CHAP. IV] RESIDUE OF (U^1—l)/p MODULO p. Ill L. Bastien320 verified that (1) holds for p<50 only for p=43, a= 19, and for Jacobi's2 cases. He stated that, if p= 4p± 1 is a prime, -tesl+l/3+l/5+...+l/(2fc--l) (mod p). W. Meissner33 gave a table showing the least positive residue of (2' — l)/p modulo p for each prune p<2000, where t is the exponent to which 2 belongs modulo p. In particular, 2P—2 is divisible by the square of the prune p = 1093, contrary to Proth7 and Grave,29 but for no other p<2000. In the chapter on Fermat's last theorem will be given not only the condition #2=0 (mod p) of Wieferich26 but also #3=0 (mod p), etc., with citations to D. Mirimanoff, Comptes Rendus Paris, 150,1910, 204-6, and Jour, fur Math., 139, 1911, 309-324; H. S. Vandiver, ibid., 144, 1914, 314-8; G. Frobenius, Sitzungsber. Ak. Wiss. Berlin, 1910, 200-8; 1914, 653-81. These papers give further properties of qu. P. Bachmann34 employed the identity = 2ci(a+6)p-1-(a-5rH+2c3{(a+6)^3~(a for a = 6 = 1, c = 2 or 1 to get expressions for q2 or #3, whence +...< for an odd prime p. Comparing this with the value of (3P—3)/p obtained by expanding (2+l)p, we see that _Z- ==2p-1+£-2p-2+f-2p-3+... H-------~2 (mod p). p p-1 Again, * _1 \ 2 summed for all sets of solutions of s2=£2+l (mod p). Finally, where r is a primitive pth root of unity. *H. Brocard36 commented on ap~l^l (mod pn). *H. G. A. Verkaart36 treated the divisibility of ap — a by p. E. Fauquembergue37 checked that 2P=2 (mod p2) for p = 1093. N. G. W. H. Beeger38 tabulated all roots of x*~l=l (mod p2) for each prime p<200. If co is a primitive root of p2, the absolutely least residue 82aSphinx-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. MSitzungsber. Ak. Wiss. Berlin, 1913, 663-7. "Jour, fttr Math., 142, 1913, 41-50. "Revista de la Sociedad Mat. Espafiola, 3, 1913-4, 113-4. 86Wiskundig Tijdschrift, vol. 2, 1906, 23S-240. 87L'interme*diaire des math., 1914, 33. 38Messenger Math., 43, 1913-4, 72-84.