  A binary string representation of prime occurrences is a sequence of bits,
where $1$ entries encode positions of prime numbers. This is a convenient
representation for analysis of prime distribution, since it allows for
application of a broad range of existing string-analysis algorithms to problems
in number theory. Binary strings of prime occurrences can be also generated
with simple algorithms. Here we discuss three such algorithms and we
demonstrate their applicability using the example of proving Goldbach's
hypothesis for some limited sets of even numbers. This work formulates three
open questions (conjectures) regarding the distribution of primes.
