open problems with monetary rewards Friday Afternoon Mathematics Undergraduate Seminar Dana C. Ernst Northern Arizona University October 24, 2014

open problems with monetary rewards There is a history of individuals and organizations offering monetary rewards for solutions—affirmative or negative—to difficult open problems. 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, etc. In the words of the Clay Mathematics Institute: “…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; to emphasize the importance of working towards a solution of the deepest, most difficult problems; and to recognize achievement in mathematics of historical magnitude.” 1

millennium prize problems

millennium prize problems The Millennium Prize Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute in 2000. 1. Poincaré Conjecture 2. P versus NP 3. Hodge Conjecture 4. Riemann Hypothesis 5. Yang–Mills Existence and Mass Gap 6. Navier–Stokes Existence and Smoothness 7. Birch and Swinnerton–Dyer Conjecture A correct solution to any of the problems results in a $1,000,000 prize. The Poincaré Conjecture was solved by Grigori Perelman, but he declined the award in 2010. The other problems remain unsolved. 3

wolfskehl and fermat’s last theorem

wolfskehl and fermat’s last theorem Theorem (Fermat’s Last Theorem) No three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two. “I have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain.” – Fermat (but in Latin) Paul Friedrich Wolfskehl (1856–1906), was a physician and mathematician. He bequeathed 100,000 marks (about 1,000,000 pounds in 1997) to the first person to prove Fermat’s Last Theorem. On June 27, 1997, the prize was finally won by Andrew Wiles, but due to hyperinflation Germany, the award had dwindled to 30,000 pounds. 5

the beal conjecture

beal’s conjecture Conjecture If ax + by = cz, where a, b, c, x, y, and z are positive integers and x, y and z are all greater than 2, then a, b, and c must have a common prime factor. Texas billionaire D. Andrew Beal stated this conjecture in 1993. Beal has been trying to prove his theorem ever since, offering cash rewards in steadily increasing amounts: ∙ 1997: $5,000 ∙ 2000: $100,000 ∙ 2013: $1,000,000 7

conway’s problems

john h. conway John H. Conway (born 26 December 1937) is a British mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory, and coding theory. Conway is currently Professor of Mathematics at Princeton University. Conway maintains a list of open problems and for each problem on the list, he is offering $1,000 to the first person that provides a correct solution. 9

the angel problem The Angels and Devils game is played by two players called the angel and the devil. It is played on an infinite chessboard. The angel has a power k, where 1 ≤ k ∈ N, specified before the game starts. The board starts empty with the angel at the origin. On each turn, the angel jumps to a different empty square that can be reached by at most k moves of a chess king. The devil, on its turn, may add a block on any single square not containing the angel. The angel may leap over blocked squares, but cannot land on them. The devil wins if the angel is unable to move. The angel wins by surviving indefinitely. Problem: Can an angel with high enough power win? 10

the angel problem Conway offered a reward for a general solution to this problem: ∙ $100 for a winning strategy for an angel of sufficiently high power, ∙ $1000 for a proof that the devil can win irrespective of the angel’s power. In 2006, 4 independent and almost simultaneous proofs emerged that the angel has a winning strategy. 11

sylver coinage game The Sylver Coinage Game is a game in which 2 players alternately name positive integers that are not the sum of nonnegative multiples of previously named integers. The person who names 1 is the loser! Named after Sylvester, who proved the game terminates. 12

sylver coinage game Sample game between A and B ∙ A opens with 5. Now neither player can name 5, 10, 15, . . . ∙ B names 4. Now neither player can name 4, 5, 8, 9, 10, or any number greater than 11. ∙ A names 11. Now the only remaining numbers are 1, 2, 3, 6, and 7. ∙ B names 6. Now the only remaining numbers are 1, 2, 3, and 7. ∙ A names 7. Now the only remaining numbers are 1, 2, and 3. ∙ B names 2. Now the only remaining numbers are 1 and 3. ∙ A names 3, leaving only 1. ∙ B is forced to name 1 and loses. Problem: If player 1 names 16, and both players play optimally thereafter, then who wins? 13

climb to a prime Let n be a positive integer. Write the prime factorization in the usual way, e.g., 60 = 22 · 3 · 5, in which the primes are written in increasing order, and exponents of 1 are omitted. Then drop exponents straight down and omit all multiplication signs, obtaining a number f(n). Now repeat. So, for example, f(60) = f(22 · 3 · 5) = 2235. Next, because 2235 = 3 · 5 · 149, it maps, under f, to 35149, and since 35149 is prime, it maps to itself. Thus 60 → 2235 → 35149 → 35149, so we have climbed to a prime, and we stop there forever. Problem: Is it true that every number eventually climbs to a prime? The number 20 has not been verified to do so. Observe that 20 → 225 → 3252 → 223271 → · · · , eventually getting to more than 100 digits without yet reaching a prime! 14

the thrackle problem A doodle on a piece of paper is called a thrackle if it consists of certain distinguished points, called spots, and some differentiable (i.e., smooth) curves, called paths, ending at distinct spots and so that every pair of paths hit exactly once, where hit means having a common point at which they have distinct tangents and which is either an endpoint of both or an interior point of both. The thrackle to the left has 6 spots and 6 paths. Problem: Can a thrackle have more paths than spots? 15

99-graph Problem: Is there a graph with 99 vertices in which every edge (i.e., pair of joined vertices) belongs to a unique triangle and every non-edge (pair of unjoined vertices) to a unique quadrilateral?! 16

sources

sources http://mathoverflow.net/questions/66084/ open-problems-with-monetary-rewards http://www.claymath.org http://en.wikipedia.org/wiki/Millennium_Prize_Problems http://en.wikipedia.org/wiki/Paul_Wolfskehl http://en.wikipedia.org/wiki/John_Horton_Conway http://www.cheswick.com/ches/conway1000.pdf http://en.wikipedia.org/wiki/Angel_problem http://en.wikipedia.org/wiki/Sylver_coinage http://www.thrackle.org 18