7 Math Problems

[Ronald Frison]

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Since the old days, many mathematicians have been attaching monetary rewards to problems they admit are difficult. Their reasons could be to draw other mathematicians' attention, to express their belief in the magnitude of the difficulty of the problem, to challenge others, "to elevate in the consciousness of the general public the fact that in mathematics, the frontier is still open and abounds in important unsolved problems.1", etc.

Question: What others are there? To put some order into the answers, let's put a threshold prize money of 100 USD. I expect there are more mathematicians who have tucked problems in their web-pages with some prizes.

P.S. Some may be interested in the psychological aspects of money rewards. However, to keep the question focused, I hope this topic won't be ignited here. One more, I understand that mathematicians do not work merely for money.

Let $f = t^2d + f_1 t^2d-1 + f_2 t^2d-2+ \cdots f_d t^d + \cdots+ f_2 t^2 +f_1 t + 1$ be a palindromic polynomial, so the roots of $f$ are of the form $\lambda_1$, $\lambda_2$, ..., $\lambda_d$, $\lambda_1^-1$, $\lambda_2^-1$, ..., $\lambda_d^-1$. Set $r_k = \prod_j=1^d (\lambda_j^k-1)(\lambda_j^-k -1)$.

Motivation: Obviously, this is a model of evolution, and one which (some) biologists actually use. Allman and Rhodes have shown that, if you know generators for the ideal for this particular case, then they can tell you generators for every possible evolutionary history. (More descendants, known sequence of branching, etc.) There are techniques in statistics where knowing this Zariski closure would be helpful progress.

Addendum: There is a paper by Fan Chung similar to this book by Chung and Graham, Open problems of Paul Erdős in graph theory. She says there, "In November 1996, a committee of Erdős' friends decided no more such awards will be given in Erdős' name." But the same article says that Chung and Graham decided to still sponsor questions in graph theory, and this article in Science Magazine implies that they are still sponsoring the Erdős problems in general.

Some 19 years ago I collected a list of Erdős prize problems and posted them to Usenet. The problems were from "A Tribute to Paul Erdős" (1990) and "Paths, Flows, and VLSI Layout" (1980). I can repeat the problems here, although I have no idea which ones may have been solved.

infinitely often. (The wording in the source does not clearly indicate that the money will be awarded if the conjecture is disproved, only if it is proved.) [Answered: By Kevin Ford, Ben Green, Sergei Konyagin, and Terence Tao, and independently by James Maynard, both groups in August 2014.]

$\$3000$. (T3N) Divergence implies arithmetic progressions. If the sum of the reciprocals of a set of positive integers is infinite, must the set contain arbitrarily long finite arithmetic progressions?

$\$500$. (P2) Sets with distinct subset sums. Is there a real number $c$ such that, given a set of n positive integers whose subsets all have distinct sums, the largest element is at least $c2^n$? (As in problem T1N, no prize is mentioned.)

There is always the chance of earning $327.68 from Donald Knuth. It is stretching things more than a bit to include that in and of itself, but the linked article is amusing and the general considerations are pertinent.

Paul Erdős was famous for offering money for solutions to math problems. My understanding is that those prizes are still being administered by Ron Graham, even though Erdős passed away several years ago.

The origin of this precise question seems to be obscure, although Lothar Collatz made similar conjectures during the 1930s1. For example, Bryan Thwaites claims to have been the first to make this conjecture in 19522, and this does not seem to have been decisively refuted. (The 1937 dates in the Wikipedia and Mathworld articles are missing citations - the Wikipedia edit dates to 7 September 2004.)

Yesterday, Terence Tao offered a 10.000$ prize to answer a question on large gaps in prime numbers. In particular, it's on improving their recent findings - and Terence Tao says that "this appears to be the limit of current technology". As far as I know it's the first kind of such a prize he offers.

Gerard Cornuejols offers $5000 for the first proof (or refutation) of each of the 18 conjectures in his 2001 book "Combinatorial Optimization: Packing and Covering". Six of the conjectures have been resolved so far, five - by Maria Chudnovsky, Paul Seymour and coauthors.

I remember reading in Havil's book Gamma that supposidly Hardy was willing to offer his Savilian Chair at Oxford University to anyone who could prove that the Euler Mascheroni constant is irrational. I wonder if this offer still stands?

As it is pointed out in a footnote in A. Zorich's survey "Flat surfaces", Anatole Katok promised (on behalf of the Center for Dynamics and Geometry of Penn State University) a prize of 10,000 euros for the solution of the problem of finding periodic orbits and describing the behavior of generic orbits of billiards in (all/almost all) triangular tables. See page 13 of the ArXiv version of the survey for more comments.

Jeffrey Shallit offers \$100 for any significantly improved bounds on the distinguishing strings problem. Which is: Given two words of length at most $n$, what is the size of the smallest DFA that accepts one and rejects the other? Current bounds are that $\Omega(\log n)$ states are needed and $O(n^2/5 (\log n)^3/5)$ are sufficient.

I'm a computer scientist by trade, but I really just enjoy working on open mathematical problems in my free time. Kimberling's page is pretty nice as you mentioned, I was able to knock one of them out (a minor one, the Swappage Problem), and always look forward to when new ones are posted.

One of the best places I have found for open problems is probably open problem garden. The site is frequently updated with new problems ranging from graph theory, theoretical computer science, algebra, etc. The nice thing is that the problems are also ranked by relative difficulty.

As for cash: A number of the problems DO offer some type of cash bounty (as clearly indicated in the summary section for the problem next to "Prize" text if it exists). Problems such as The Erdos-Turan conjecture on additive bases offer cash incentives for solving.

There are also other problems listed that offer monetary compensation and are posted periodically throughout the site. However, sifting through the problems can be time consuming as many of them do not offer cash incentives.

For example ongoing (December 2023 - January 2024: "Santa 2023 - The Polytope Permutation Puzzle") is challenge to propose algorithms/solutions for Rubik's cube like puzzles for a given list of configurations. Mathematically speaking it means - we are asked to find the shortest path(s) between two given nodes of the Cayley graph of the Rubik's cube group (or coset, or some of its modifications). For puzzles of large size - optimal algorithms are - unknown - so the solutions are potentially contributing to the resolving that open problem. About 390 pairs of nodes are given. Participants are evaluated how short paths they propose - the shorter the better - sum of lengths over all pairs is a score. Some relevant papers are: , , etc...

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