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From the Fun Fact files, here is a Fun Fact at the Easy level:

# Goldbach's Conjecture

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.

Su, Francis E., et al. "Goldbach's Conjecture." Math Fun Facts. <http://www.math.hmc.edu/funfacts>.

References:
Paulo Ribenboim, The Little Book of Big Primes, Springer-Verlag, 1991, pp.154-155.

Keywords:    unsolved problem, number theory
Subjects:    number theory
Level:    Easy
Fun Fact suggested by:   Lesley Ward
Suggestions? Use this form.
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