section_path: Scientific work/Algebra and number theory/Further investigations
section_name: Further investigations
paragraph_text: 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 integral logarithm.
  
  When Olbers encouraged Gauss in 1816 to compete for a prize from the French Academy on proof for Fermat's Last Theorem (FLT), he refused because of his low esteem on this matter. However, among his left works a short undated paper was found with proofs of FLT for the cases n = 3 and n = 5. 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.
  
  Gauss contributed to solving the Kepler conjecture in 1831 with the proof that a greatest packing density of spheres in the three-dimensional space is given when the centers of the spheres form a cubic face-centered arrangement, when he reviewed a book of Ludwig August Seeber on the theory of reduction of positive ternary quadratic forms. 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.
  
  In two papers on biquadratic residues (1828, 1832) Gauss introduced the ring of Gaussian integers Z {\displaystyle \mathbb {Z} }, showed that it is a unique factorization domain. and generalized some key arithmetic concepts, such as Fermat's little theorem and Gauss's lemma. The main objective of introducing this ring was to formulate the law of biquadratic reciprocity – as Gauss discovered, rings of complex integers are the natural setting for such higher reciprocity laws.
  
  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 sums) can be applied to prove higher reciprocity laws.