David Hilbert was a famous mathematician. He was born on January 23, 1862 in the Province of Prussia and died on February 14, 1943. He received a doctorate in 1885 from the University of Koenigsberg. He was a member of the university staff from 1886 to 1895, becoming a full professor in 1893. He made friends with Minkowski and Hurwitz, who were influential to his ideas in mathematics.
Hilbert proved a basis theory in invariant theory in 1888. He improved on the work of Kummer, Kronecker, and Dedekind in a paper named ‘Zahlbericht’ in 1897. It started the current field of class field theory. He published twenty-one axioms based on the works of Euclidean geometry discovered by Euclid in about 300 BC. He made contributions to number theory, mathematical logic, differential equations, and the Three Body Problem. He proved Waring’s Theorem.
Hilbert presented ten unsolved problems at the Second International Congress of Mathematicians in Paris on August 8, 1900. There are a total of twenty-three he submitted later in print. Some of these problems have not been solved. The eighth problem on the list is the famous Riemann’s Hypothesis. Clay Institute included it in its list of seven unsolved problems on May 24, 2000, all of which have a one million dollar prize for the solution or refutation.
Hilbert’s eighth problem included the Riemann Hypothesis, Goldbach Conjecture, and Twin Primes Conjecture. The Goldbach conjecture claims that any even integer greater than two can be expressed as the sum of two prime numbers. Examples are 2+2=4, 5+7=12, 13+23=36, etc. It has been verified by computer up to millions of digits. The Twin Prime Conjecture claims that the twin primes are infinite. A twin prime is two consecutive odd primes. Examples are 3, 5; 11, 13; and 17, 19. The Riemann Hypothesis was presented in a lecture by Bernhard Riemann in 1859. It requires a little more knowledge of mathematics. It claims that the zeros of the Riemann Zeta Function all must lie on the “critical line”, which has all real parts equal to one half.
The first problem is only partially solved. It is “Cantor’s problem of the cardinal number of the continuum”. The second problem has not been proven to date. It is the “compatibility of the arithmetical axioms”. The third problem has been solved. It is “give two tetrahedra that cannot be decomposed into congruent tetrahedra directly or by adjoining congruent tetrahedra.” The fourth problem has been solved. It is about Euclidean geometry and the parallel postulate. The fifth problem is only partially solved. It asks about the differentiability of functions. The sixth problem is the question “can physics be axiomatized?” It has not been proven to date. The seventh problem has only been partially solved. It is about transcendental numbers.
The aforementioned eighth problem is the unsolved Riemann Hypothesis. The ninth problem has only been partially solved.
It is “construct generalizations of the reciprocity theorem of number theory.” The tenth problem has been solved. It is about Diophantine Equations. The eleventh problem has been solved. It extends the results for quadratic fields to arbitrary integer algebraic fields. The twelfth problem has not been solved. It asks if a theorem by Kronecker can be extended to arbitrary algebraic fields. The thirteenth problem has been solved. It shows the impossibility of solving a seventh degree general equation by functions of two variables. The fourteenth problem has been solved. A refutation to the finiteness of systems of relatively integral functions was found.
The fifteenth problem has been partially solved. It asks for Schubert’s enumerative geometry to be justified. The sixteenth problem has not been solved. It asks for the study of the topology of real algebraic curves and surfaces. The seventeenth problem has been solved. It asks for the representation of definite form by squares. The eighteenth problem has been solved. It asks for a building of spaces using congruent polyhedral.
The nineteenth problem has been solved. It asks for an analysis of the analytic character of solutions to variational problems. The twentieth problem has been solved. It asks for a solution to general boundary problems. The twenty-first problem has been solved. It was found correct sometimes and incorrect others depending on a more exact explanation of the problem. It asks for differential equations to be solved using a monodromy group. The twenty-second problem has been solved. It is called uniformization. The twenty-third problem has not been solved. It asks for the methods of the calculus of variables to be extended.