Easy Hard Problems

[packbat]

"What's the easiest unsolved math problem to explain?" I asked my dad tonight, just out of curiosity. I asked because the two obvious, famous answers - Fermat's last theorem and the four-color problem - are both (probably) solved.

Well, I can't guarantee the actual answer is here, but a few candidates he pointed me to:

• The P = NP problem: if the answer to a computational yes-no question can be checked quickly (in polynomial time), does that imply it may be answered quickly (in polynomial time)? This is a marginal case, as a lot of people don't know what "polynomial time" is, so two better candidates are...
  
• Goldbach's conjecture: that every even integer greater than 2 can be written as the sum of two primes, and...
  
• The twin prime conjecture: that there exist an infinite number of twin primes - primes separated by two (like 3 and 5). (Bonus: this is a special case of Polignac's conjecture.) However, there are a pair which do not even require understanding primes...
  
• The existence of (a) an infinite number of even perfect numbers and/or (b) the existence of any odd perfect number. Perfect numbers being, in these examples, those which equal the sum of all the divisors smaller than themselves - such as 6, equal to 1+2+3, and 28, equal to 1+2+4+7+14.

(Now, one could argue that an even easier hard problem to state is "how come things fall", but that's physics!)