What are the hardest math rational number questions?

Answer 1

There is no such thing. Something that one person might find hard, another may find easy - and conversely. Within rational numbers, there are some questions concerning primes which are so hard that mathematicians have not found an answer.

One such is the Goldbach conjecture. In 1742, Goldbach wrote to Euler stating that "every number that is greater than 2 is the sum of three primes". Note that in those days the number 1 was considered to be a prime: that convention that is no longer followed. As re-expressed by Euler, an equivalent form of Goldbach's conjecture asserts that all positive even integers >=4 can be expressed as the sum of two primes.

By early 2012, the Goldbach conjecture has been shown to be true for integers up to four quintillion (4*10^18), but has yet to be proved. If you can solve this, I believe that there is a USD1,000,000 prize from the Clay Mathematics Institute.