CHAP, xxvi] FERMAT'S LAST THEOREM. 755

A. Eieke137 again attempted to prove xp+yp=zp impossible, but again confused (pp. 251-2) algebraic and arithmetical divisibility, even for p = 3 (p. 253).

E. Lucas138 proved (p. 267, p. 275) the theorem of Cauchy,29 and (p. 370-1) the formulas (1), (3), (4) of Legendre17, with the aim to show that, when x, y, z are relatively prune in pairs, no one of them is a prime or a power of a prune [cf . Markoff157]. He proved (p. 341) the first result due to Jaquemet.3

D. Mirimanoff139 found in terms of the units a necessary and sufficient condition that the second factor [Kummer61] of the class number be divisible by X. He treated in detail the case X = 37.

J. Rothholz140 used the theorem of Kummer25 on the divisors of an±&n to show (?) that xZn^LyZn=zZn has no integral solutions if n is a prune 4fc+3 or if one of the numbers x, y, z is a prime and n is an odd prime; xn+yn =zn is impossible if x, y or z is a power of a prime, the prune not being = I (mod n), while n is an odd prime; xn+yn- (2p)n is impossible if n and p are odd primes; xn±yn=zn is impossible if x} y or z has one of the values 1, • • -, 202. The history of the theorem is discussed at length. On p. 29 are pointed out two errors hi the proof by Rieke.134

* W. L. A. Tafelmacher141 proved Abel's formulas and congruencial corollaries from them. In the second paper he proved that Fermat's equation is impossible for n = 3, 5, 11, 17, 23, 29 and, in case x+y— 2^=0 (mod 7i4) for 71= 7, 13, 19, 31 [tut with proofs valid only when no one of x, yy z is divisible by n, since the argument pp. 273-8 does not suffice to exclude the case in which one of these numbers is divisible by n].

H. Teege142 proved that £5-f?/5=l has no rational solutions by setting

s. Then

Since z is rational, (4s+l)2-4(s-l)(4s+l)==m2. Set m=5ju. Then 4s +1 = 5/A Let /i = 6/a, where a and 6 are relatively prune. Thus

Hence a2 divides 5p5. The impossibility of this equation is proved by considering the cases a divisible or not divisible by 5.

H. W. Curjel143 proved that if $*— y*= 1 and x, y are primes, then z is a prime, t is a power of 2, and x or y equals 2.

Several144 proved by use of cube roots of unity the known result that, if n is odd and not a multiple of 3, (x+y)n — xn —yn is divisible by xz+xy+y2.

S. Levanen145 discussed £5+t/5=2mz5, for x, y, z without common factor,

187 Zeitschr. Math. Phys., 36, 1891, 249-254. Error indicated in 37, 1892, 57, 64.

188 ThSorie des nombrea, 1891. References in Introduction, p. xxix, where it is stated falsely

that Kummer proved Fennat's theorem for all even exponents. «» Jour, ftir Math., 109, 1892, 82-88.

140 Beitrage zum Fennatschen Lehrsatz. Diss. (Giessen), Berlin, 1892.

141 Anales de la Universidad de Chile, Santiago, 82, 1892, 271-300, 415-37. Report from

Lind,**1 p. 50.

^Zeitschr. Math. Naturw. Untemcht, 24, 1893, 272-3. "» Math. Quest. Educ. Times, 58, 1893, 25 (quest, by J. J. Sylvester). *« Ibid., 112.