Unsolved mathematical problems are the cornerstone of ongoing research and exploration in the field. They represent the frontiers of human knowledge, challenging mathematicians to push the boundaries of what is known and understood. Here, we will know ten such problems, each posing unique challenges and opportunities for discovery:

1. Riemann Hypothesis: Proposed by Bernhard Riemann in 1859, this conjecture concerns the distribution of the non-trivial zeros of the Riemann zeta function. It predicts that all non-trivial zeros have a real part equal to 1/2. The Riemann Hypothesis has profound implications for number theory, particularly the distribution of prime numbers, and remains one of the most famous unsolved problems in mathematics.
2. Birch and Swinnerton-Dyer Conjecture: This conjecture, formulated in the 1960s, relates to the rational points on elliptic curves defined over the rational numbers. It suggests a deep connection between the arithmetic properties of these curves and the behavior of certain L-functions associated with them. The Birch and Swinnerton-Dyer Conjecture remains unproven, although significant progress has been made in special cases.
3. P vs. NP Problem: A central question in computer science and mathematics, the P vs. NP problem asks whether every problem whose solution can be quickly verified by a computer can also be quickly solved by a computer. In essence, it explores the fundamental complexity of computational tasks and has wide-ranging implications for cryptography, optimization, and algorithmic efficiency.
4. Yang-Mills Existence and Mass Gap: The Yang-Mills existence problem is a foundational question in theoretical physics and mathematics. It seeks to establish the existence of quantum field theories known as Yang-Mills theories with a certain property called the “mass gap.” Resolving this problem would deepen our understanding of fundamental forces in nature, such as electromagnetism and the strong nuclear force.
5. Navier-Stokes Existence and Smoothness: The Navier-Stokes equations describe the motion of viscous fluid flows and play a crucial role in understanding fluid dynamics. However, the question of whether smooth solutions to these equations exist globally in three dimensions remains open. This problem is fundamental in physics and engineering and has eluded resolution for over a century.
6. Goldbach’s Conjecture: Proposed by Christian Goldbach in 1742, this conjecture posits that every even integer greater than 2 can be expressed as the sum of two prime numbers. Despite extensive computational verification for large numbers, a rigorous proof remains elusive, making it one of the oldest unsolved problems in number theory.
7. Collatz Conjecture: Also known as the “3n + 1 problem” or “hailstone sequence,” the Collatz Conjecture involves iterating a simple arithmetic operation on positive integers. Starting with any positive integer n, if it is even, divide it by 2; if it is odd, multiply it by 3 and add 1. The conjecture suggests that this process always leads to the cycle 4, 2, 1, regardless of the initial value of n. Despite extensive computational testing, no counterexample has been found, but a proof remains elusive.
8. Bounded Gaps Conjecture: Proposed by Yitang Zhang in 2013, the bounded gaps conjecture asserts that there are infinitely many pairs of consecutive prime numbers with a finite difference between them (bounded gaps). While Zhang made significant progress by demonstrating the existence of such pairs within a finite bound, the full conjecture remains unproven.
9. The Twin Prime Conjecture: This conjecture posits that there are infinitely many pairs of prime numbers that are two units apart, such as (3, 5), (11, 13), and (17, 19). Despite numerous attempts to prove or disprove it, including recent progress by mathematicians like Yitang Zhang and the Polymath Project, a definitive resolution to this conjecture remains elusive.
10. Perfect Cuboid Problem: Also known as Euler’s brick problem, this question concerns the existence of a cuboid (rectangular box) with integer side lengths a, b, and c, such that the diagonal d = √(a^2 + b^2 + c^2) is also an integer. While various examples exist, a solution for a perfect cuboid with distinct side lengths has yet to be found, making it an intriguing open problem in number theory and geometry.

In conclusion, these ten unsolved mathematical problems represent some of the most intriguing and challenging questions in the field. They embody the spirit of inquiry and exploration that drives mathematicians to push the boundaries of knowledge, seeking deeper insights into the structure and nature of mathematics and the universe. While progress may be incremental, each step brings us closer to unraveling the mysteries of these enigmatic problems and advancing our understanding of the mathematical world.