Unsolved problems in number theory and news about them

In summary, number theory is a branch of mathematics that studies the properties of numbers and their relationships. Despite centuries of research, there are still numerous unsolved problems in this field, such as the Goldbach conjecture and the Collatz conjecture. However, recent news includes progress in solving some of these long-standing problems, such as the resolution of the twin prime conjecture and advancements in the study of prime gaps. Additionally, the ongoing research in this area has led to the discovery of new and intriguing mathematical phenomena, highlighting the importance of continued exploration and investigation in number theory.
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trees and plants
Hello people of physicsforums. Has anyone proved these problems in number theory :Riemann hypothesis, Goldbach conjecture, twin primes conjecture, infinitude of prime numbers? If you want make discussion about them.

Thank you for allowing me participate in physicsforums and for wanting to make discussions on topics of math, physics and other sciences. Have a good day.
 
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I’m closing this thread now.

These unsolved problems are easy to look up via google.
 
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1. What is the Riemann Hypothesis and why is it important?

The Riemann Hypothesis is a famous unsolved problem in number theory, proposed by mathematician Bernhard Riemann in 1859. It states that all non-trivial zeros of the Riemann zeta function lie on the critical line of 1/2. This hypothesis has significant implications in various fields of mathematics, including prime number distribution and the distribution of prime numbers in arithmetic progressions.

2. What is the Goldbach Conjecture and why is it important?

The Goldbach Conjecture is another famous unsolved problem in number theory, proposed by mathematician Christian Goldbach in 1742. It states that every even integer greater than 2 can be expressed as the sum of two prime numbers. This conjecture has important implications in number theory and has been extensively studied by mathematicians, but it has yet to be proven.

3. What is the Twin Prime Conjecture and why is it important?

The Twin Prime Conjecture is a well-known unsolved problem in number theory that states there are infinitely many pairs of prime numbers that differ by 2 (e.g. 41 and 43). This conjecture has important implications in prime number distribution and has been studied by mathematicians for centuries, but a proof has yet to be found.

4. What is the Collatz Conjecture and why is it important?

The Collatz Conjecture, also known as the 3n+1 problem, is an unsolved problem in number theory that has intrigued mathematicians for decades. It states that for any positive integer n, if n is even, divide it by 2; if n is odd, multiply it by 3 and add 1. The conjecture suggests that this process will eventually lead to the number 1 for any starting value of n. While this conjecture has been verified for many numbers, a general proof has yet to be found.

5. What is the current progress on solving these unsolved problems in number theory?

Despite decades, or even centuries, of research and study, these unsolved problems in number theory still remain a mystery. Many mathematicians have made significant progress and developed new techniques and theories, but a complete proof for these conjectures has yet to be found. However, with the advancement of technology and collaboration among mathematicians, there is still hope that one day these problems will be solved.

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