Chap. IV] RESIDUE OF {U^^ — l)/p MODULO p. Ill 
' L. Bastien^^" verified that (1) holds for p< 50 only for p= 43, a= 19, and 
for Jacobi's^ cases. He stated that, if p= 4p='= 1 is a prime, 
-1^2=1 + 1/3+1/5+... +1/(2/1-1) (mod p). 
W. Meissner^^ gave a table showing the least positive residue of (2' — l)/p 
modulo p for each prime p< 2000, where t is the exponent to which 2 belongs 
modulo p. In particular, 2^ — 2 is divisible by the square of the prime 
p = 1093, contrary to Proth^ and Grave,^^ but for no other p<2000. 
In the chapter on Fermat's last theorem will be given not only the con- 
dition q2—0 (mod p) of Wieferich^^ but also q^^O (mod p), etc., with cita- 
tions to D. Mirimanoff, Comptes Rendus Paris, 150, 1910, 204-6, and Jour, 
fiir Math., 139, 1911, 309-324; H. S. Vandiver, ibid., 144, 1914, 314-8; 
G. Frobenius, Sitzungsber. Ak. Wiss. Beriin, 1910, 200-8; 1914, 653-81. 
These papers give further properties of q^. 
P. Bachmann^^ employed the identity 
(a-\-h-\-cy-{a+b-cy+ia-h-cy-{a-h+cy 
= 2(^)cl(a+6r-^-(a-6r-n+2(|)c^l(a+6r-^-(a-6r-^f + ... 
for a = 6 = 1, c = 2 or 1 to get expressions for ^2 or q^, whence 
for an odd prime p. Comparing this with the value of (3^ — 3)/p obtained 
by expanding (2+1)^, we see that 
?!ll2_2P-i_^i.2P-2_|_i.2P-3+ , . . +-^-2 (mod p). 
p p-l 
Again, 
92^2- (^) Vssfirn.(s-0 {-ly+'+'s (mod p), 
summed for all sets of solutions of s^=f^+l (mod p). Finally, 
g2^s'|(r''-r-'')2(r2'"'-l)-i|, 
h=l[ " J 
where r is a primitive pth root of unity. 
*H. Brocard^^ commented on a^^=l (mod p""). *H. G. A. Verkaart^^ 
treated the divisibility of a^ — a by p. E. Fauquembergue^^ checked that 
2^=2 (mod p2) for p = 1093. 
N. G. W. H. Beeger^^ tabulated all roots of a;^~^= 1 (mod p^) for each 
prime p<200. If w is a primitive root of p^, the absolutely least residue 
32aSphinx-Oedipe, 7, 1912, 4-6. It is stated that G. Tarry had verified in 1911 that 2P-2 is 
not divisible by a prime p < 1013. 
"Sitzungsber. Ak. Wiss. Berlin, 1913, 663-7. 
"Jour, fiir Math., 142, 1913, 41-50. 
'^Revista de la Sociedad Mat. Espanola, 3, 1913-4, 113-4. 
'"Wiskundig Tijdschrift, vol. 2, 1906, 238-240. 
"L'interm6diaire des math., 1914, 33. 
38Messenger Math., 43, 1913-4, 72-84. 
