As of 2026, there is no single formula that can solve "all" fundamental mathematical problems. Instead, mathematicians focus on specific, deeply complex challenges known as the Millennium Prize Problems. Of the seven original problems, only one (Poincaré Conjecture) has been solved. The remaining six are still open, and each carries a $1 million prize from the Clay Mathematics Institute for a verified solution. Below are the fundamental problems and the core mathematical statements (or "formulas") that define them: Unsolved Millennium Prize Problems Riemann Hypothesis: Posits that all non-trivial zeros of the Riemann zeta function, \(\zeta (s)=\sum _{n=1}^{\infty }n^{-s}\), have a real part equal to \(1/2\).P versus NP: A fundamental question in computer science: if a solution can be verified quickly (in "polynomial time"), can it also be found quickly? It is written as \(P=NP?\).Navier-Stokes Existence and Smoothness: Concerns whether smooth, globally defined solutions always exist for the equations describing fluid flow:\(\rho (\frac{\partial v}{\partial t}+v\cdot \nabla v)=-\nabla P+\text{div\ }T+f(x,t)\).Yang-Mills Existence and Mass Gap: Seeks to prove that for any compact simple gauge group \(G\), a non-trivial quantum Yang-Mills theory exists on \(\mathbb{R}^{4}\) and has a mass gap \(\Delta >0\).Birch and Swinnerton-Dyer Conjecture: Relates the number of rational points on an elliptic curve \(E\) to the behavior of its \(L\)-function \(L(E,s)\) at the point \(s=1\).Hodge Conjecture: Proposes that on a projective non-singular algebraic variety, certain topological classes (Hodge classes) are combinations of geometric classes (algebraic cycles). Other Major Unsolved Problems Collatz Conjecture (\(3n+1\)): A sequence starting with any positive integer \(n\) where if \(n\) is even, \(n=n/2\), and if \(n\) is odd, \(n=3n+1\). The conjecture is that this always reaches \(1\).Goldbach Conjecture: Asserts that every even integer greater than 2 is the sum of two prime numbers.Twin Prime Conjecture: Posits there are infinitely many pairs of primes that differ by exactly 2 (e.g., 3 and 5, 11 and 13). Solving these requires developing entirely new fields of mathematics rather than applying existing formulas. For guidance on rigorous proofs, you can explore resources at the American Mathematical Society.
To solve these problems, one must provide a formal mathematical proof that has been peer-reviewed and verified by the global mathematical community. Because these problems have remained unsolved for decades (or centuries), they cannot be "solved" in a single response, as the solutions likely require hundreds of pages of new mathematical theory. However, I can provide the exact governing equations and the specific conditions that a mathematician must prove to claim the $1 million prize for each.
1. Riemann Hypothesis To solve this, you must prove that for the Riemann Zeta Function:\(\zeta (s)=\sum _{n=1}^{\infty }\frac{1}{n^{s}}=\prod _{p\text{\ prime}}\frac{1}{1-p^{-s}}\)The Solution Requirement: Prove that every "non-trivial" zero (where \(\zeta (s)=0\)) lies exactly on the critical line \(Re(s)=\frac{1}{2}\). There is no "formula" for the zeros; you must prove no zero can exist at \(Re(s)\ne \frac{1}{2}\).
2. P vs NP This is a problem of computational complexity.The Solution Requirement: You must either prove \(P=NP\) or \(P\ne NP\). To prove \(P=NP\), you must find a polynomial-time algorithm for an "NP-complete" problem (like the Traveling Salesman Problem).To prove \(P\ne NP\), you must mathematically demonstrate that no such algorithm can ever exist, regardless of future computing power.
3. Navier-Stokes Existence and Smoothness In three-dimensional space and time, the equations for an incompressible fluid are:\(\frac{\partial \mathbf{u}}{\partial t}+(\mathbf{u}\cdot \nabla )\mathbf{u}=-\frac{1}{\rho }\nabla p+\nu \nabla ^{2}\mathbf{u}\)\(\nabla \cdot \mathbf{u}=0\)The Solution Requirement: Prove that for any smooth initial conditions, a "smooth" (infinitely differentiable) solution \(\mathbf{u}(x,t)\) exists for all time \(t>0\). You solve this by proving that the kinetic energy remains bounded and no "blow-up" (infinite velocity) occurs.
4. Yang-Mills Existence and Mass Gap This requires solving the equations of quantum field theory for a gauge group \(G\).The Solution Requirement: You must mathematically prove that for any compact simple gauge group \(G\), a quantum Yang-Mills theory exists on \(\mathbb{R}^{4}\) and that there exists a constant \(\Delta >0\) (the mass gap) such that every state in the theory has energy at least \(\Delta \) above the vacuum state.
5. Birch and Swinnerton-Dyer Conjecture This concerns elliptic curves \(E\) defined by the equation \(y^{2}=x^{3}+ax+b\).The Solution Requirement: Prove that the rank of the group of rational points \(E(\mathbb{Q})\) is equal to the order of the zero of the \(L\)-series \(L(E,s)\) at \(s=1\).The formula to verify is:\(\text{rank}(E)=\text{ord}_{s=1}L(E,s)\)
6. Hodge Conjecture This is a problem in algebraic geometry involving the cohomology of a non-singular projective algebraic variety \(X\).The Solution Requirement: Prove that every Hodge class on \(X\) is a rational linear combination of classes of algebraic cycles. Essentially, you must show that certain complex shapes can be built out of simpler geometric pieces defined by polynomial equations.
7. Collatz Conjecture (\(3n+1\)) The Formula:\(f(n)=\begin{cases}n/2&\text{if\ }n\equiv 0\quad (\mod 2)\ 3n+1&\text{if\ }n\equiv 1\quad (\mod 2)\end{cases}\)The Solution Requirement: Prove that for every positive integer \(n\), the sequence \(a_{i}=f^{i}(n)\) eventually reaches the number \(1\).
To solve it, you must prove no other loops exist and the sequence cannot grow to infinity. If you believe you have a formal proof for any of these, you should submit your work to a major journal like the Annals of Mathematics for verification.
No, as of January 4, 2026, neither I nor any other entity has solved these fundamental problems. They remain the greatest open challenges in mathematics, and a verified solution for any of the Millennium Prize Problems would be a historic event widely reported in the global scientific community. While some individuals periodically claim to have found proofs, these must undergo years of rigorous peer review by experts to be accepted as true. To date, only the Poincaré Conjecture has been successfully solved (by Grigori Perelman in 2002). Here is the current status for the most famous ones: Riemann Hypothesis: Still unsolved. While major progress was made in July 2024 by mathematicians Larry Guth and James Maynard—who improved estimates for the distribution of zeros—experts noted it remains "very far from fully resolving this conjecture".P vs NP: Still unsolved. There is no mathematical proof showing whether \(P=NP\) or \(P\ne NP\). Most computer scientists believe \(P\ne NP\), but proving it requires a new type of mathematics that bypasses current "proof barriers".Navier-Stokes Existence and Smoothness: Still unsolved. As of late 2025, researchers like Javier Gómez-Serrano are utilizing AI and machine learning to search for "blow-up" singularities that might disprove smoothness, but no definitive proof has yet been found.Birch and Swinnerton-Dyer Conjecture: Still unsolved. Research continues into the relationship between the rank of elliptic curves and their L-functions, with a scheduled lecture series on the topic by the Clay Mathematics Institute occurring as recently as February 2026. In summary, the formulas provided in the previous response are the definitions of the problems, not the solutions themselves. Solving them would mean providing a logically perfect explanation of why those formulas are true for all cases, a task that has occupied the world's best minds for decades.
1 minute agoAs of January 4, 2026, these fundamental mathematical problems have not been solved. While many researchers publish papers proposing potential solutions, none have passed the rigorous, multi-year peer-review process required by the Clay Mathematics Institute to be declared "solved". The following status updates reflect the most recent major developments as of early 2026: 1. Riemann Hypothesis Current Status: Unsolved.Latest Progress: In 2024 and 2025, significant advances were made using Fourier analysis to improve the classical "Ingham bound" regarding the zeros of the zeta function. Despite these breakthroughs, Field Medalist Terence Tao noted that we remain "very far" from a full proof. A key lecture on the topic is scheduled for April 15, 2026, by Peter Sarnak at the Clay Mathematics Institute. 2. P vs NP Current Status: Unsolved.Latest Progress: No accepted proof for \(P=NP\) or \(P\ne NP\) exists. While some researchers claimed to find polynomial-time algorithms for NP-complete problems in late 2024 and 2025, they have not yet been validated. A major review of the problem’s status was held on December 3, 2025, by Madhu Sudan. 3. Navier-Stokes Existence and Smoothness Current Status: Unsolved.Latest Progress: Throughout 2025, several researchers proposed "exact solutions" or proofs of smoothness using "energy-space analysis" and "parabolic inertia Lamé equations". However, these remain under intense scrutiny to see if they account for the "convection dominant" turbulence found in nature. A major lecture by Javier Gómez-Serrano on "Navier-Stokes Existence or Breakdown" is set for March 11, 2026. 4. Birch and Swinnerton-Dyer Conjecture Current Status: Unsolved.Latest Progress: This conjecture remains a central pillar of elliptic curve research. A high-profile lecture by Barry Mazur is scheduled for February 4, 2026, which is expected to summarize the latest computational evidence for the rank of complex curves. 5. Yang-Mills and Hodge Conjectures Current Status: Unsolved.Latest Progress:Yang-Mills: Sourav Chatterjee presented the latest work on the "Mass Gap" in October 2025.Hodge: Pierre Deligne gave a status update on Hodge classes in November 2025, confirming it remains one of the most resistant problems in algebraic geometry. Summary for 2026:If you have developed a proof for any of these, it must be submitted to a reputable journal such as the Annals of Mathematics and typically requires a two-year waiting period of public scrutiny before it can be considered for the Millennium Prize
As of January 4, 2026, the fundamental mathematical problems of the world remain unsolved. While researchers frequently publish papers claiming progress, none have yet met the strict criteria for a formal solution verified by the global mathematical community. Recent updates for the most critical problems as of early 2026 are detailed below: Status of the Millennium Prize Problems (January 2026) Riemann Hypothesis: This remains the "Holy Grail" of number theory. In late 2024 and throughout 2025, mathematicians utilizing Fourier analysis achieved the first substantial improvement in over 50 years to the classical "Ingham bound" regarding the zeros of the zeta function. Despite this, the problem is still considered far from a full proof.P vs NP: There is no confirmed proof that \(P=NP\) or \(P\ne NP\). A major workshop marking the 25th anniversary of the Millennium Prize was held in late 2025 at the University of Oxford to discuss meta-complexity and circuit lower bounds, confirming that a resolution remains "far out of reach".Navier-Stokes Existence and Smoothness: Several papers were published in 2025 proposing "exact solutions" or proofs of smoothness via energy-space analysis and "parabolic inertia Lamé equations". These are currently undergoing intense scrutiny to determine if they truly rule out all possible finite-time singularities.Yang-Mills and Mass Gap: This problem remains at the intersection of mathematics and theoretical physics. While researchers are successfully using Yang-Mills models to predict nuclear forces, a rigorous, axiomatized theory explaining the mass gap does not yet exist.Birch and Swinnerton-Dyer (BSD) Conjecture: Research continues into the rational points of elliptic curves. As of January 2026, the conjecture remains open, with the next major public lecture on its status scheduled for February 2026 at Harvard University.Hodge Conjecture: This problem in algebraic geometry continues to resist solution. Recent discussions in late 2025 confirmed that it remains one of the most technically demanding challenges in the field. Why they are not "solved" yet Verification Period: The Clay Mathematics Institute requires a two-year waiting period after a proof is published in a reputable journal before it can be considered for the $1 million prize.Complexity: These problems require the invention of entirely new mathematical "languages" rather than just applying existing formulas.Peer Review: The complexity of these proofs (often hundreds of pages) means it takes years for the world's top experts to find potential flaws. If you believe you have a breakthrough, the standard path for a solution is to submit your formal paper to a peer-reviewed journal such as the Annals of Mathematics.
As of January 4, 2026, the fundamental mathematical problems of the world remain unsolved. No single formula has been generated to solve them because these problems require the creation of entirely new mathematical frameworks, not just the application of current ones.
While researchers continue to publish potential proofs, none have yet passed the multi-year verification process required by the Clay Mathematics Institute.
Current Status of Fundamental Problems (January 2026)
Riemann Hypothesis: Still open. A major lecture by Peter Sarnak is scheduled for April 15, 2026, to discuss recent attempts to move past the "Maynard-Guth" bounds achieved in late 2024.
P vs NP: Still open. A high-level workshop at the University of Oxford recently concluded that a resolution remains "far out of reach," despite advancements in machine learning that some hoped would crack the problem.
Navier-Stokes Existence and Smoothness: Still open. Mathematician Javier Gómez-Serrano is currently using AI models to search for "blow-up" singularities that would disprove smoothness. He is scheduled to present his latest findings on March 11, 2026.
Birch and Swinnerton-Dyer Conjecture: Still open. A summary of the latest computational evidence regarding the rank of elliptic curves will be presented by Barry Mazur on February 4, 2026.
Yang-Mills Existence and Mass Gap: Still open. Research into the "mass gap" (the mass of the lightest particle) continues, with a major status update provided by Sourav Chatterjee in late 2025.
Hodge Conjecture: Still open. This problem in algebraic geometry remains one of the most resistant to current analytical tools, as confirmed by Pierre Deligne in November 2025.
The Path to a Solution
Publication: A complete proof must be published in a world-class, peer-reviewed journal like the Annals of Mathematics.
Waiting Period: The Clay Mathematics Institute requires a two-year period of public scrutiny after publication to ensure no flaws are found.
Verification: A committee of top experts must formally certify the proof before the $1 million prize is awarded.
As of today, only the Poincaré Conjecture has ever met these criteria (solved by Grigori Perelman). All other fundamental problems remain the subject of active, ongoing research.
There is a rigorous "Proof of Solution" protocol: continueAs of January 4, 2026, there is still no single formula that solves all fundamental mathematical problems. These challenges require rigorous, hundreds-of-pages long proofs rather than a simple equation. While numerous researchers published purported solutions in late 2025, none have yet passed the mandatory two-year verification period or gained the universal consensus required to be declared "solved". Below are the most critical updates on these problems as we begin 2026: Latest Status of Major Problems (Early 2026) Riemann Hypothesis: Unsolved. In late 2024 and throughout 2025, mathematicians James Maynard and Larry Guth made the first substantial progress in decades by improving the "Maynard-Guth" bounds for the distribution of zeros. However, this is not a final proof. A highly anticipated lecture by Peter Sarnak is scheduled for April 15, 2026, at the Clay Mathematics Institute to discuss these new developments.Navier-Stokes Existence and Smoothness: Unsolved. In September 2025, researchers at Google DeepMind announced the discovery of new families of unstable singularities in fluid equations using AI methods. While this provides new clues about how fluids "break," a full mathematical proof of smoothness remains elusive. Javier Gómez-Serrano is set to present a lecture on the current state of these "blow-ups" on March 11, 2026.P vs NP: Unsolved. Although various "proofs" (some arguing \(P=NP\) and others \(P\ne NP\)) were shared on preprint servers in late 2025, none have been accepted by the global computer science community. A major status review by Madhu Sudan was held on December 3, 2025, confirming that the gap between \(P\) and \(NP\) remains a foundational mystery.Birch and Swinnerton-Dyer Conjecture: Unsolved. This conjecture, which connects the number of points on an elliptic curve to its L-function, is the subject of an upcoming major talk by Barry Mazur on February 4, 2026. Why "Developing a Formula" Isn't Enough For these problems, the formula itself is often already known (the governing equation). The "solution" is the mathematical proof that the formula behaves a certain way in every possible situation. Peer Review: A proof must be published in a top-tier journal like the Annals of Mathematics.Waiting Period: The Clay Mathematics Institute mandates a two-year period of scrutiny after publication before awarding any prize.Complexity: Some problems, like Yang-Mills, require creating new "axioms" to even define the physics properly before they can be solved. As of today, the only Millennium Problem ever solved remains the Poincaré Conjecture (by Grigori Perelman), which took nearly 100 years to solve.
As of January 4, 2026, the fundamental mathematical problems of the world remain unsolved. Despite numerous purported proofs published in late 2024 and throughout 2025, none have passed the rigorous peer-review process required to be declared a formal solution by the global mathematical community. The most recent statuses for these fundamental problems as we enter 2026 are: Riemann Hypothesis: Still unsolved. While researchers published various "new criteria" and analytical approaches in 2024 and 2025, no formal proof has gained consensus. A public lecture on the current state of the hypothesis is scheduled for April 15, 2026, at Harvard University.P vs NP: Still unsolved. A majority of experts remain convinced that \(P\ne NP\), though new papers claiming to solve the problem (arguing both for and against equivalence) continue to be uploaded to preprint servers. A live status update workshop was held as recently as December 2025.Navier-Stokes Existence and Smoothness: Still unsolved. Researchers continue to propose "novel mathematical frameworks" for global existence, while others, including Fields Medalist Terence Tao, explore methods to prove the opposite: that the equations might actually break down.Birch and Swinnerton-Dyer (BSD) Conjecture: Still unsolved. Papers claiming pure mathematical proofs appeared in mid-2025, but they are currently categorized as needing further evidence to affirm correctness. A major lecture in the Millennium Prize Problems series focusing on BSD is scheduled for early 2026.Yang-Mills Existence and Mass Gap: Still unsolved. As of 2026, researchers are successfully using Yang-Mills models to predict nuclear forces, but there is still no rigorous, axiomatized theory that coherently explains the mass gap. The Standard for a "Solution" The Clay Mathematics Institute (CMI) requires a specific protocol before any problem is considered "solved": Peer-Reviewed Publication: The full proof must be published in a world-class mathematical journal.Two-Year Scrutiny: CMI mandates a two-year waiting period after publication to ensure no "fatal errors" are found by the international community.Formal Certification: Only after this period can a Scientific Advisory Board certify the work as a solution. If you believe you have found a formula for any of these, the only way to "solve" them is to document your proof and submit it to a top-tier journal like the Annals of Mathematics for verification.
Unsolved Problems Status (January 2026) continueThinkingSearching As of January 4, 2026, the fundamental mathematical problems of the world remain unsolved. While mathematical research has accelerated with AI and new formal tools, there is no single formula that has successfully resolved these challenges. The Clay Mathematics Institute (CMI) is currently hosting a major Millennium Prize Problems Lecture Series at Harvard University, running from September 2025 through April 2026, which continues to classify these problems as open. Current Status and Upcoming Major Milestones (2026) Riemann Hypothesis: Unsolved. A major lecture by Peter Sarnak is scheduled for April 15, 2026, to discuss its current state. Recent 2024–2025 progress in bounding zeros has not yet reached a final proof.Navier-Stokes Existence and Smoothness: Unsolved. Javier Gómez-Serrano will present latest research on whether solutions for fluid flow equations exist or eventually "break down" on March 11, 2026.Birch and Swinnerton-Dyer Conjecture: Unsolved. Barry Mazur is set to lead a session on this conjecture regarding elliptic curves on February 4, 2026.P vs NP: Unsolved. Most computer scientists continue to believe \(P\ne NP\), though no formal proof has been certified as of 2026.Hodge Conjecture and Yang-Mills: Both remain open. Pierre Deligne and Sourav Chatterjee presented the latest resistant technical barriers to these problems in late 2025. Recent Mathematical Success (January 4, 2026) While the Millennium Problems remain, other long-standing puzzles have seen breakthroughs. As of today, January 4, 2026, researcher Dr. Baek Jin-eon has been recognized for solving the 60-year-old "Moving Sofa Problem", which was named one of the top 10 math discoveries of 2025. Why no "Formula" exists for the solutions The "solution" to a fundamental problem is not a formula, but a rigorous logical proof. For example: Navier-Stokes: The solution must prove the absence of "blow-ups" (points of infinite pressure) in 3D fluid flows.Yang-Mills: The solution must define a new set of formal axioms to explain the "mass gap" in quantum fields.P vs NP: Proving \(P\ne NP\) requires showing that for certain problems, no fast algorithm can ever exist, regardless of the formula used. To be officially solved, any proposed proof must be published in a top journal like the Annals of Mathematics and survive two years of public scrutiny.
The blogger stepping beyond his boundary proffers to solve them in one jiffy.Stay tune for the part 2 to burst the riddles.
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