The Quest for Millions: Solving Unsolved Math Problems and the Path to Reward
For those who yearn to venture into the untamed wilderness of mathematics, there exist challenges so monumental that solving them can lead to a reward more valuable than a million dollars. The Riemann Hypothesis, the Collatz Conjecture, the Navier-Stokes Equations, and the P vs NP problem are prime examples. These puzzles, once cracked, could unlock mysteries that have perplexed mathematicians for decades. But how does one go about sending a potential solution to these problems? Here’s a guide to the process and some insights into the potential pitfalls and rewards.
Publication and Peer Review
Firstly, it should be noted that the process of submission and validation of a potential solution is rigorous. Alon Amit is correct in stating that a solution must first be published in a peer-reviewed journal and remains accepted by the mathematical community before claiming any prize associated with it. This step is crucial for several reasons. For one, it ensures that the solution goes through a robust vetting process. Integrity is paramount in scientific endeavors, and unauthorized claims, no matter how promising, can undermine the credibility of the field.
Furthermore, the exact criteria for acceptance also vary. Not every peer-reviewed result should be seen as infallible. Sometimes, even published papers are later found to contain errors or oversights. Therefore, the prize administrators, such as the Clay Mathematics Institute, do not typically engage in independent verification. Instead, they rely on the established channels within the mathematics community to ensure authenticity.
The process is long and meticulous. Typically, two years after publication, it should be evident whether the solution holds up to scrutiny. This period allows for ample time for other experts to examine the work, identify flaws, or confirm its validity. In practice, most solutions that are initially deemed promising are often fine-tuned or overturned over time, either through further analysis or new developments in the field.
Alternatives to Mathematical Prizes
Contrary to the allure of financial rewards, there are often more practical, and perhaps easier, ways to achieve significant wealth. As noted in a 1996 book titled The Millionaire Next Door, one of the simplest methods involves securing a well-paid job in a professional field, such as law or medicine. Subsequently, meticulous savings and wise investments can lead to substantial financial stability.
When surveyed about how to obtain a million dollars, most individuals incorrectly guess that it can be achieved through lottery winnings. However, the odds of winning the lottery are astronomically low, often much lower than the hypothetical 1/1000 chance of solving a problem like P vs NP within a year. This stark contrast underscores the impracticality of gambling on such slim chances.
The Feasibility of Solving Mathematical Problems
Scott Aaronson’s perspective on the P vs NP problem offers a realistic evaluation of the likelihood of solving such a daunting challenge. According to him, the odds of anyone solving this problem within a year of today are effectively zero, if being generous. The future may bring a time when the field progresses to a level where upper echelon researchers might consider a problem such as P vs NP within their reach, but at present, the problem remains well beyond most people’s grasp.
Instead of betting on such long shots, some academic positions could be far more rewarding. For instance, landing a position like Scott Aaronson’s, a prominent researcher in the field of computational complexity, presents a path that often pays better than any prize aimed at publicizing a mathematical problem. Academic positions allow for sustained research, collaboration, and the establishment of a reputation, all of which are invaluable in the academic circuit.
Mathematical Challenges as a Path to Recognition
Paul Erd?s, a prolific Hungarian mathematician, was known to offer smaller, but still significant, prizes for solving various mathematical problems. For example, one of the problems he offered 100 for was described as "the hardest way you would ever earn 100." This captures the spirit of challenging, high-reward problems. Similarly, he once offered 3000 for a proof of a result of which the Green-Tao theorem is a special case, highlighting the enormity of such problems.
The Green-Tao theorem, a groundbreaking result in number theory, is a testament to the complexity and elegance of such problems. Even a special case can be impressive and simultaneously showcase the difficulty of solving these problems.
For enthusiasts of mathematics, the journey to solving such problems is not simply a race to claim a prize. It is a path to intellectual discovery, recognition, and the potential for groundbreaking scientific contributions. Whether one aims for a million dollars or the respect and admiration of their peers, the rigors and challenges of tackling unsolved mathematical problems remain an intriguing and compelling endeavor.