Future of number theory

In summary, If Riemann's Hypothesis is proved as true, it would not cause number theory to collapse. This is because number theory is mostly based on prime numbers, and proving the hypothesis true would simply allow us to predict the pattern of prime numbers. In fact, it would lead to even more questions and discoveries in the field of mathematics. This is similar to how in science, one answer often leads to many more questions and advancements. This can be seen in the example of physics, where what was once thought to be the end of the field in the 19th century eventually gave rise to even more complex and fascinating theories in the form of Quantum and Relativistic Physics. Therefore, the confirmation or refutation of the R
  • #1
l-1j-cho
104
0
If Riemann's Hypothesis is proved as true, would number theory collapse?
 
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  • #2
No. Why would you think that?
 
  • #3
well, I have absolutely no idea what Riemann hypothesis is, but apparently, if it is proved as true, then we can predict the pattern of prime numbers, of which number theory mostly consists of prime numbers
 
  • #4
In science, when you get an answer, you get also several new questions.
At the end of 19th century, physicists thought that Physics was nearly terminated.
All the relevant questions had been yet studied and understood.
It only left some "secondary" aspects that would be quickly dispatched...
These secondary points gave birth to both Quantum and Relativistic Physics..

If Riemann's Hypothesis is confirmed, thousands of more interesting questions will arise.
(And if it is refuted, then mathematicians will have a lot of work to do !)

That's why science is so wonderful...
 
  • #5


I cannot predict the future of number theory with certainty. However, if Riemann's Hypothesis is proven to be true, it would certainly have a significant impact on the field of number theory. It would provide a deeper understanding of the distribution of prime numbers and potentially lead to the development of new mathematical techniques and theories.

It is unlikely that number theory would collapse if Riemann's Hypothesis is proven to be true. Number theory is a vast and complex field that encompasses many other areas of mathematics, such as algebra and geometry. Even if one conjecture is proven, there will always be new questions and problems to explore and solve.

In fact, the proof of Riemann's Hypothesis would open up new avenues for research and could potentially lead to the discovery of other important conjectures and theorems. It would also have practical applications in fields such as cryptography and computer science.

Therefore, while the proof of Riemann's Hypothesis would certainly be a monumental achievement, it would not mark the end of number theory. It would instead pave the way for further advancements and discoveries in this fascinating field of mathematics.
 

1. What is the current state of number theory research?

The current state of number theory research is very active and diverse. There are many different areas of study within number theory, including analytic number theory, algebraic number theory, and computational number theory. Researchers are constantly making new discoveries and building upon existing theories.

2. What advancements can we expect in the future of number theory?

In the future, we can expect advancements in the use of computational techniques to solve long-standing problems in number theory. Additionally, there may be new breakthroughs in understanding the connections between different areas of number theory, such as the link between analytic and algebraic number theory.

3. How does number theory relate to other branches of mathematics?

Number theory has connections to many other branches of mathematics, including algebra, geometry, and analysis. In particular, it has applications in cryptography, coding theory, and physics. Many mathematicians believe that number theory is the foundation of all mathematics.

4. What impact will technology have on the future of number theory?

Technology, especially advancements in computing power, will have a significant impact on the future of number theory. It will allow for faster and more accurate calculations, which can aid in solving complex problems and testing new theories. Additionally, technology has enabled mathematicians to collaborate and share ideas more easily, leading to further progress in the field.

5. How can the future of number theory benefit society?

The future of number theory has the potential to benefit society in many ways. As number theory has applications in cryptography and coding theory, advancements can lead to stronger and more secure communication systems. Additionally, a better understanding of number theory can aid in solving real-world problems, such as optimizing traffic flow or predicting patterns in financial markets.

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