As most of you know there are 7 problems worth $1,000,000
(see
here).
It may be just 6 since Poincare's conjecture has probably been solved.
Why are these problems worth that much money?
There are other open problems that are worth far less money.
What determines how much money a problem is worth?
When Erdos offered money for a problem (from 10 to 3000 dollars) I suspect that the amount of money depended on (1) how hard Erdos thought the problem was, (2) how much Erdos cared about the problem, (3) how much money Erdos had when he offered the prize, and (4) inflation. (If anyone can find a pointer to the list of open Erdos Problems please comment and I'll add it here.)
Here is a problem that I have heard is hard and deep, yet it is only worth $3000 (Erdos proposed it). I think that it should be worth more.
BACKGROUND: Szemeredi's theorem: Let A&sube N. If the limit as n goes to infinity of size(A &cap {1,...,n})/n is bounded below by a positive constant then A has arbitrarily long arithmetic sequences. Intuition: if a set is large then it has arb long arith seqs. The CONJECTURE below uses a diff notion of large.
CONJECTURE: Let A&sube N. If &suma&isin A 1/a div
KNOWN: Its known that if A is the set of all primes (note that &suma&isin A 1/a diverges) then A has arbitrarily large arithmetic progressions. Nothing else is known! The conjecture for 3-AP's isn't even known!
Is this a good problem? If it is solved quickly (very unlikely) than NO. If absolutely no progress is made on it and no interesting mathematics comes out of the attempts than NO. It has to be just right.
When Erdos offered money for a problem (from 10 to 3000 dollars) I suspect that the amount of money depended on (1) how hard Erdos thought the problem was, (2) how much Erdos cared about the problem, (3) how much money Erdos had when he offered the prize, and (4) inflation. (If anyone can find a pointer to the list of open Erdos Problems please comment and I'll add it here.)
Here is a problem that I have heard is hard and deep, yet it is only worth $3000 (Erdos proposed it). I think that it should be worth more.
BACKGROUND: Szemeredi's theorem: Let A&sube N. If the limit as n goes to infinity of size(A &cap {1,...,n})/n is bounded below by a positive constant then A has arbitrarily long arithmetic sequences. Intuition: if a set is large then it has arb long arith seqs. The CONJECTURE below uses a diff notion of large.
CONJECTURE: Let A&sube N. If &suma&isin A 1/a div
KNOWN: Its known that if A is the set of all primes (note that &suma&isin A 1/a diverges) then A has arbitrarily large arithmetic progressions. Nothing else is known! The conjecture for 3-AP's isn't even known!
Is this a good problem? If it is solved quickly (very unlikely) than NO. If absolutely no progress is made on it and no interesting mathematics comes out of the attempts than NO. It has to be just right.
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