1 10 History of the Theory of Numbers. [Chap, iv 
H. F. Baker" extended Sylvester's theorem to any modulus N: 
A' »-l N — TTli 
where the m, denote the integers <A^ and prime to A'", N'N=1 (mod r), 
and \k\ is the least positive residue modulo r of k. 
Lerch^ extended Mirimanofif's^^ formula to the case of a composite 
modulus m. Set 
m 
Let a belong to the exponent <f){m)/e. Then q{a, m)=e'Za/^ (mod m), 
where /? ranges over the residues of the incongruent powers of a, and 
wa+j3=0 (mod a), 0^a<a. As an extension of Sylvester's theorem, 
T T ' 
q(a, w)=2-= —2-^ (mod m), 
V V 
where v ranges over the integers < m and prime to m, while 
7nr,-\-v=0, wr/ — ?'=0 (mod a), 0^r,<a, 0^r,'<a. 
For m = mi. . .rtik, where the rrij are relatively prime, 
k 
q{a, m) = 2 njn/(l){nj)q{a, nij) (mod m), 
where m = mjnj, n/n/=l (mod mj). 
H. Hertzer-'* verified that, for a<p<307, a^^ — 1 is di\'isible by p^ only 
for a = 68, p = 113; a = 3, 9, p = ll. He examined all the primes between 
307 and 751, but only for a and p — a when a<y/p, finding only p = 113, 
a = 68. Removing the restriction a< Vp^ be found only the solutions 
p = ll,a = 3; p = 331, a = 18, 71; p = 353,a = 14; 
p = 487, a = 10, 175; p = 673, a = 22, 
together with the square of each a. 
A. Friedmann and J. Tamarkine^^ gave formulas connecting q^ with 
Bernoullian numbers and [u/p]. 
A. Wieferich^® proved that if x^+y^-\-z^ = Q is satisfied by integers 
X, y, z prime to p, where p is an odd prime, then 2""^ = 1 (mod p^). Shorter 
proofs were given by D. IMirimanoff-^ and G. Frobenius.'* 
D. A. Grave- ^ gave the residue of q^ for each prime p< 1000 and thought 
he could prove that 2^ — 2 is never divisible by p^ (error, Meissner^). 
A. Cunningham^" verified that 2^ — 2 is not divisible by p^ for any prime 
p< 1000, and^^ that 3^ - 3 is not divisible by p^ for a prime p = 2''3''+ 1< 100. 
W. H. L. Janssen van Raay^^ noted that 2^ — 2 is not divisible by p^ in 
general. 
»Proc. London Math. Soc, (2), 4, 1906, 131-5. "Comptes Rendus Paris, 142, 1906, 35-38. 
"Archiv Math. Phys., (3), 13, 1908, 107. «Jour. fur Math., 135, 1909, 146-156. 
»Jour. fiir Math., 136, 1909, 293-302. "L'enseignement math., 11, 1909, 455-9. 
"Sitzungsber. Ak. Wiss. Berlin, 1909, 1222-4; reprinted in Jour, fiir Math., 137, 1910, 314. 
**An elementary text on the theory of numbers (in Russian), I^ev, 1909, p. 315; Kiev Izv. Univ., 
1909, Nos. 2-10. 
"Report British Assoc, for 1910, 530. L'interm^diaire des math., 18, 1911, 47; 19, 1912, 159. 
Proc. London Math. Soc, (2), 8, 1910, xiii. 
"L'interm^diaire des math., 18, 1911, 47. Cf., 20, 1913, 206. 
"Nieuw Archief voor Wiskunde, (2), 10, 1912, 172-7. 
