborators around the world.
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埃尔德什数的概念最初是由埃尔德什的朋友们提出来的，以赞扬保罗·埃尔德什的巨大成就。后来，它演变为研究数学家如何通过合作来解决问题的的工具而受到重视。有几个项目致力于使用埃尔德什数为代表方法来研究人员之间的连通性。<ref name="Erdős Number Project">{{cite web|url=http://www.oakland.edu/enp|title=Erdös Number Project|publisher=Oakland University}}</ref>例如，埃尔德什合作图可以告诉我们作者是如何聚集在一起的，每篇论文的共同作者数量随时间变化或新理论的产生如何传播的。<ref>{{cite web|url=http://www.oakland.edu/enp/trivia/|title=Facts about Erdös Numbers and the Collaboration Graph|work=Erdös Number Project|publisher=Oakland University}}</ref>
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保罗·埃尔德什Paul Erdős(1913年至1996年）是一位在业界产生有影响力的匈牙利数学家，其一生中大量的时间都在与很多同事合作撰写论文，致力于解决困扰已久的疑难数学问题。他一生中所发表的论文（至少1,525篇）比历史上其他任何数学家都多。莱昂哈德·欧拉Leonhard Euler发表过的数学论文页数更多，但单独的论文却较少（大约800篇）。而埃尔德什的大部分时间都在旅居中，其拜访过全球500多个合作者。
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多项研究表明，领先的数学家往往具有极低的埃尔德什数。<ref name="trails">{{cite journal
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The idea of the Erdős number was originally created by the mathematician's friends as a tribute to his enormous output. Later it gained prominence as a tool to study how mathematicians cooperate to find answers to unsolved problems. Several projects are devoted to studying connectivity among researchers, using the Erdős number as a proxy. For example, Erdős [[collaboration graph]]s can tell us how authors cluster, how the number of co-authors per paper evolves over time, or how new theories propagate.
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The idea of the Erdős number was originally created by the mathematician's friends as a tribute to his enormous output. Later it gained prominence as a tool to study how mathematicians cooperate to find answers to unsolved problems. Several projects are devoted to studying connectivity among researchers, using the Erdős number as a proxy.<ref name="Erdős Number Project">{{cite web|url=http://www.oakland.edu/enp|title=Erdös Number Project|publisher=Oakland University}}</ref> For example, Erdős [[collaboration graph]]s can tell us how authors cluster, how the number of co-authors per paper evolves over time, or how new theories propagate.<ref>{{cite web|url=http://www.oakland.edu/enp/trivia/|title=Facts about Erdös Numbers and the Collaboration Graph|work=Erdös Number Project|publisher=Oakland University}}</ref>
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埃尔德什数的概念最初是由埃尔德什的朋友们提出来的，以赞扬保罗·埃尔德什的巨大成就。后来，它演变为研究数学家如何通过合作来解决问题的的工具而受到重视。有几个项目致力于使用埃尔德什数为代表方法来研究人员之间的连通性。例如，埃尔德什合作图可以告诉我们作者是如何聚集在一起的，每篇论文的共同作者数量随时间变化或新理论的产生如何传播的。
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Several studies have shown that leading mathematicians tend to have particularly low Erdős numbers. The median Erdős number of [[Fields Medalists]] is 3. Only 7,097 (about 5% of mathematicians with a collaboration path) have an Erdős number of 2 or lower. As time passes, the smallest Erdős number that can still be achieved will necessarily increase, as mathematicians with low Erdős numbers die and become unavailable for collaboration. Still, historical figures can have low Erdős numbers. For example, renowned Indian mathematician [[Srinivasa Ramanujan]] has an Erdős number of only 3 (through [[G. H. Hardy]], Erdős number 2), even though Paul Erdős was only 7 years old when Ramanujan died.
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Several studies have shown that leading mathematicians tend to have particularly low Erdős numbers.<ref name="trails">{{cite journal
   
  |last1      = De Castro
 
  |last1      = De Castro
 
  |first1      = Rodrigo
 
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  |archive-url  = https://web.archive.org/web/20150924054224/http://www.oakland.edu/upload/docs/Erdos%20Number%20Project/trails.pdf
 
  |archive-url  = https://web.archive.org/web/20150924054224/http://www.oakland.edu/upload/docs/Erdos%20Number%20Project/trails.pdf
 
  |archive-date = 2015-09-24
 
  |archive-date = 2015-09-24
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}} Original Spanish version in ''Rev. Acad. Colombiana Cienc. Exact. Fís. Natur.'' '''23''' (89) 563–582, 1999, {{MR|1744115}}.</ref> The median Erdős number of [[Fields Medalists]] is 3. Only 7,097 (about 5% of mathematicians with a collaboration path) have an Erdős number of 2 or lower.<ref name="paths"/> As time passes, the smallest Erdős number that can still be achieved will necessarily increase, as mathematicians with low Erdős numbers die and become unavailable for collaboration. Still, historical figures can have low Erdős numbers. For example, renowned Indian mathematician [[Srinivasa Ramanujan]] has an Erdős number of only 3 (through [[G. H. Hardy]], Erdős number 2), even though Paul Erdős was only 7 years old when Ramanujan died.<ref name=":0" />
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}} Original Spanish version in ''Rev. Acad. Colombiana Cienc. Exact. Fís. Natur.'' '''23''' (89) 563–582, 1999, {{MR|1744115}}.</ref>菲尔兹奖得主Fields Medalists的埃尔德什中位数是3。只有7,097名（拥有合作经历的数学家中约5％）的埃尔德什数为2或更低。随着时间的流逝，低埃尔德什数的数学家因死亡而无法进行协作，所能达到的最小埃尔德什数必然会增加。历史人物仍可能一直具有较低的埃尔德什数。例如，印度著名数学家Srinivasa Ramanujan的埃尔德什数仅为3（通过与G. H. Hardy合作，其埃尔德什数为2），尽管Ramanujan去世时保罗·埃尔德什只有7岁。<ref name="paths"/> As time passes, the smallest Erdős number that can still be achieved will necessarily increase, as mathematicians with low Erdős numbers die and become unavailable for collaboration. Still, historical figures can have low Erdős numbers. For example, renowned Indian mathematician [[Srinivasa Ramanujan]] has an Erdős number of only 3 (through [[G. H. Hardy]], Erdős number 2), even though Paul Erdős was only 7 years old when Ramanujan died.<ref name=":0" />
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== 数学的定义与应用 ==
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多项研究表明，领先的数学家往往具有极低的埃尔德什数。费尔兹奖得主Fields Medalists的埃尔德什中位数是3。只有7,097名（拥有合作经历的数学家中约5％）的埃尔德什数为2或更低。随着时间的流逝，低埃尔德什数的数学家因死亡而无法进行协作，所能达到的最小埃尔德什数必然会增加。历史人物仍可能一直具有较低的埃尔德什数。例如，印度著名数学家Srinivasa Ramanujan的埃尔德什数仅为3（通过与G. H. Hardy合作，其埃尔德什数为2），尽管Ramanujan去世时保罗·埃尔德什只有7岁。
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== Definition and application in mathematics 数学的定义与应用 ==
      
[[文件:Erdosnumber.png|缩略图|右|如果爱丽丝在一张纸上与保罗·埃尔德什合作，在另一张纸上与鲍勃合作，但是鲍勃从未与埃尔德什本人合作，那么爱丽丝的埃尔德什数为1，而鲍勃的埃尔德什数为2，因为他离埃尔德什有两步。]]
 
[[文件:Erdosnumber.png|缩略图|右|如果爱丽丝在一张纸上与保罗·埃尔德什合作，在另一张纸上与鲍勃合作，但是鲍勃从未与埃尔德什本人合作，那么爱丽丝的埃尔德什数为1，而鲍勃的埃尔德什数为2，因为他离埃尔德什有两步。]]
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To be assigned an Erdős number, someone must be a coauthor of a research paper with another person who has a finite Erdős number.  Paul Erdős has an Erdős number of zero. Anybody else's Erdős number is {{math|''k'' + 1}} where {{math|''k''}} is the lowest Erdős number of any coauthor.  The [[American Mathematical Society]] provides a free online tool to determine the Erdős number of every mathematical author listed in the ''[[Mathematical Reviews]]'' catalogue.
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要分配一个埃尔德什数，某人必须与另一个具有有限埃尔德什数的人共同撰写研究论文。保罗·埃尔德什的埃尔德什数为零。其他人的埃尔德什数为''k+1''，其中''k''是任何合著者中最低的埃尔德什数。美国数学学会提供免费的在线工具来确定《数学评论》目录中列出的每个数学作者的埃尔德什数。<ref name=":0">{{cite web|url=https://www.ams.org/mathscinet/collaborationDistance.html|title= Collaboration Distance|work=[[MathSciNet]]|publisher=American Mathematical Society}}</ref>
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To be assigned an Erdős number, someone must be a coauthor of a research paper with another person who has a finite Erdős number.  Paul Erdős has an Erdős number of zero. Anybody else's Erdős number is {{math|''k'' + 1}} where {{math|''k''}} is the lowest Erdős number of any coauthor.  The [[American Mathematical Society]] provides a free online tool to determine the Erdős number of every mathematical author listed in the ''[[Mathematical Reviews]]'' catalogue.<ref name=":0">{{cite web|url=https://www.ams.org/mathscinet/collaborationDistance.html|title= Collaboration Distance|work=[[MathSciNet]]|publisher=American Mathematical Society}}</ref>
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要分配一个埃尔德什数，某人必须与另一个具有有限埃尔德什数的人共同撰写研究论文。保罗·埃尔德什的埃尔德什数为零。其他人的埃尔德什数为''k+1''，其中''k''是任何合著者中最低的埃尔德什数。美国数学学会提供免费的在线工具来确定《数学评论》目录中列出的每个数学作者的埃尔德什数。
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Erdős wrote around 1,500 mathematical articles in his lifetime, mostly co-written. He had 511 direct collaborators; these are the people with Erdős number 1. The people who have collaborated with them (but not with Erdős himself) have an Erdős number of 2 (11,009 people as of 2015), those who have collaborated with people who have an Erdős number of 2 (but not with Erdős or anyone with an Erdős number of 1) have an Erdős number of 3, and so forth. A person with no such coauthorship chain connecting to Erdős has an Erdős number of [[infinity]] (or an [[defined and undefined|undefined]] one). Since the death of Paul Erdős, the lowest Erdős number that a new researcher can obtain is 2.
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Erdős wrote around 1,500 mathematical articles in his lifetime, mostly co-written. He had 512 direct collaborators;<ref name="Erdős Number Project"/> these are the people with Erdős number 1.  The people who have collaborated with them (but not with Erdős himself) have an Erdős number of 2 (12,600 people as of 7 August 2020<ref name="Erdős Number Project File Erdos2">[https://www.oakland.edu/enp/thedata/erdos2/ Erdos2], Version 2020, 7 August 2020.</ref>), those who have collaborated with people who have an Erdős number of 2 (but not with Erdős or anyone with an Erdős number of 1) have an Erdős number of 3, and so forth. A person with no such coauthorship chain connecting to Erdős has an Erdős number of [[infinity]] (or an [[defined and undefined|undefined]] one). Since the death of Paul Erdős, the lowest Erdős number that a new researcher can obtain is 2.
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埃尔德什一生撰写了约1500篇数学文章，其中大部分是合作的。他有511个直接合作者；这些是埃尔德什数为1的人。与这些人合作（但未与埃尔德什本人合作）的人所拥有的埃尔德什数为2（截至2020年8月7日为12,600人），而与埃尔德什数为2的人合作的人（但与埃尔德什或埃尔德什数为1的任何人无合作关系），其埃尔德什数为3，依此类推。没有此类共同作者链接能指向埃尔德什的人，其埃尔德什数为无穷大（或未定义）。自保罗·埃尔德什逝世以来，新研究员可获得的最低埃尔德什数为2。
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埃尔德什一生撰写了约1500篇数学文章，其中大部分是合作的。他有511个直接合作者<ref name="Erdős Number Project"/>；这些是埃尔德什数为1的人。与这些人合作（但未与埃尔德什本人合作）的人所拥有的埃尔德什数为2（截至2020年8月7日为12,600人<ref name="Erdős Number Project File Erdos2">[https://www.oakland.edu/enp/thedata/erdos2/ Erdos2], Version 2020, 7 August 2020.</ref>），而与埃尔德什数为2的人合作的人（但与埃尔德什或埃尔德什数为1的任何人无合作关系），其埃尔德什数为3，依此类推。没有此类共同作者链接能指向埃尔德什的人，其埃尔德什数为无穷大（或未定义）。自保罗·埃尔德什逝世以来，新研究员可获得的最低埃尔德什数为2。
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费尔兹奖获得者的埃尔德什中位数低至3。<ref name="paths"/>埃尔德什排名第二的奖牌获得者包括Atle Selberg，Kunihiko Kodaira，Klaus Roth，Alan Baker，Enrico Bombieri，David Mumford，Charles Fefferman，William Thurston，Shing-Tung Tung，Jean Bourgain，Richard Borcherds，Manjul Bhargava，Jean-Pierre Serre和陶哲轩。费尔兹奖获得者中没有人的埃尔德什数为1。<ref name="project">{{cite web|url=http://www.oakland.edu/enp/erdpaths/|title=Paths to Erdös|work=The Erdös Number Project|publisher=Oakland University}}</ref>但是，恩德雷·塞梅雷迪（Endre Szemerédi）是阿贝尔奖获得者，其埃尔德什数为1。<ref name="trails"/>
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菲尔兹奖获得者的埃尔德什中位数低至3。<ref name="paths"/>埃尔德什排名第二的奖牌获得者包括Atle Selberg，Kunihiko Kodaira，Klaus Roth，Alan Baker，Enrico Bombieri，David Mumford，Charles Fefferman，William Thurston，Shing-Tung Tung，Jean Bourgain，Richard Borcherds，Manjul Bhargava，Jean-Pierre Serre和陶哲轩。菲尔兹奖获得者中没有人的埃尔德什数为1。<ref name="project">{{cite web|url=http://www.oakland.edu/enp/erdpaths/|title=Paths to Erdös|work=The Erdös Number Project|publisher=Oakland University}}</ref>但是，恩德雷·塞梅雷迪（Endre Szemerédi）是阿贝尔奖获得者，其埃尔德什数为1。<ref name="trails"/>
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== Most frequent Erdős collaborators 最频繁的埃尔德什合作者 ==
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== 最频繁的埃尔德什合作者 ==
    
虽然埃尔德什与数百位合著者合作，但其中一些人与他合作过数十篇论文。以下是最经常与埃尔德什合作的十人列表，以及与埃尔德什合作的论文数量（即合作数量）。<ref>Grossman, Jerry, [https://files.oakland.edu/users/grossman/enp/Erdos0p.html Erdos0p],  Version 2010, ''[http://www.oakland.edu/enp The Erdős Number Project]'', [[Oakland University]], US, October 20, 2010.</ref>
 
虽然埃尔德什与数百位合著者合作，但其中一些人与他合作过数十篇论文。以下是最经常与埃尔德什合作的十人列表，以及与埃尔德什合作的论文数量（即合作数量）。<ref>Grossman, Jerry, [https://files.oakland.edu/users/grossman/enp/Erdos0p.html Erdos0p],  Version 2010, ''[http://www.oakland.edu/enp The Erdős Number Project]'', [[Oakland University]], US, October 20, 2010.</ref>
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== Related fields 相关领域 ==
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== 相关领域 ==
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{{As of|2016}}, all [[Fields Medal]]ists have a finite Erdős number, with values that range between 2 and 6, and a median of 3. In contrast, the median Erdős number across all mathematicians (with a finite Erdős number) is 5, with an extreme value of 13. The table below summarizes the Erdős number statistics for [[Nobel Prize|Nobel prize]] laureates in Physics, Chemistry, Medicine and Economics. The first column counts the number of laureates. The second column counts the number of winners with a finite Erdős number. The third column is the percentage of winners with a finite Erdős number. The remaining columns report the minimum, maximum, average and median Erdős numbers among those laureates.
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{{As of|2016}}, all [[Fields Medal]]ists have a finite Erdős number, with values that range between 2 and 6, and a median of 3. In contrast, the median Erdős number across all mathematicians (with a finite Erdős number) is 5, with an extreme value of 13.<ref>{{Cite web|url=http://wwwp.oakland.edu/enp/trivia/|title=Facts about Erdös Numbers and the Collaboration Graph - The Erdös Number Project- Oakland University|website=wwwp.oakland.edu|access-date=2016-10-27}}</ref> The table below summarizes the Erdős number statistics for [[Nobel Prize|Nobel prize]] laureates in Physics, Chemistry, Medicine and Economics.<ref>{{Cite journal|last=López de Prado|first=Marcos|title=Mathematics and Economics: A reality check|journal=The Journal of Portfolio Management|volume=43|issue=1|pages=5–8|doi=10.3905/jpm.2016.43.1.005|year=2016}}</ref> The first column counts the number of laureates. The second column counts the number of winners with a finite Erdős number. The third column is the percentage of winners with a finite Erdős number. The remaining columns report the minimum, maximum, average and median Erdős numbers among those laureates.
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截至2016年，所有费尔兹奖章获得者都有一个有限的埃尔德什数，其值在2到6之间，中位数为3。相反，所有数学家的埃尔德什数的中位数（有限的埃尔德什数）为5，极限值为13。下表总结了物理，化学，医学和经济学方面的诺贝尔奖得主的埃尔德什数统计。第一列计算获奖人数。第二列计算的是具有有限埃尔德什数的获胜者数量。第三列是具有有限埃尔德什数的获胜者的百分比。其余各列表示了这些获奖者中埃尔德什数的最小，最大，平均和中位数。
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截至2016年，所有菲尔兹奖章获得者都有一个有限的埃尔德什数，其值在2到6之间，中位数为3。相反，所有数学家的埃尔德什数的中位数（有限的埃尔德什数）为5，极限值为13。<ref>{{Cite web|url=http://wwwp.oakland.edu/enp/trivia/|title=Facts about Erdös Numbers and the Collaboration Graph - The Erdös Number Project- Oakland University|website=wwwp.oakland.edu|access-date=2016-10-27}}</ref>下表总结了物理，化学，医学和经济学方面的诺贝尔奖得主的埃尔德什数统计。<ref>{{Cite journal|last=López de Prado|first=Marcos|title=Mathematics and Economics: A reality check|journal=The Journal of Portfolio Management|volume=43|issue=1|pages=5–8|doi=10.3905/jpm.2016.43.1.005|year=2016}}</ref>第一列计算获奖人数。第二列计算的是具有有限埃尔德什数的获胜者数量。第三列是具有有限埃尔德什数的获胜者的百分比。其余各列表示了这些获奖者中埃尔德什数的最小，最大，平均和中位数。
    
{| class="wikitable sortable"
 
{| class="wikitable sortable"
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=== Physics 物理领域 ===
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=== 物理领域 ===
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Among the Nobel Prize laureates in Physics, [[Albert Einstein]] and [[Sheldon Lee Glashow]] have an Erdős number of 2. Nobel Laureates with an Erdős number of 3 include [[Enrico Fermi]], [[Otto Stern]], [[Wolfgang Pauli]], [[Max Born]], [[Willis E. Lamb]], [[Eugene Wigner]], [[Richard P. Feynman]], [[Hans A. Bethe]], [[Murray Gell-Mann]], [[Abdus Salam]], [[Steven Weinberg]], [[Norman F. Ramsey]], [[Frank Wilczek]], and [[David Wineland]]. Fields Medal-winning physicist [[Ed Witten]] has an Erdős number of 3.
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Among the Nobel Prize laureates in Physics, [[Albert Einstein]] and [[Sheldon Lee Glashow]] have an Erdős number of 2. Nobel Laureates with an Erdős number of 3 include [[Enrico Fermi]], [[Otto Stern]], [[Wolfgang Pauli]], [[Max Born]], [[Willis E. Lamb]], [[Eugene Wigner]], [[Richard P. Feynman]], [[Hans A. Bethe]], [[Murray Gell-Mann]], [[Abdus Salam]], [[Steven Weinberg]], [[Norman F. Ramsey]], [[Frank Wilczek]], and [[David Wineland]]. Fields Medal-winning physicist [[Ed Witten]] has an Erdős number of 3.<ref name="paths">{{Cite web |title = Some Famous People with Finite Erdős Numbers |url = http://www.oakland.edu/enp/erdpaths/ |publisher = [[Oakland University|oakland.edu]] |access-date = 4 April 2014 }}</ref>
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在诺贝尔物理学奖获得者中，爱因斯坦Albert Einstein和谢尔登·李·格拉肖Sheldon Lee Glashow的埃尔德什数为2。诺贝尔奖获得者中埃尔德什数为3的有: Enrico Fermi，Otto Stern，Wolfgang Pauli，Max Born，Willis E.Lamb，Eugene Wigner，Richard P.Feynman，Hans A.Bethe，Murray Gell-Mann，Abdus Salam，Steven Weinberg，Norman F.Ramsey，Frank Wilczek, and David Wineland。获得菲尔兹奖的物理学家Ed Witten的埃尔德什数为3。
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在诺贝尔物理学奖获得者中，爱因斯坦Albert Einstein和谢尔登·李·格拉肖Sheldon Lee Glashow的埃尔德什数为2。诺贝尔奖获得者中埃尔德什数为3的有: Enrico Fermi，Otto Stern，Wolfgang Pauli，Max Born，Willis E.Lamb，Eugene Wigner，Richard P.Feynman，Hans A.Bethe，Murray Gell-Mann，Abdus Salam，Steven Weinberg，Norman F.Ramsey，Frank Wilczek, and David Wineland。获得菲尔兹奖的物理学家Ed Witten的埃尔德什数为3。<ref name="paths">{{Cite web |title = Some Famous People with Finite Erdős Numbers |url = http://www.oakland.edu/enp/erdpaths/ |publisher = [[Oakland University|oakland.edu]] |access-date = 4 April 2014 }}</ref>
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=== Biology 生物学领域 ===
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=== 生物学领域 ===
    
[[computational biology|Computational biologist]] [[Lior Pachter]] has an Erdős number of 2. [[Evolutionary biology|Evolutionary biologist]] [[Richard Lenski]] has an Erdős number of 3, having co-authored a publication with Lior Pachter and with mathematician [[Bernd Sturmfels]], each of whom has an Erdős number of 2.
 
[[computational biology|Computational biologist]] [[Lior Pachter]] has an Erdős number of 2. [[Evolutionary biology|Evolutionary biologist]] [[Richard Lenski]] has an Erdős number of 3, having co-authored a publication with Lior Pachter and with mathematician [[Bernd Sturmfels]], each of whom has an Erdős number of 2.
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计算生物学家Lior Pachter的埃尔德什数为2。进化生物学家Richard Lenski的埃尔德什数为3，与Lior Pachter和数学家Bernd Sturmfels共同撰写了出版物的每位作者埃尔德什数为2。
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计算生物学家Lior Pachter的埃尔德什数为2。<ref name="erdos2">{{cite web |title=List of all people with Erdos number less than or equal to 2 |url=https://files.oakland.edu/users/grossman/enp/ErdosA.html |work=The Erdös Number Project |publisher=Oakland University |date=14 July 2015 |access-date=25 August 2015}}</ref>进化生物学家Richard Lenski的埃尔德什数为3，与Lior Pachter和数学家Bernd Sturmfels共同撰写了出版物的每位作者埃尔德什数为2。<ref>{{cite web|url=http://telliamedrevisited.wordpress.com/2015/05/28/erdos-with-a-non-kosher-side-of-bacon|title=Erdös with a non-kosher side of Bacon|author=Richard Lenski|date=May 28, 2015}}</ref>
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[[computational biology|Computational biologist]] [[Lior Pachter]] has an Erdős number of 2.<ref name="erdos2">{{cite web |title=List of all people with Erdos number less than or equal to 2 |url=https://files.oakland.edu/users/grossman/enp/ErdosA.html |work=The Erdös Number Project |publisher=Oakland University |date=14 July 2015 |access-date=25 August 2015}}</ref> [[Evolutionary biology|Evolutionary biologist]] [[Richard Lenski]] has an Erdős number of 3, having co-authored a publication with Lior Pachter and with mathematician [[Bernd Sturmfels]], each of whom has an Erdős number of 2.<ref>{{cite web|url=http://telliamedrevisited.wordpress.com/2015/05/28/erdos-with-a-non-kosher-side-of-bacon|title=Erdös with a non-kosher side of Bacon|author=Richard Lenski|date=May 28, 2015}}</ref>
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=== Finance and economics 财经领域 ===
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===财经领域 ===
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=== Philosophy 哲学领域 ===
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=== 哲学领域 ===
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=== Law 法律领域 ===
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=== 法律领域 ===
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=== Politics 政治领域 ===
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=== 政治领域 ===
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=== Engineering 工程领域 ===
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=== 工程领域 ===
    
工程的某些领域，尤其是通信理论和密码学，直接利用了埃尔德什数主要涉及的离散数学。因此，这些领域的从业人员的埃尔德什数低就不足为奇了。例如，加州理工学院电气工程学教授Robert McEliece与埃尔德什本人合作，其埃尔德什数为1。<ref>{{cite journal |author=Erdős, Paul, Robert McEliece, and Herbert Taylor |title=Ramsey bounds for graph products |journal=[[Pacific Journal of Mathematics]] |volume=37 |issue=1 |date=1971 |pages=45–46 |url=https://msp.org/pjm/1971/37-1/pjm-v37-n1-p07-p.pdf |doi=10.2140/pjm.1971.37.45|doi-access=free }}</ref>RSA密码系统的发明者，密码学家Ron Rivest，Adi Shamir和Leonard Adleman的埃尔德什数均为2。<ref name="erdos2"/>
 
工程的某些领域，尤其是通信理论和密码学，直接利用了埃尔德什数主要涉及的离散数学。因此，这些领域的从业人员的埃尔德什数低就不足为奇了。例如，加州理工学院电气工程学教授Robert McEliece与埃尔德什本人合作，其埃尔德什数为1。<ref>{{cite journal |author=Erdős, Paul, Robert McEliece, and Herbert Taylor |title=Ramsey bounds for graph products |journal=[[Pacific Journal of Mathematics]] |volume=37 |issue=1 |date=1971 |pages=45–46 |url=https://msp.org/pjm/1971/37-1/pjm-v37-n1-p07-p.pdf |doi=10.2140/pjm.1971.37.45|doi-access=free }}</ref>RSA密码系统的发明者，密码学家Ron Rivest，Adi Shamir和Leonard Adleman的埃尔德什数均为2。<ref name="erdos2"/>
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==影响 ==
 
==影响 ==
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[[文件:Paul Erdos with Terence Tao.jpg|缩略图|右|1985年，保罗·埃尔德什在阿德莱德大学任教，他的学生陶哲轩（Terence Tao）当时只有10岁。陶后来成为加州大学洛杉矶分校的数学教授，于2006年获得费尔兹奖，并于2007年当选为皇家学会会员。他的埃尔德什数为2。]]
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[[文件:Paul Erdos with Terence Tao.jpg|缩略图|右|1985年，保罗·埃尔德什在阿德莱德大学任教，他的学生陶哲轩（Terence Tao）当时只有10岁。陶后来成为加州大学洛杉矶分校的数学教授，于2006年获得菲尔兹奖，并于2007年当选为皇家学会会员。他的埃尔德什数为2。]]
    
多年以来，埃尔德什数在数学家之间一直盛行。在千年之交的所有在职数学家中，都伴随着一个有限埃尔德什数，数字范围最大为15，中位数为5，平均值为4.65。<ref name="Erdős Number Project"/>几乎每个具有有限埃尔德什数的人其数字都小于8。由于当今科学领域跨学科合作的频率很高，因此许多其他科学领域的大量非数学家也具有有限的埃尔德什数。<ref>{{cite web |url=http://www.oakland.edu/enp/erdpaths/ |title=Some Famous People with Finite Erdős Numbers | first=Jerry | last=Grossman |access-date=1 February 2011}}</ref>例如，政治学家Steven Brams的埃尔德什数为2。在生物医学研究中，统计学家通常是出版物的作者，许多统计学家可以通过John Tukey（其埃尔德什数为2）与埃尔德什链接。同样，著名的遗传学家Eric Lander和数学家Daniel Kleitman在论文上进行了合作，<ref>{{cite journal | pmid = 10582576 | doi=10.1089/106652799318364 | volume=6 | title=A dictionary-based approach for gene annotation | year=1999 | journal=J Comput Biol | pages=419–30 | last1 = Pachter | first1 = L | last2 = Batzoglou | first2 = S | last3 = Spitkovsky | first3 = VI | last4 = Banks | first4 = E | last5 = Lander | first5 = ES | last6 = Kleitman | first6 = DJ | last7 = Berger | first7 = B| issue=3–4 }}</ref><ref>{{cite web|url=http://www-math.mit.edu/~djk/list.html|title=Publications Since 1980 more or less|first=Daniel|last=Kleitman|author-link=Daniel Kleitman|publisher=[[Massachusetts Institute of Technology]]}}</ref>由于Kleitman的埃尔德什数为1，<ref>
 
多年以来，埃尔德什数在数学家之间一直盛行。在千年之交的所有在职数学家中，都伴随着一个有限埃尔德什数，数字范围最大为15，中位数为5，平均值为4.65。<ref name="Erdős Number Project"/>几乎每个具有有限埃尔德什数的人其数字都小于8。由于当今科学领域跨学科合作的频率很高，因此许多其他科学领域的大量非数学家也具有有限的埃尔德什数。<ref>{{cite web |url=http://www.oakland.edu/enp/erdpaths/ |title=Some Famous People with Finite Erdős Numbers | first=Jerry | last=Grossman |access-date=1 February 2011}}</ref>例如，政治学家Steven Brams的埃尔德什数为2。在生物医学研究中，统计学家通常是出版物的作者，许多统计学家可以通过John Tukey（其埃尔德什数为2）与埃尔德什链接。同样，著名的遗传学家Eric Lander和数学家Daniel Kleitman在论文上进行了合作，<ref>{{cite journal | pmid = 10582576 | doi=10.1089/106652799318364 | volume=6 | title=A dictionary-based approach for gene annotation | year=1999 | journal=J Comput Biol | pages=419–30 | last1 = Pachter | first1 = L | last2 = Batzoglou | first2 = S | last3 = Spitkovsky | first3 = VI | last4 = Banks | first4 = E | last5 = Lander | first5 = ES | last6 = Kleitman | first6 = DJ | last7 = Berger | first7 = B| issue=3–4 }}</ref><ref>{{cite web|url=http://www-math.mit.edu/~djk/list.html|title=Publications Since 1980 more or less|first=Daniel|last=Kleitman|author-link=Daniel Kleitman|publisher=[[Massachusetts Institute of Technology]]}}</ref>由于Kleitman的埃尔德什数为1，<ref>
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根据亚历克斯·洛佩兹·奥尔蒂斯Alex Lopez-Ortiz的说法，在1986年至1994年的三个周期中，所有费尔兹奖Fields和内凡琳娜奖Nevanlinna prize得主的埃尔德什数最多为9。
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根据亚历克斯·洛佩兹·奥尔蒂斯Alex Lopez-Ortiz的说法，在1986年至1994年的三个周期中，所有菲尔兹奖Fields和内凡琳娜奖Nevanlinna prize得主的埃尔德什数最多为9。
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== Variations 演变 ==
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== 演变 ==
    
目前出现了很多对该概念进行变型的提议以应用于其他领域。
 
目前出现了很多对该概念进行变型的提议以应用于其他领域。
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在国际象棋中，Morphy number描述了一个棋手与Paul Morphy的联系，Paul Morphy被广泛认为是他那个时代最伟大的棋手，也是非官方的第二位国际象棋国际象棋世界冠军。<ref>{{Cite web|last=Kingston|first=Taylor|title=Your Morphy Number Is Up|url=http://www.chesscafe.com/text/skittles258.pdf|url-status=live|archive-url=https://web.archive.org/web/20060613225534/http://www.chesscafe.com/text/skittles258.pdf|archive-date=13 June 2006|access-date=9 December 2020|website=Chesscafe}}</ref>
 
在国际象棋中，Morphy number描述了一个棋手与Paul Morphy的联系，Paul Morphy被广泛认为是他那个时代最伟大的棋手，也是非官方的第二位国际象棋国际象棋世界冠军。<ref>{{Cite web|last=Kingston|first=Taylor|title=Your Morphy Number Is Up|url=http://www.chesscafe.com/text/skittles258.pdf|url-status=live|archive-url=https://web.archive.org/web/20060613225534/http://www.chesscafe.com/text/skittles258.pdf|archive-date=13 June 2006|access-date=9 December 2020|website=Chesscafe}}</ref>
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== See also 其他参考资料 ==
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== 参见 ==
    
* 科学计量学
 
* 科学计量学
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