relate to prime numbers. As an instance of one which does not
relate to prime numbers we may refer to the question whether
there exists an odd perfect number. A perfect number is a
natural number which is equal to the sum of its aliquot parts.
Thus 6 is perfect because it is equal to 1 + 2 + 3, and 28 is
perfect because it is equal to 1 + 2 + 4 + 7 + 14. Euclid
stated a formula which gives all the even perfect numbers, but
no one has ever succeeded in proving either the existence or
the non-existence of an odd perfect number. A considerable
number of properties of odd perfect numbers are known in case
such numbers exist.

In fact, a very noted professor in Berlin University developed
a series of properties of odd perfect numbers in his lectures
on the theory of numbers, and then followed these developments
with the statement that it is not known whether any such
numbers exist. This raises the interesting philosophical
question whether one can know things about what is not known to
exist; but the main interest from our present point of view
relates to the fact that the meaning of odd perfect number is
so very elementary that all can easily grasp it, and yet no one
has ever succeeded in proving either the existence or the
non-existence of such numbers.

It would not be difficult to increase greatly the number of the
given illustrations of unsolved questions relating directly to
the natural numbers. In fact, the well-known greater Fermat
theorem is a question of this type, which does not appear more
important intrinsically than many others but has received
unusual attention in recent years on account of a very large