Pollock's Conjecture

Pollock’s Conjecture (1850) states that every number is the sum of at most 5 tetrahedral numbers; this conjecture has not yet been proven. It is also conjectured that there are exactly 241 numbers that cannot be written as the sum of 4 or fewer tetrahedral numbers.

The conjecture is almost certainly true, but has not yet been proven.

http://mathworld.wolfram.com/PollocksConjecture.html