Dana Ernst
October 28, 2014
9.3k

# Open problems with monetary rewards

There is a history of individuals and organizations offering monetary rewards for solutions, either in the affirmative or negative, to difficult mathematically-oriented problems. For example, the Millennium Prize Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute in 2000. A correct solution to any of the problems results in a \$1,000,000 prize being awarded by the institute. To date, only one of the problems has been solved (the Poincaré Conjecture was solved by Grigori Perelman, but he declined the award in 2010). These are hard problems! The renowned mathematician John Conway (Princeton) maintains a list of open problems and for each problem on the list, he is offering \$1000 to the first person that provides a correct solution. In this talk, we will explore a few of Conway’s problems, and in the unlikely event we come up with a solution, we’ll split the money.

This talk was given at the 2014 Northern Arizona University High School Math Day on Tuesday, October 28, 2014.

October 28, 2014

## Transcript

1. open problems with monetary rewards
2014 NAU High School Math Day
Dana C. Ernst
Northern Arizona University
October 28, 2014

2. 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.”
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3. millennium prize problems

4. 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.
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5. wolfskehl and fermat’s last theorem

6. 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. In 1905, he bequeathed 100,000 marks (about
\$1,700,000 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 as a
result of currency reform in Germany in 1948, the award was only
worth 75,000 marks.
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7. the beal conjecture

8. 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
∙ 1997: \$5,000
∙ 2000: \$100,000
∙ 2013: \$1,000,000
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9. conway’s problems

10. 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.
If you solve one of his problems, you can reach him by sending snail
mail (only) in care of the Department of Mathematics at Princeton
University.
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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!
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12. sylver coinage game
Sample game between A and B
∙ A opens with 5. Can’t play: 5, 10, 15, . . .
∙ B names 4. Can’t play: 4, 5, 8, 9, 10, 12→. Remaining: 1, 2, 3, 6, 7, 11.
∙ A names 11. Remaining: 1, 2, 3, 6, 7.
∙ B names 6. Remaining: 1, 2, 3, 7.
∙ A names 7. Remaining: 1, 2, 3.
∙ B names 2. Remaining: 1, 3.
∙ A names 3. Remaining: 1.
∙ B is forced to name 1 and loses.
Problem
If player 1 names 16, and both players play optimally thereafter, then
who wins?
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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!
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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?
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15. sources

16. 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://www.ams.org/notices/199710/barner.pdf
http://www.bealconjecture.com
http://en.wikipedia.org/wiki/John_Horton_Conway
http://www.cheswick.com/ches/conway1000.pdf
http://en.wikipedia.org/wiki/Sylver_coinage
http://www.thrackle.org
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