Abstract
The denominators of the Bernoulli numbers can be completely determined. This is due to Clausen [26] and von Staudt [96]. More precisely, the “fractional part” of B n is given by the following theorem. This result gives a foundation for studying p-adic properties of the Bernoulli numbers. It also plays a fundamental role in the theory of p-adic modular forms through the Eisenstein series [82].
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Notes
- 1.
Thomas Clausen (born on January 16, 1801 in Snogebaek, Denmark—died on May 23, 1885 in Dorpat, Russia (now Tartu, Estonia)).
- 2.
Karl Georg Christian von Staudt (born on January 24, 1798 in Imperial Free City of Rothenburg (now Rothenburg ob der Tauber, Germany)—died on June 1, 1867 in Erlangen, Bavaria (now Germany).
- 3.
Ferdinand Gotthold Max Eisenstein (born on April 16, 1823 in Berlin, Germany—died on October 11, 1852 in Berlin, Germany).
- 4.
Throughout the book, the term “natural number” means a positive integer.
- 5.
Pierre de Fermat (born on August 17, 1601 in Beaumont-de-Lomagne, France—died on January 12, 1665 in Castres, France).
- 6.
Ernst Eduard Kummer (born on January 29, 1810 in Sorau, Brandenburg, Prussia (now Germany)—died on May 14, 1893 in Berlin, Germany).
- 7.
- 8.
Heinrich Christian Schumacher (born on September 3, 1780 in Bramstedt, Germany—died in 1850). His first degree was in law, and after that he studied astronomy under Gauss and became an astronomer. He launched the journal Astronomische Nachrichten, which is famous for the contributions of Abel and Jacobi on the theory of elliptic functions, as the managing editor. Incidentally, Abel once met Clausen during his stay in Hamburg (cf. [19]).
- 9.
Joseph von Fraunhofer (born on March 6, 1787 in Straubing, Germany—died on June 7, 1826 in Munich, Germany).
- 10.
Friedrich Wilhelm Bessel (born on July 22, 1784 in Minden, Westphalia (now Germany)—died on March 17, 1846 in Königsberg, Prussia (now Kaliningrad, Russia)), an astronomer, who is famous for the Bessel function.
- 11.
Johann Carl Friedrich Gauss (born on April 30, 1777 in Brunswick, Duchy of Brunswick (now Germany)—died on February 23, 1855 in Göttingen, Hanover (now Germany)).
- 12.
\(_{2}F_{1}(\alpha,\beta,\alpha +\beta + \frac{1} {2}; x)^{2} = _{3}F_{2}{\biggl ({ 2\alpha,2\beta,\alpha +\beta \atop \alpha +\beta +\frac{1} {2},2\alpha +2\beta }; x\biggr )}\)
- 13.
It was correct up to 248 decimal places. This was the world record from 1847 to 1853. For a history of computation of π, see for example [13].
- 14.
Fermat conjectured that all the numbers of the form \(2^{2^{n} } + 1\) (which are called Fermat numbers) are prime. This is true for n ≤ 4, but Euler showed in 1732 that this is not the case for n = 5 by giving the factorization. As of November 2013, it is known that every Fermat number \(2^{2^{n} } + 1\) with 5 ≤ n ≤ 32 is not prime. (Complete factorization is known up to n = 11.)
- 15.
Max Noether (born on September 24, 1844 in Mannheim, Germany—died on December 13, 1921 in Erlangen, Germany).
- 16.
Under the same title “De numeris Bernoullianis”, Erlangen, 1845.
- 17.
Heinrich Ferdinand Scherk (born on October 27, 1798 in Poznan, Poland—died on October 4, 1885 in Bremen, Germany), he also wrote a paper on Bernoulli numbers [81].
- 18.
“On expansions of powers of cosine and sine by cosine and sine with their arguments multiplied.”
- 19.
Ferdinand Joachimsthal (born on March 9, 1818 in Goldberg, Prussian Silesia (now Zlotoryja, Poland)—died on April 5, 1861 in Breslau, Germany (now Wroclaw, Poland)).
- 20.
Johann Peter Gustav Lejeune Dirichlet (born on February 13, 1805 in Düren, French Empire (now Germany)—died on May 5, 1859 in Göttingen, Hanover (now Germany)).
- 21.
Dirichlet’s wife Rebecca Mendelssohn is a younger sister of the composer Felix Mendelssohn .
- 22.
Lazarus Immanuel Fuchs (born on May 5, 1833 in Moschin, Prussia (now Poznan, Poland)—died on April 26, 1902 in Berlin, Germany).
- 23.
Paul David Gustav du Bois-Reymond (born on December 2, 1831 in Berlin, Germany—died on April 7, 1889 in Freiburg, Germany).
- 24.
Paul Albert Gordan (born on April 27, 1837 in Breslau, Germany (now Wroclaw, Poland)—died on December 21, 1912 in Erlangen, Germany).
- 25.
Paul Gustav Heinrich Bachmann (born on June 22, 1837 in Berlin, Germany—died on March 31, 1920 in Weimar, Germany).
- 26.
Karl Herman Amandus Schwarz (born on January 25, 1843 in Hermsdorf, Silesia (now Poland)—died on November 30, 1921 in Berlin, Germany).
- 27.
Georg Ferdinand Ludwig Philipp Cantor (born on March 3, 1845 in St. Petersburg, Russia—died on January 6, 1918 in Halle, Germany).
- 28.
Arthur Moritz Schönflies (born on April 17, 1853 in Landsberg an der Warthe, Germany (now Gorzów, Poland)—died on May 27, 1928 in Frankfurt am Main, Germany).
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Ibukiyama, T., Kaneko, M. (2014). Theorem of Clausen and von Staudt, and Kummer’s Congruence. In: Bernoulli Numbers and Zeta Functions. Springer Monographs in Mathematics. Springer, Tokyo. https://doi.org/10.1007/978-4-431-54919-2_3
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