420 
MATHEMATICS: H. S. VANDIVER 
Proc. iST. A. 
+ + = 0 (13) 
is impossible in integers prime to each other if xyz = 0 (mod p). 
I shall now show that this criterion constitutes no extension over the 
one given by Kummer to the effect that (13) is impossible in integers if 
is a regular prime. For if the former applies to other than regular primes 
as exponents, then p must be a prime such that 
iP - 3)/2 
nBi = 0 (mod p), 
i = 1 
This being the case, it follows from (12) that ki = 0 (mod p), and by Kum- 
mer's theorem,^ k is also divisible by p. Hence /z] = 0 (mod p^) con- 
trary to Bernstein's assumption that the class number of il{e^^'^^^^) is not 
divisible by p"^. 
In a paper cited above the writer pointed out errors in Kummer' s memoir 
of 1857 on the relation (13). Considering this in connection with the 
failure of Bernstein's criterion it follows that Fermat's last theorem has 
not been rigorously proved for all non-regular primes less than 100. These 
non-regular primes are 37, 59 and 67. Mirimanoff^ gave an adequate 
proof for the case p = 37, which leaves the cases ^ = 59 and 67 still to be 
disposed of. 
In connection with the cyclotomic class number it may be noted that 
Furtwangler^ proved that the class number of 12(e2tv/:/?«) jg divisible by 
the prime p if, and only if, the first factor of the class number of the field 
12(/*''/^) is divisible by p. 
2. Bernstein^ also gave a criterion in connection with (13). for the case 
where x, y and z are prime to ^. He makes two assumptions the first 
being to the effect that the second factor of the class number of 12 (e^^'"^^) 
is divisible by p, and states /. <;., p. 507, that his criterion includes that, 
of Kummer' s 1857 memoir as a special case. This is not correct, how- 
ever, as there are no primes, p less than 100 such that the 2nd factor of 
the class number is divisible by p. (This follows from some computations 
by Kummer^ in connection with certain investigations regarding the last 
theorem.) This being the case, Bernstein's criterion can at most supple- 
ment Kummer's which latter criterion stated that if (13) is satisfied in 
integers such that xyz is prime to p then _ 3)/3 and _ 53/2 are divis- 
ible^o by p. 
This criterion would eliminate, for example, all ^'s less than 100, which 
Bernstein's condition does not do. 
3. Kummer rigorously established the theorem that (13) cannot be 
satisfied in integers prime to each other if ^ is a regular prime, and noted 
that all primes < 100 were regular except 37, 59 and 67. But Kummer 
also showed that the first factor of the class numbers of ^{e^^"^^^) for all 
primes p, where 100 < ^ < 167, excepting p = 101, 103, 131, 149 and 
157, was in each case prime to p.^^ It follows that the primes p which are 
