FERMAT'S LAST THEOREM, the cele brated proposition that the equation XI) + Yn = Zn cannot be satisfied by integral values of X, Y and Z, and when n is an integer greater than 2. It was stated, though without proof, by the French mathematician Pierre de Fermat, about 250 years ago. Proofs have been found for many other remarkable theorems in the theory of numbers that were given by Fermat in the same manner, but this one has resisted all attempts at demonstration. There is no sufficient reason to believe it false, and it has indeed been proved to be true for every value of n from 3 up to about 97, and also for many special values greater than this; but no general. proof, valid for all values of has yet been given. Many interesting things about the equa tion have been established, however. Only cases where the highest common factor of X and Y, of Y and Z, and of X and Z is 1, need to be considered. It is true for n, for example, if it is true for any factor of n; and this has led mathematicians to limit their study of it to the case in which n is a prime number. When n is prime, it is easy to show that the equation can not be satisfied if any one of the three numbers X, Y and Z is prime. It is also easy to show
that when n is prime there is no solution unless X + Y — Z is divisible by n. Many other simi lar properties are also known, but the general demonstration of the proposition does not ap pear to be possible by any of the methods with which mathematicians are now familiar. A prize of 100,000 marks has been offered by the Royal Academy of Sciences at Gottingen for the solution of the problem. This prize, which does not expire until about the end of the 20th century, has caused the solution of the problem to be undertaken by a large number of people from all walks of life, in the same manner as the old problem of the quadrature of the circle. Of course, the greater part of these attempts are not of the slightest value. However, Kummer developed his valuable theory of ideals in con nection with this theorem, and Zermelo and Mirimanoff have brought the problem distinctly nearer solution. Consult Dickson, L. E., (Fer mat's Last Theorem and the Origin and Nature of the Theory of Algebraic Numbers,' in Annals of Mathematics, Vol. 18 (Lancaster, Pa., 1916-17).