A popular magazine announced a contest to solve Goldbach's Conjecture. Don't expect much enthusiasm from the mathematical community.
Goldbach's Conjecture. Any even integer greater than 4 is the sum of two odd primes.
Vinogradov (1937): There is an integer N such that any odd integer greater than N is the sum of three primes.
Chen and Wang (1989): N ≤ e e 11503.
Liu and Wang (2002): N ≤ e 3100.
Why aren't mathematician's thrilled?
Over the years, many of us have received purported proofs of famous conjectures or recently proven theorems. Examples
- The four colour theorem: The shortest accepted proofs so far (Haken and Appel, Seymour) have 500 or more cases. No mathematican expects that somebody will find a two-page solution in the near future.
- Fermat's last theorem: Andrew Wiles solved this (with a little help on one piece) after a seven-year effort. The proof is several hundred pages long. Again, no short proof is expected.
- Angle trisection, duplication of the cube, squaring the circle: These cannot be done with ruler and compass. It is extremely hard to convince a non-mathematician of this. However, the proof is understandable to students in undergraduate mathematics programs.
If I left out your favorite problem, you don't need to contact me.
What we all think will happen with the Goldbach award:
Many amateurs will construct what they believe are proofs. Because the contest requires the winning proof to be acceptable to a standard mathematics journal, the journals will receive many submissions on this topic. Then, the journals send those submissions to [unpaid] referees, who will most certainly not appreciate the increase in the workload. Perhaps the magazine is willing to pay referee fees? To top it off, most submissions by amateurs are not well-written, and most amateurs do not believe that the referees cannot follow their proofs. Some submitters will go so far as to try to sue the journals/referees for stealing their work. Don't laugh, it has happened.
Last modified April 11, 2000, by S.C. Locke.