**Goldbach Conjecture:**Goldbach asserts that all positive even integers >=4 can be expressed as the sum of two primes. Two primes (p,q) such that p+q=2n for n a positive integer are sometimes called a Goldbach partition (Oliveira e Silva).**Riemann Hypothesis:**neither the Riemann zeta function nor any Dirichlet L-series has a zero with real part larger than 1/2.**Twin prime conjecture:**the conjecture that there are an infinite number of twin primes**The 196-algorithm:**proof that the 196-algorithm does not terminate when applied to the number 196**Proof that 10 is a solitary number:**A solitary number is a number which does not have any friends. Solitary numbers include all primes, prime powers, and numbers; some numbers can be proved not to be solitary by finding another integer with the same index, although sometimes the smallest such number is fairly large; a proof appears to be extremely difficult**Determining if any odd perfect numbers exist:**Perfect numbers are positive integers n such that n = s(n), where s(n) is the restricted divisor function (i.e., the sum of proper divisors of n), or equivalently sigma(n)=2n, where sigma (n) is the divisor function (i.e., the sum of divisors of n including n itself); for example, the first few perfect numbers are 6, 28, 496, 8128

Ex.) 6 = 1 + 2 + 3

Ex.) 28 = 1 + 2 + 4 + 7 + 14

Ex.) 496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248

(Source: Wolfram MathWorld)