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 H. Poincaré 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.
Although the proof or the disproof of such theorems may not appear to be of great consequence, yet the interdependence of mathematical theorems is most marvelous, and the mathematical investigator is attracted by such difficulties of long standing. These particular difficulties are mentioned here mainly because they seem to be among the simplest illustrations of the fact that mathematics is teeming with classic unknowns as well as with knowns. By classic unknowns we mean here those things which are not yet known to any one, but which have been objects of study on the part of mathematicians for some time. As our elementary mathematical text-books usually confine themselves to an exposition of what has been fully established, and hence is known, the average educated man is led to believe too frequently that modern mathematical investigations relate entirely to things which lie far beyond his training.
It seems very unfortunate that there should be, on the part of educated people, a feeling of total isolation from the investigations in any important field of knowledge. The modern mathematical investigator seems to be in special danger of isolation, and this may be unavoidable in many cases, but it can be materially lessened by directing attention to some of the unsolved mathematical problems which can be most easily understood. Moreover, these unsolved problems should have an educational value since they serve to exhibit boundaries of modern scientific achievements, and hence they throw some light on the extent of these achievements in certain directions.
