Page:The Scientific Monthly vol 1 (IA scientificmonthl01ameruoft).pdf/105

 Both of the given instances of unanswered classic questions 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 nonexistence 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 prize offered for its solution. In view of the fact that those who have become interested in this theorem often experience difficulty in finding the desired information in any English publication, we proceed to give some details about this theorem and the offered prize. The following is a free translation of a part of the announcement made in regard to this prize by the Königliche Gesellschaft der Wissenschaften, Gottingen, Germany:

On the basis of the bequest left to us by the deceased Dr. Paul Wolskehl, of Darmstadt, a prize of 100,000 mk., in words, one hundred thousand marks, is hereby offered to the one who will first succeed to produce a proof of the great Fermat theorem. Dr. Wolfskehl remarks in his will that Fermat had maintained that the equation

could not be satisfied by integers whenever λ is an odd prime number. This Fermat theorem is to be proved either generally in the sense of Fermat, or, in supplementing the investigations by Kummer, published in Crelle's Journal, volume 40, it is to be proved for all values of λ for which it is actually true. For further literature consult Hilbert's report