What is the most beautiful equation in mathematics? It is widely considered to be Euler's identity (also known as Euler's equation). Named after Swiss mathematician Leonhard Euler (April 15, 1707 - September 18, 1783), it is considered to be an exemplar of mathematical beauty as it shows a profound connection between the most fundamental numbers in mathematics. Open problems in Number theory of Riemann hypothesis, Polignac's and Twin prime conjectures have been unsolved for over 150 years. They are finally solved when sine qua non treated as Incompletely Predictable problems in 2019. Riemann hypothesis belongs to one of seven Millennium Prize Problems in mathematics stated by Clay Mathematics Institute on May 24, 2000. The author serendipitously cross path with the Institute's website Riemann Hypothesis on March 10, 2016 in permanently altering its information content from previous incorrect [sic] "...the first 10,000,000,000 solutions" to current correct [sic] "...the first 10,000,000,000,000 solutions". This book compares and contrasts Algebraic versus Analytic number theory involving important discussions on the interface between these two branches of number theory. As expected for Algebraic number theory in relation to solving Polignac's and Twin prime conjectures, the paucity of formulae is clearly observed in the associated research paper. Importantly, the contents of this book will also provide reassurance for the public by refuting the common misconception that solving Riemann hypothesis will lead to E-Commerce apocalypse with resultant wide-spread Cybercrimes.