266 
MATHEMATICS: H. S. VANDIVER Proc. N. A. S. 
ON RUMMERS MEMOIR OF 1857 CONCERNING FERMATS 
LAST THEOREM 
By H. S. Vandiver 
Department op Mathematics, CorneivL University 
Communicated by L. B. Dickson, March 6, 1920 
The article referred to in the title appeared in the Mathematische Ab- 
kandlungen of the Berlin Academy for the year 1857, pp. 41-74. In the 
present note it will be pointed out that the proofs of the results given by 
Kummer are inaccurate and incomplete in several respects. The dis- 
cussion will be confined exclusively to the indication of those parts of the 
argument which are deficient or incorrect, and in no case will methods for 
correcting the proofs or supplying the missing analysis be considered here. 
In the first place Kummer divides his work, as usual, into the discussion 
of the two cases arising from 
+ / + 2^ = 0 (1) 
where x, y and z are rational integers and X is an odd prime. The first case 
arises in the consideration of (1) when xyz is prime to X, and the second 
case when xyz is divisible by X. 
For the first case he shows that if (1) is satisfied then B^x — i)/^ and 
j5(x_5)/2 are each divisible by X, the B's being BernouUian numbers, 
= Ve, = V30, etc. 
The proof is given on pages 61-65 of the article in question. No criti- 
cisms will be made here of this part of the work. 
For the second case Kummer concludes that the relation (1) is impossible 
under the three assumptions: 
1. The first factor of the class number H is divisible by X but not by X^ 
2. If B^ = O(mod X), V2(X~ 1)2, there exists an ideal in the field Q{a), 
CK^ = 1, a =±= 1, with respect to which as a modulus the unit 
k = o 
is not congruent to the \th power of an integer in 
3. The BernouUian number B^x ^ot divisible by X^. 
In the above assumptions H — hi, h2, where 
P , D X— 1 
hi — J, hi = —y p, = , 
(2X)'^-^ A 2 
and P, D and A are defined as follows : 
Let i3 be a primitive root of /3^~ ^ = 1 and X a primitive root of X. Then 
P = n0(/3'^'-^),0(« = 1 + ^^^ + ^2/32-f .... -f 7x-2/3'~' 
i = 1 
where 7,- is the least positive residue of 7* modulo X Also 
1 (1 - - a~) 
