Unsolved Problems

Goldbach's Conjecture

Every even integer greater than 2 can be expressed as the sum of two primes.

One of the oldest and best-known unsolved problems in number theory and all of mathematics. The conjecture has been shown to hold for all integers less than 4 × 10^18, but remains unproven despite considerable effort.


8 = 5 + 3
18 = 11 + 7

The image also illustrates a way of visualizing the conjecture.

Hailstone Sequence (Collatz Conjecture)

Start with any number:

If the number is even then divide by 2

If the number is odd then multiply by 3 and then add 1.


The conjecture is that the sequence will always end up at 1.