The key is: how to prove the solution?

Carl Friedrich Gauss (1777-1855)

Prime numbers from Gauss to Riemann. And from RSA encryption to quantum computers that can break any code in the world. 

Carl Friedrich Gauss (1777-1855)(1) was a mathematician who spent a couple of hours each day trying to create prime numbers. His student Bernhard Riemann  (1826-1866) (2) created his "Riemann hypothesis" (3) that is also known as Riemann's conjecture that used to create the series of prime numbers. The question is where Gauss and Riemann needed those prime numbers? 

Those two famous mathematicians were not the first persons who were try to create so many prime numbers as they could. There were many people like Stanislaw "Stan" Ulam (1909-1984) (4). That man is better known for his work with Edward Teller and the hydrogen bomb. Who tried to create the geometrical model of how the prime numbers divide into spiral structures. That model is called Ulam's spiral(5). 

Even if the answer for Riemann's conjecture is 1/2 there is the possibility to chain the algorithms. The thing is that in point 1/2 the Riemann's conjecture is giving non-trivial zeros. If there are points where the answer is not a prime number or the answers are turning easy to predict that is the end of the use of pure Riemann's conjecture in cryptography. But that conjecture can connect with other mathematical formulas. 

So when we are going to Riemann's conjecture that is giving prime numbers all the time that the formula is driven there is also the possibility that there are prime numbers also outside the series that the Riemann's conjecture is giving. When the series of the prime number ends is unknown. That thing is possible because when the numbers are turning bigger. And there might be some prime numbers between the answers generated by using Riemann's conjecture. 

The sequence between those solutions is turning longer. And that means there is the possibility. That Riemann's conjecture is leaving some prime numbers away from the series that the formula is giving. The thing that makes that formula so impressive, interesting and mysterious is, where Riemann or Gauss needed those numbers. The modern RSA encryption requires Riemann's conjecture. 

If Riemann's conjecture is used purely. That makes secrecy quite easy to break. Purely used conjecture is easy to break simply by using fast computers that are calculating prime numbers. And then the system can simply try every prime number to the captured message. That is called a brute force attack. But if Riemann's conjecture is connected with other mathematical algorithms and functions. That thing makes the algorithm safer and harder to break. 

(1) https://en.wikipedia.org/wiki/Carl_Friedrich_Gauss

(2) https://en.wikipedia.org/wiki/Bernhard_Riemann

(3)https://en.wikipedia.org/wiki/Riemann_hypothesis

(4)https://en.wikipedia.org/wiki/Stanislaw_Ulam

(5) https://en.wikipedia.org/wiki/Ulam_spiral

Image: https://en.wikipedia.org/wiki/Carl_Friedrich_Gauss

And finally, what people should do when they try to solve mathematical problems? 

The key element in mathematical problems is to show people how to get the answer? How to prove that some solution is right or some solution is wrong. Must be done by using a methodology that is accepted in mathematics. 

That thing means that only the answer is not enough. Proving the solution is the key. And the thing that is making that thing. Is that every stage of the calculation must write to the paper. The idea is that the inspector can retake those calculations. And the solution should be the original numbers. 

When some person is making those calculations. The thing is that every part of the formula must make exactly correctly. If something is left outside the mark the answer is always wrong. And the thing is that if the person makes a little bit too much work. 

That thing gives better or correct answers than just removing the "unnecessary parts". Like some "meanless" square root marks from the calculations. In formulas or algorithms is not unnecessary marks. 

And the thing is that proving the thing is the key element. If somebody presses some button unnecessarily that thing is not a mistake. The mistake is that if some mark is lost. And that thing causes horrible errors in the calculation. In mathematics is many things that are sure or they are not sure. But there are also unsolved answers.