The great mathematician David Hilbert at the turn of the 20th century believed that proving Riemann Hypothesis would shed light on many other mathematical mysteries. For these, he mentioned Goldbach’s conjecture and the problem of the existence of infinitely many twin primes. Little did he realize that he completely missed the bull’s-eye. By a true reversal of fortune it will be plausibly indicated that proving Goldbach’s and the twin primes conjectures could definitely decide on the mathematical necessity of Riemann Hypothesis.

All these are being vindicated by the discovery of a sieve of Diophantus. Made possible by the fact that the discoverer is an outsider and could not truly insist that analytical approach to number theory is the only means to an end of a proof for Riemann Hypothesis. On the contrary it is the use of the imaginary landscape that is making all the unnecessary difficulties for prolonging ever finding a proof. As a true believer of the power of the integers, Diophantus of Alexandria expressed his complete trust more than 1700 years ago. He believed that some algebraic equations now known as Diophantine equations could have infinite number of integer solutions and some realistically have no solutions, for example, Fermat’s last theorem for integer power greater than 2. Believing n-degree polynomials should have exactly n solutions led Gauss to prove the fundamental theorem of algebra in 1799. But he used complex numbers. Fortunately, a complex number has a real part and an imaginary part. For simple binomials with only the highest degree term plus the positive constant term (e.g. x+1=0 or x²+1=0 or x³+1=0), only the even degree equations require imaginary solutions while the odd degree equations always have real solutions. But for negative constant term both even and odd degree equations have real solutions. Adding these simple binomials gives a sum as a 3-degree polynomial x³+x²+x+3=0 which in this particular case does not increase the number of real solutions.