Here's a famous unsolved problem: is every even
number greater than 2 the sum of 2 primes?
The Goldbach conjecture, dating from
1742, says that the answer is yes.
Some simple examples:
4=2+2, 6=3+3, 8=3+5, 10=3+7, ..., 100=53+47, ...
What is known so far:
Schnirelmann(1930): There is some N such that every
number from some
point onwards can be written as the sum of at most N primes.
Vinogradov(1937): Every odd number from some point onwards
can be written as the sum of 3 primes.
Chen(1966): Every sufficiently large even integer is the
sum of a prime and an "almost prime" (a number with at
most 2 prime factors).
See the reference for more details.
Presentation Suggestions:
Have students suggest answers for the first few even
numbers.
The Math Behind the Fact:
This conjecture has been numerically verified for all even
numbers up to several million.
But that doesn't make it true for all N...
see Large Counterexample for an example of
a conjecture whose first counterexample occurs for
very large N.
How to Cite this Page:
Su, Francis E., et al. "Goldbach's Conjecture."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.
