In this monograph, by using the geometric method, the author proves nine unsolved world's baffling mathematical problems in the number theory field (contains Fermat's Last Theorem, Goldbach Conjecture, Twin Primes Conjecture, Proposition-C, $n DEGREES2+1$ Conjecture, Riemann Hypothesis, Generalized Riemann Hypothesis, Hecke Conjecture and Euler's Constant $gamma$). Many mathematicians and amateurs accepted the challenge of but so far inspite of repeated attempt no convincing proofs have been achieved. The author thinks that it seems the availablemathematical tools and methods are not enough...