数学家传记
维戈·布朗是一位挪威数学家和数论学家。
维戈·布朗的父亲是Soren Markus 布朗（1838-1893），一位炮兵上尉，他的母亲是Lorentze Thaulow 朱利叶斯·佩特森（1842-1890）。现在，从我们刚给出的日期可以看出，布朗的父母在他年幼时就去世了。事实上，他的母亲在他四岁时去世，他的父亲于1893年3月4日去世，当时他七岁。然而，布朗是他父母十个孩子中最小的，他有姐姐们把他抚养长大。他于1903年进入奥斯陆大学，在那里学习数学和自然科学。他所修的课程是为了使他具备成为学校教师的资格，这是一门广泛的课程，要求他学习广泛的主题。它没有留出时间进行多少专门知识的学习。许多处于布朗位置的聪明学生都会阅读数学书籍，以进行比课程中所呈现的更深入的研究。然而，布朗的方法不同，他试图在没有老师或高级教材支持的情况下发展自己的数学思想，因此，他在还是本科生时就产生了一些相当原创的想法。
1910年，布朗前往德国哥廷根大学，这是当时世界领先的数学中心，在那里他开始研究数论中一些最困难的问题。我们应该注意到，布朗访问哥廷根没有得到任何资助，他完全自费资助了这次访问。数论学家埃德蒙·朗道在布朗到达前一年被任命为哥廷根的教授，大卫·希尔伯特和菲利克斯·克莱因也在教职员中。没有证据表明布朗与这些人中的任何一位有过任何有意义的互动，但他一定从听埃德蒙·朗道的课中受益。回到挪威后，布朗确实获得了一笔研究资助来支持他的工作，但在这个阶段他没有工作。1914年第一次世界大战爆发，挪威采取了中立立场。然而，这被证明难以维持，该国受到来自双方的压力。布朗在挪威武装部队中服役了若干年。
他攻击了两个最著名的数论问题，即哥德巴赫猜想和孪生素数猜想。克里斯蒂安·哥德巴赫的猜想是每个大于2的偶数自然数都可以表示为两个素数之和。例如，4 = 2 + 2，6 = 3 + 3，8 = 3 + 5，10 = 5 + 5，……注意，一般来说，偶数可以以不止一种方式表示为两个素数之和，例如10 = 5 + 5 = 3 + 7。孪生素数猜想是存在无限多对素数。例如，3，5；5，7；11，13；17，19；29，31；……布朗的第一个结果发表在Über das Goldbachsche Gesetz und die Anzahl der PrimzahlpaareⓉ（关于克里斯蒂安·哥德巴赫的定理和素数对的数量）（1915年）。在这篇论文中，他开始发展筛法，这将引导他得出一些非常重要的结果。这些筛法是埃拉托色尼筛法的改进，基于容斥原理。它们本质上是初等的，很少有人相信它们会导致重大结果。然而，布朗在这篇论文中引入的思想，由他进一步发展，后来由其他人发展，将导致数论的革命。1919年，布朗发表了一个显著的结果，他证明了孪生素数的倒数和是有限的，即
是有限的。他在论文La série où les dénominateurs sont "nombres premiers jumeaux" est convergent ou finieⓉ（关于级数1/5 + 1/7 +1/11 + 1/13 + 1/17 + 1/19 + 1/29 + 1/31 + 1/41 + 1/43 + 1/59 + 1/61 + ...其中分母是‘孪生素数’是收敛或有限的）（1919年）中证明了这一点，该论文发表在Bulletin des Sciences Mathématiques上。证明使用了今天所谓的‘布朗筛法’。注意，这个结果与素数倒数和是无限的定理形成鲜明对比，后者由莱昂哈德·欧拉在1737年首次证明。现在，人们自然会问的第一个问题是“孪生素数倒数和的近似值是多少”？不幸的是，对于究竟求和的是哪个级数，没有公认的标准。‘布朗常数’最常见的定义是：
而对的最佳估计值，于2002年得出，是1.902160583104。
然而，正如我们从布朗论文的标题中可以看到的，他对自己的级数采取了稍有不同的起点
而其他人则取
省略一个，因为5出现在两个素数对中。当然，这些变体的和很容易与相关联。注意，是有理数还是无理数仍然是一个未解决的问题。然而，已知的是，如果被证明是无理数，那么就会推出存在无穷多个孪生素数。如果被证明是有理数，这对是否存在无穷多个孪生素数没有影响。对素数的研究难道不是一个最引人入胜的课题吗！
1920年，布朗发表了Le crible de Eratosthène et le théorème de GoldbachⓉ（埃拉托色尼筛法和哥德巴赫定理），其中他使用他的筛法证明了克里斯蒂安·哥德巴赫猜想和孪生素数猜想的较弱形式。在这篇论文中，他证明了（i）存在无穷多个整数，使得和都至多有九个素因子；（ii）每个充分大的偶数都是两个数之和，每个数至多有九个素因子。后来的数学家使用基于布朗最初发展的方法加强了这些结果，但这两个猜想仍然未解决。
让我们回到布朗的职业生涯。他于1921年在奥斯陆被任命为应用数学助理。两年后，他被任命为特隆赫姆技术大学的教授。他于1940年与Laura Elise Michelsen（1902-2004）结婚。Laura出生于挪威阿克什胡斯的弗龙，父母是Hjalmar Fredrik Bernhard Michelsen和Eva Gloersen，她在奥斯陆经营一所裁缝学校。我们注意到，她的家乡弗龙与德勒巴克相邻，布朗的父亲曾在德勒巴克买下一栋老房子。布朗和他的妻子Laura [3]：-
……住在德勒巴克一栋古老的木房子里，这栋房子建于1770年，他的父亲于1888年买下。近年来，他和妻子积极帮助保护德勒巴克的那些老房子。他喜欢在森林里散步，在那里他会收集奇特的木块或树根，并用它们制作精美的手工艺品。
1946年，布朗 被任命为奥斯陆大学讲席，任职九年，直到1955年七十岁时退休。
退休后，布朗 出版了两本书，即 The Art of Calculating in Old Norway until the Time of Abel（挪威语）（1962年）和 All is Number, a History of Mathematics from Antiquity to the Renaissance（挪威语）（1964年）。奥斯丁·欧尔 写道[1]：-
近年来，布朗 写了两本关于数学史的书，一本论述挪威的早期历史，另一本论述该学科的总体历史。两本书都非常初等，但作者提出了许多有趣的观察。
布朗 在序言中解释说，在 尼尔斯·阿贝尔 之前，挪威数学：-
……在大多数情况下是一门计算技艺而非数学。我所提到的学者中，不超过四位可以称得上数学家这一称号。
这四位是 Fredrich Christian Holberg Arentz（1736-1825）、Diderich Christian Fester（1732-1811）、Jens Kraft（1720-1756）和 卡斯帕尔·韦塞尔（1745-1818）。
Christoph J Scriba 在一篇评论中写道：-
……作者概述了古挪威和冰岛初等数学的知识。除了带有几何图案的考古发现外，揭示计算技艺知识的最古老文献包含在12世纪和13世纪的《古拉庭法》、《弗罗斯塔庭法》和《国王宝鉴》[《皇家镜子》，约1260年]中。生活在约1300年挪威和冰岛的Hauk Erlendsson编写了一部《算法》，主要基于约翰尼斯·德·萨克罗博斯科的版本。冰岛算术《Rymbegla》被简要讨论。书中还叙述了学校中的算术教学、中世纪及之后挪威学生在国外的学习，以及17和18世纪丹麦和挪威高中和技术学校的建立。
Christoph J Scriba 还评论了布朗的All is Number，写道：-
这本挪威语平装书包含了在奥斯陆大学讲授的数学史讲座的实质内容。史前、埃及、巴比伦、印度和阿拉伯数学的概述占据了前80页，而接下来的100页描述了大约二十多位希腊数学家的生平和工作。其中，伟大的阿基米德得到了最充分的论述（34页）——作者用同样多的页数论述了欧洲中世纪和文艺复兴时期的数学，从波爱修斯到吉罗拉莫·卡尔达诺和洛多维科·费拉里。作者抵制了在有限篇幅内塞入过多事实的诱惑。他对数学细节的呈现与关于所讨论时代数学和数学家的更一般性信息交织在一起。他处理的一些内容不在标准书籍中：挪威的《国王宝鉴》，可能由大主教Einar Gunnarsson撰写，他可能在巴黎与约翰尼斯·德·萨克罗博斯科有过个人接触；Hauk Erlendsson的《算法》[14世纪初]，也深受约翰尼斯·德·萨克罗博斯科的影响；以及冰岛算术《Rymbegla》[12-14世纪？]。
除了这些数学史书籍外，布朗还撰写了多篇关于该主题的论文。例如，他用挪威语写了以下内容：Quadrature of the circle（1941）；The study of the prime numbers from antiquity to our time（1942）；Wallis's and Brouncker's formulas for π（1951）；Niels Henrik Abel（1953）；The manuscript of Abel's Paris treatise found（1953）；（与伯格·耶森合作）A letter by Niels Henrick Abel from his youth（1958）。他还出版了卡尔·斯特默（1957）、卡斯帕尔·韦塞尔（1959）、索菲斯·李（1967）和阿克塞尔·图厄（1977）的传记。布朗感兴趣的另一主题是音乐理论。在Music and ternary continued fractions （1950）中，他讨论了八度划分为等程音程的问题。论文Music and Euclidean algorithms（挪威语）（1961）审视了音乐理论中的四个不同问题。人们可能会想象布朗有音乐才能，但[3]：-
布朗自己称这是命运的讽刺：他非常没有音乐才能，却要写关于音乐的文章。尽管没有音乐天赋，布朗对和谐与几何对称有强烈的感知。他讲授了一些关于数学和美学的课程……
Christoph Scriba 于 1955 年在德国黑森林的 Oberwolfach 数学研究中心举行的一次数学史会议上遇到了布朗。Scriba 这样评价布朗 [3]：-
他广泛的兴趣和人文主义情怀，加上他的谦逊、善良和完全没有教授架子，给我留下了最深刻的印象。……在他漫长而富有成果的一生中，他还参与政治并为和平而努力……
由于他的杰出贡献，布朗获得了许多荣誉。他于 1939 年获得弗里乔夫·南森卓越奖。该奖项以挪威探险家、科学家和外交家弗里乔夫·南森命名，自 1903 年以来一直颁发，上一届获奖者是 1938 年的陶拉尔夫·斯科伦。1946 年，布朗获得了挪威理工学院创始人奖，1958 年他获得了挪威皇家科学与文学院的 Gunnerus 奖章。该奖章以挪威皇家科学院的创始人 Johan Ernst Gunnerus 命名。布朗还于 1966 年获得了汉堡大学的荣誉博士学位。他当选为奥斯陆、特隆赫姆、乌普萨拉和芬兰科学院的科学学会或科学院成员。
Viggo Brun's father was Soren Markus Brun (1838-1893), an artillery captain, and his mother was Lorentze Thaulow Petersen (1842-1890). Now, from the dates we have just given we can see that Viggo's parents died when he was young. In fact his mother died when he was four and his father died on 4 March 1893 when he was seven. However, Viggo was the youngest of his parents' ten children and he had older sisters who brought him up. He entered the University of Oslo in 1903 where he studied mathematics and natural sciences. The course he took was to qualify him to become a school teacher and it was a broad course requiring him to study a wide range of topics. It did not allow time for much in the way of specialised knowledge. Many bright pupils in Brun's position would have read mathematics books to take them into deeper studies than those being presented in their courses. Brun's approach was, however, different and he tried to develop his own mathematical ideas without having the support of teachers or advanced texts and, as a consequence, he produced some rather original ideas while still an undergraduate.
In 1910 Brun went to Göttingen University in Germany, the leading mathematics centre in the world at this time, and while he was there he began to work on what were some of the most difficult problems in number theory. We should note that Brun received no financial support for his visit to Göttingen and he funded the visit entirely from his own funds. The number theorist Edmund Landau had been appointed to a professorship at Göttingen a year before Brun arrived and Hilbert and Klein were also on the staff. There is no evidence that Brun interacted in any meaningful way with any of these, but he must have benefited from listening to Edmund Landau. Returning to Norway, Brun did receive a research grant to support his work but at this stage he had no job. The outbreak of World War I in 1914 saw Norway adopt a position of neutrality. This, however, proved difficult to maintain and the country was under pressure from both sides. Brun served for a number of years in the Norwegian armed forces.
He had attacked two of the most famous number theory problems, namely Goldbach's conjecture and the twin prime conjecture. Goldbach's conjecture is that every even natural number greater than 2 can be expressed as the sum of two primes. For example, 4 = 2 + 2, 6 = 3 + 3, 8 = 3 + 5, 10 = 5 + 5, ... Note that, in general, even numbers will be expressible as the sum of two primes in more than one way, for example 10 = 5 + 5 = 3 + 7. The twin prime conjecture is that there are infinitely many prime pairs . For example, 3, 5; 5, 7; 11, 13; 17, 19; 29, 31; ... Brun's first results were given in Über das Goldbachsche Gesetz und die Anzahl der Primzahlpaare Ⓣ (1915). In this paper he began to develop the sieve methods which would lead him to some very important results. These sieve methods were refinements, based on the inclusion-exclusion principle, of the sieve of Eratosthenes. They are essentially elementary in nature and few believed that they would lead to significant results. However, the ideas that Brun introduced in this paper, further developed by him and later by others, would lead to a revolution in number theory. In 1919 Brun published a remarkable result when he proved that the sum of the reciprocals of twin primes is finite, that is
is finite. He proved this in the paper La série où les dénominateurs sont "nombres premiers jumeaux" est convergent ou finie Ⓣ (1919) which he published in the Bulletin des Sciences Mathématiques. The proof uses what today is called 'Brun's sieve'. Note that this result is in sharp contrast with the theorem that the sum of the reciprocals of the primes is infinite, first proved by Leonhard Euler in 1737. Now the first question one would naturally ask is "What is the approximate value of the sum of the reciprocals of twin primes"? Unfortunately there is no accepted standard for exactly what series is being summed. The most common definition of 'Brun's constant' is:
and the best estimate for , produced in 2002, is 1.902160583104.
However, as we can see from the title of Brun's paper, he took a slightly different start to his series
while others take
omitting one since 5 appears in two prime pairs. Of course, the sums of these variants are easily related to . Note that it is still an open question whether is rational or irrational. It is known, however, that if were proved irrational it would follow that there are infinitely many twin primes. If were proved rational it would have no bearing on whether there were infinitely many twin primes. Isn't a study of the primes a most fascinating topic!
In 1920 Brun published Le crible de Eratosthène et le théorème de Goldbach Ⓣ in which he used his sieve methods to prove weaker forms of both the Goldbach conjecture and the twin prime conjecture. In this paper he proved (i) there exist infinitely many integers such that both and have at most nine prime factors; and (ii) every sufficiently large even integer is the sum of two numbers each having at most nine prime factors. Later mathematicians have strengthened these results, using methods based on those first developed by Brun, but the two conjectures are still open.
Let us return to Brun's career. He was appointed as an assistant in applied mathematics in Oslo in 1921. Two years later, he was appointed as a professor at the Technical University of Trondheim. He married Laura Elise Michelsen (1902-2004) in 1940. Laura, who was born in Frogn, Akershus, Norway, to parents Hjalmar Fredrik Bernhard Michelsen and Eva Gloersen, ran a dressmaking school in Oslo. We note that her home town of Frogn was adjacent to Drobak where Brun's father had bought an old house. Viggo and his wife Laura [3]:-
... lived in an ancient wooden house in Drobak, which had been built in 1770 and which his father had bought in 1888. In recent years he and his wife were active in helping to preserve those old houses in Drobak. He loved to go for walks in the forest where he would collect curious pieces of wood or roots from which he produced nice handiwork.
In 1946 Brun was appointed to a chair at the University of Oslo which he occupied for nine years until he retired in 1955 at the age of seventy.
Brun published two books after he retired, namely The Art of Calculating in Old Norway until the Time of Abel (Norwegian) (1962) and All is Number, a History of Mathematics from Antiquity to the Renaissance (Norwegian) (1964). Øystein Ore writes [1]:-
Viggo Brun has in recent years written two books on the history of mathematics, one dealing with the early history in Norway, the other with the general story of the subject. Both are on a very elementary level, but with many interesting observations by the author.
Brun explains in his Preface that, before Niels Abel, Norwegian mathematics:-
... was in most cases an art of calculating rather than mathematics. No more than four of the scholars I mention could lay claim to the title of mathematician.
These four are Fredrich Christian Holberg Arentz (1736-1825), Diderich Christian Fester (1732-1811), Jens Kraft (1720-1756) and Caspar Wessel (1745-1818).
Christoph J Scriba, writes in a review that:-
... the author gives a summary of the knowledge in elementary mathematics in old Norway and Iceland. Apart from archaeological finds bearing geometric designs, the oldest sources revealing knowledge in the art of calculating are contained in the Gulating Law, the Frostating Law and the Kongespeilet [The royal mirror, c. 1260], from the 12th and 13th centuries. Hauk Erlendsson, who lived in Norway and Iceland around 1300, composed an "Algorismus" mainly based on that of Sacrobosco. The Icelandic arithmetic "Rymbegla" is briefly discussed. There is also an account of the teaching of arithmetic in schools, the study of Norwegian students abroad in the Middle Ages and later on, and the establishment of high and technical schools in Denmark and Norway in the 17th and 18th centuries.
Christoph J Scriba also reviewed Brun's All is Number, writing:-
This paperback in the Norwegian language contains the substance of lectures on the history of mathematics which have been given at the University of Oslo. A summary of pre-historic, Egyptian, Babylonian, Hindu and Arabic mathematics fills the first 80 pages, while on the next 100 pages the life and work of about two dozen Greek mathematicians is described. Of these, the great Archimedes receives the fullest treatment (34 pages) - the same number of pages in which the author deals with medieval and Renaissance mathematics in Europe, from Boethius to Cardano and Ferrari. The author has resisted the temptation to crowd too many facts into the limited space. His presentation of mathematical details is intermingled with information of a more general nature concerning mathematics and mathematicians of the times under discussion. Some items treated by him are not contained in the standard books: The Norwegian "Kongespeilet", perhaps composed by Archbishop Einar Gunnarsson, who may have been in personal contact with Sacrobosco at Paris; the "Algorismus" of Hauk Erlendsson [early 14th century], which is strongly influenced by Sacrobosco, too; and the Icelandic arithmetic "Rymbegla" [12th -14th centuries?].
In addition to these books on the history of mathematics, Brun wrote many papers on the subject. For example, he wrote the following in Norwegian: Quadrature of the circle (1941); The study of the prime numbers from antiquity to our time (1942); Wallis's and Brouncker's formulas for π (1951); Niels Henrik Abel (1953); The manuscript of Abel's Paris treatise found (1953); (with Borge Jessen) A letter by Niels Henrick Abel from his youth (1958). He also published biographies of Carl Stormer (1957), Caspar Wessel (1959), Sophus Lie (1967), and Axel Thue (1977). Another topic that interested Brun was the theory of music. In Music and ternary continued fractions (1950) he discussed the division of the octave into equally tempered intervals. The paper Music and Euclidean algorithms (Norwegian) (1961) looks at four different problems from the theory of music. One might imagine that Brun was musical but [3]:-
Brun himself called it an irony of fate that he, who was very unmusical, should write about music. Though not musically gifted, Viggo Brun had a strong sense for harmony and geometric symmetry. He gave some lectures about mathematics and aesthetics ...
Christoph Scriba met Brun in 1955 at a conference on the history of mathematics at the Oberwolfach mathematics research centre in the Black Forest, Germany. Scriba says of Brun [3]:-
His wide interests and his humanistic outlook, combined with his modesty, kindness, and totally un-professorial habits made the strongest impression upon me. ... During his long and fruitful life, he also engaged in politics and worked for peace ...
Many honours were given to Brun for his outstanding contributions. He was awarded the Fridtjof Nansen Award for Excellence in 1939. This award, named after the Norwegian explorer, scientist, and diplomat Fridtjof Nansen, had been awarded since 1903 and the previous winner, in 1938, had been Thoralf Albert Skolem. In 1946 Brun was awarded the Norwegian Institute of Technology Founder's Prize, and in 1958 he received the Gunnerus medal of the Royal Norwegian Academy of Science and Letters. This medal is named after Johan Ernst Gunnerus, the founder of the Royal Norwegian Academy. Brun also received an honorary doctorate from the University of Hamburg in 1966. He was elected a member of the scientific societies or academies of Oslo, Trondheim, Uppsala, and the Finnish Academy of Sciences.
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