Questions on problems that have yet to be completely solved by current mathematical methods.
There are many unsolved problems in mathematics. Some prominent outstanding unsolved problems (as well as some which are not necessarily so well known) include
The Goldbach conjecture.
The Riemann hypothesis.
The conjecture that there exists a Hadamard matrix for every positive multiple of 4.
The twin prime conjecture (i.e., the conjecture that there are an infinite number of twin primes).
Determination of whether NP-problems are actually P-problems.
The Collatz problem.
Proof that the $196$-algorithm does not terminate when applied to the number $196$.
Proof that $10$ is a solitary number.
Finding a formula for the probability that two elements chosen at random generate the symmetric group $S_n$.
Solving the happy end problem for arbitrary $n$.
Finding an Euler brick whose space diagonal is also an integer.
Proving which numbers can be represented as a sum of three or four (positive or negative) cubic numbers.
Lehmer's Mahler measure problem and Lehmer's totient problem on the existence of composite numbers $n$ such that $\phi(n)|(n-1)$, where $\phi(n)$ is the totient function.
Determining if the Euler-Mascheroni constant is irrational.
Deriving an analytic form for the square site percolation threshold.
Determining if any odd perfect numbers exist.