One of the earliest opportunities to impress on the student the
fact that mathematical knowledge is very limited in certain
directions presents itself in connection with the study of
prime numbers. Among the small prime numbers there appear many
which differ only by 2. For instance, 3 and 5, 5 and 7, 11 and
13, 17 and 19, 29 and 31, constitute such pairs of prime
numbers. The question arises whether there is a limit to such
pairs of primes, or whether beyond each such pair of prime
numbers there must exist another such pair.

This question can be understood by all and might at first
appear to be easy to answer, yet no one has succeeded up to the
present time in finding which of the two possible answers is
correct. It is interesting to note that in 1911 E. Poincare
transmitted a note written by M. Merlin to the Paris Academy of
Sciences in which a theorem was announced from which its author
deduced that there actually is an infinite number of such prime
number pairs, but this result has not been accepted because no
definite proof of the theorem in question was produced.

Another unanswered question which can be understood by all is
whether every even number is the sum of two prime numbers. It
is very easy to verify that each one of the small even numbers
is the sum of a pair of prime numbers, if we include unity
among the prime numbers; and, in 1742, C. Goldbach expressed
the theorem, without proof, that every possible even number is
actually the sum of at least one pair of prime numbers. Hence
this theorem is known as Goldbach's theorem, but no one has as
yet succeeded in either proving or disproving it.