The theory of numbers, by Robert D. Carmichael ...

OTHER TOPICS

91

~ 47. THE EQUATION x"+yn=zn.
The following theorem, which is commonly known as Fermat's Last Theorem, was stated without proof by Fermat
in the seventeenth century:
If n is an integer greater than 2 there do not exist integers
x, y, z, all different from zero, such that
x+yn=Z (I)
No general proof of this theorem has yet been given. For
various special values of n the proof has been found; in particular, for every value of n not greater than Ioo.
In the study of equation (i) it is convenient to make some
preliminary reductions. If there exists any particular solution
of (i) there exists also a solution in which x, y, z are prime
each to each, as one may show readily by the method employed
in the first part of ~ 46. Hence in proving the impossibility
of equation (I) it is sufficient to treat only the case in which
x, y, z are prime each to each.
Again, since n is greater than 2 it must contain the factor
4 or an odd prime factor p. If n contains the factor p we write
n = mp, whence we have
(Xm) + (ym)P = (z M)P
If n contains the factor 4 we write n = 4m, whence we have
(XI)I + (y" I = (Z )4
From this we see that in order to prove the impossibility of
(I) in general it is sufficient to prove it for the special cases when
n is 4 and when n is an odd prime p. For the latter case the
proof has not been found. For the former case we give a
proof below. The theorem may be stated as follows:
I. There are no integers x, y, z, all different from zero, such
that
X4 +y4 = 4.