  It is the purpose of this thesis to enunciate and prove a collection of
explicit results in the theory of prime numbers.
  First, the problem of primes in short intervals is considered. We prove that
there is a prime between consecutive cubes $n^3$ and $(n+1)^3$ for all $n \geq
\exp(\exp(33.3))$. To prove this, we first derive an explicit version of the
Riemann--von Mangoldt explicit formula. We then assume the Riemann hypothesis
and show that there will be a prime in the interval $(x-4/ \pi \sqrt{x} \log x,
x]$ for all $x > 2$. Moreover, we show that the constant $4/\pi$ can be reduced
to $(1+\epsilon)$ for all sufficiently large values of $x$.
  Using explicit results on primes in arithmetic progressions, we prove two new
results in additive number theory. First, we prove that every integer greater
than 2 can be written as the sum of a prime and a square-free number. We then
work similarly to prove that every integer greater than 10 and not congruent to
$1$ modulo $4$ can be written as the sum of the square of a prime and a
square-free number.
  Finally, we provide new explicit results on an arcane inequality of
Ramanujan.
