Editing Carl Friedrich Gauss (section)

==== Further investigations ====
One of Gauss's first results was the empirically found conjecture of 1792 – the later called [[prime number theorem]] – giving an estimation of the number of prime numbers by using the [[Logarithmic integral function|integral logarithm]].<ref>{{cite book | last1 = Koch | first1 = H. | author-link1 = Herbert Koch | last2 = Pieper | first2 = H. | title = Zahlentheorie | publisher = VEB Deutscher Verlag der Wissenschaften | place = Berlin | date = 1976 | pages = 6, 124}}</ref>{{efn|Gauss told the story later in detail in a letter to [[Johann Franz Encke|Encke]].{{sfn|Bachmann|1922|p=4}}}}

In 1816, [[Heinrich Wilhelm Matthias Olbers|Olbers]] encouraged Gauss to compete for a prize from the French Academy for a proof for [[Fermat's Last Theorem]]; he refused, considering the topic uninteresting. However, after his death a short undated paper was found with proofs of the theorem for the cases ''n'' = 3 and ''n'' = 5.<ref>{{cite journal | last1 = Kleiner | first1 = I. | year = 2000 | title = From Fermat to Wiles: Fermat's Last Theorem Becomes a Theorem | journal = [[Elemente der Mathematik]] | volume = 55 | pages = 19–37 | url = http://math.stanford.edu/~lekheng/flt/kleiner.pdf | doi = 10.1007/PL00000079
 | s2cid = 53319514 | url-status = dead | archive-url = https://web.archive.org/web/20110608052614/http://math.stanford.edu/~lekheng/flt/kleiner.pdf | archive-date = 8 June 2011}}</ref> The particular case of ''n'' = 3 was proved much earlier by [[Leonhard Euler]], but Gauss developed a more streamlined proof which made use of [[Eisenstein integers]]; though more general, the proof was simpler than in the real integers case.{{sfn|Bachmann|1922|pp= 60&ndash;61}}

Gauss contributed to solving the [[Kepler conjecture]] in 1831 with the proof that a [[Close-packing of equal spheres|greatest packing density]] of spheres in the three-dimensional space is given when the centres of the spheres form a [[Cubic crystal system|cubic face-centred]] arrangement,<ref>{{Cite journal | last = Hales | first = Thomas C. | author-link = Thomas Callister Hales | title = Historical overview of the Kepler conjecture | doi = 10.1007/s00454-005-1210-2 | mr = 2229657 | year = 2006 | journal = [[Discrete & Computational Geometry]] | issn = 0179-5376 | volume=36 | issue = 1 | pages = 5–20| doi-access = free}}</ref> when he reviewed a book of [[Ludwig August Seeber]] on the theory of reduction of positive ternary quadratic forms.<ref>{{Cite book | last = Seeber | first = Ludwig August | year = 1831 | title = Untersuchungen über die Eigenschaften der positiven ternaeren quadratischen Formen | place = Mannheim  | url = https://books.google.com/books?id=QKJGAAAAcAAJ}}</ref> Having noticed some lacks in Seeber's proof, he simplified many of his arguments, proved the central conjecture, and remarked that this theorem is equivalent to the Kepler conjecture for regular arrangements.<ref>{{cite journal | date = July 1831 | title = Untersuchungen über die Eigenschaften der positiven ternären quadratischen Formen von Ludwig August Seeber | url = https://babel.hathitrust.org/cgi/pt?id=mdp.39015064427944&seq=387 | journal = [[Göttingische gelehrte Anzeigen]] | issue = 108 | pages = 1065–1077}}</ref>

In  two papers on [[Quartic reciprocity|biquadratic residue]]s (1828, 1832) Gauss introduced the [[ring theory|ring]] of [[Gaussian integers]] <math>\mathbb{Z}[i]</math>, showed that it is a [[unique factorization domain]],<ref name="Kleiner1998">{{cite journal | url = https://ems.press/journals/em/articles/664 | title = From Numbers to Rings: The Early History of Ring Theory | first1 = Israel | last1 = Kleiner | author-link = Israel Kleiner (mathematician) | journal = [[Elemente der Mathematik]] | volume = 53 | number = 1 | doi = 10.1007/s000170050029 | year = 1998 | pages = 18–35 | zbl = 0908.16001 | doi-access = free}}</ref> and generalized some key arithmetic concepts, such as [[Fermat's little theorem]] and [[Gauss's lemma (number theory)|Gauss's lemma]]. The main objective of introducing this ring was to formulate the law of biquadratic reciprocity<ref name="Kleiner1998" /> – as Gauss discovered, rings of complex integers are the natural setting for such higher reciprocity laws.<ref>{{cite book | last1 = Lemmermeyer | first1 = Franz | title = Reciprocity Laws: from Euler to Eisenstein | series = Springer Monographs in Mathematics | publisher = Springer | place = Berlin | year = 2000 | page = 15 | isbn = 3-540-66957-4 | doi= 10.1007/978-3-662-12893-0}}</ref>

In the second paper, he stated the general law of biquadratic reciprocity and proved several special cases of it. In an earlier publication from 1818 containing his fifth and sixth proofs of quadratic reciprocity, he claimed the techniques of these proofs ([[Gauss sum]]s) can be applied to prove higher reciprocity laws.{{sfn|Bachmann|1922|pp= 52, 57&ndash;59}}