what is the hardest math problem

Exploring the Most Challenging Mathematical Problems

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Some problems are the core of mathematics or of special parts in the canon of mathematics because they call into play the full ingenuity of mathematicians. Occasionally, the solution involves the most unlikely areas in the larger mathematical literature. For example, the solution to one problem in number theory or algebra can sometimes involve complex analysis, elliptic functions, differential equations, combinatorial, arithmetic, geometry, topology, or algebraic methods. A genuinely difficult problem is one that defies an easy solution strategy. Although proving a theorem may involve an abundance of labor, it need not be a difficult mathematical problem: once someone has outlined the necessary steps, proving the theorem requires a meticulous execution of these steps. Furthermore, the proof(s) of the theorem may be apparent from the start; it just needs time to articulate the argument. Once the time has passed, the solution can be written up without much ado.

There is an abundance of mathematical problems, many of which can be quite challenging, requiring years of study for an individual mathematician or teams of mathematicians to fully connect and hence gain a better appreciation of their inner beauty. A problem is a mathematical problem if its formal mathematical statement can appear in any publication, but exactly which open problems should an individual mathematician or groups of mathematicians decide to tackle in their research? Should the accessibility of the solution be of prime concern? Should the problem be classified according to its complexity and its significance? The choice is inevitably subjective and the decision may be influenced by practical considerations such as skill levels, affordability, and current and future directions of the evolving field of mathematics. This can generally influence what possible paths are taken in attempting settlement of these problems. By emphasizing the history, process, suspense, and challenges collectively, mathematical problem solving itself turns into a narrative.

The main purpose of this monograph is to establish several local-global principles. The classical problem is to determine the solvability of certain Diophantine equations defined by polynomials with integral coefficients by means of the corresponding equations defined by the polynomials modulo p for every prime number p. The simplest example is represented by the equations characterized by a given degree, say two or three (Fermat equations with exponent > 4 have been proved not to be solvable in integers). There are some generalizations. The determination of the solvability of the equations should be considered as a mathematical problem.

Some of the most famous existing conjectures are many years, even centuries, old, and their investigation has produced a noticeable volume of valuable results. Of the many problems outstanding today, Riemann’s hypothesis is the best known. Two conjectures concerning modular elliptic curves have received great attention during recent years. In number fields, we find the following: The class number one problem. The generalization of Landau’s theorem on simple number fields. The generalized Fermat equation, a Diophantine equation with exponent n > 2. In number theory, but in neither number fields nor elliptic curves, Etale cohomology may be considered as one of the most profound theories in pure mathematics.

The Birch and Swinnerton-Dyer Conjecture is a tetexuated relation between the rank of an elliptic curve and the order of its vanishing at s = 1. One easily defined part of the genuine difficulty thus stems from understanding the point set structure of any rational points on such elliptic curves. But even taking the simpler case of Mordell’s conjecture, for which only weak results were known and were proved by a 21st-century Fields medal winner, one already sees that the reason for the difficulty is that the structure of the Mordell-Weil group is so complicated. Even more fundamental to any statement of this conjecture is what allowed Mordell to phrase his conjecture: that every elliptic curve E is isomorphic to a cubic projective curve given by a Weierstrass equation. This is the first recall that Poincaré conjectured that the fundamental group of the complement of a hyperplane section of a smooth non-singular projective surface is necessarily trivial.

The first of the Millennium prizes, for a proof or disproof of the Riemann Hypothesis, is motivated by connecting it to the distribution of the prime numbers and mentioning a few subsidiary matters.

In this section, we define each of the problems and give a brief history of the problem. It is a useful exercise to reflect on just what makes these problems so difficult, both from the perspective of whether the problem is stated in a sufficiently clear way – such that it makes sense to ask whether a solution exists – and also the type of mathematical ideas that have been developed and might be seen to apply to such a problem.

When we explore the most challenging mathematical problems, we realize that they are so because of their difficulty even for modern computers. The increasing interest in the study of those hard problems is a reflection of the advances in the technology itself. Many important practical problems like scheduling, optimization, and strategy planning find in mathematical programming a major tool. These problems, while involving hard computations, have associated specific characteristics allowing the obtention of good quality solutions. Yet, mathematical programs involving highly degenerate or disordered non-convex structures such as topological and integer constraints are not amenable to these methodologies. Moreover, we understand the word “problem” in a very specific sense; with this, we are actually referring to any hard-to-solve mathematical task in any branch of mathematics that has generated a considerable amount of research. The word “hard” has a specific meaning of the manual labor required to go through the arguments.

It is difficult to anticipate the ultimate impact of the solution to any mathematical conjecture and, in the case of a century-old proposition such as the Riemann Hypothesis, we will probably never know the full ramifications of the mathematical solution. However, considering previous problems and theories in mathematics that were developed to answer relevant questions of a different nature and generated solutions with an enormous impact on present-day life, several important problems historically contributed significantly to the growth of mathematics. Remarkably, the study of one of the most abstract concepts, the notion of number, has played a huge practical role in such diverse areas as meteorology, military security, computer science, physiology, acoustics, and what arguably is the greatest of current challenges, the control of the environment and the resources of the planet.

One might wonder where the interest in these abstract mathematical concepts originates. From history, it is clear that interest in these questions is not purely theoretical. In the world of science and technology, we depend on engineering based on physical science. Although it is useful to have descriptive laws for scientific phenomena, based purely on observations, scientists need to do more than simply describe what happens. They are also interested in theories that allow them to predict natural phenomena or testable hypotheses, both within the known limit of the readers, and – even more interestingly – outside the known limit of the readers. For this, the systems involved must be understood at a deeper level, which is where the use of abstract mathematics becomes essential. Many scientists have found it difficult to understand how these concepts have such extraordinary predictive power, in areas that are often so far removed from our everyday world.

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