222 History of the Theory of Numbers. [Chap, vii 
Hence ±7? and =»=iV are the roots of x^=— 1 (mod p) according as 
p = S7n + l or Sm + 5. ,;, 
U. Concina"^ proved the first result by Legendre.^^ 
A. Cunningham--^ tabulated the roots of i/*=±2, 2?/"*=±l (mod p), 
for each prime p< 1000. 
Cunningham"^ listed the roots of ?/'= ± 1 (mod p"), where l^qp", 
p being an odd prime ^19, p''<10^, a = l and often also a = 2, q a factor 
of p— 1. 
A. Gcrardin and L. Valroff"7 solved 2i/=l (mod p), 1000<p<5300. 
Cunningham-^^ announced the completion of tables giving all proper 
roots of ?/'"= 1 (mod p*) for m odd ^15, and of 'y'"= — 1 (mod p*) for m even 
^ 14. These tables have since been completed up to p* < 100000 and are 
now nearly all in type. 
T. G. Creak^-^ announced the completion of like tables for m = 16 to 
50; 52, 54, 56, 63, 64, 72, 75, and 10^<p'^<10^ 
H. C. Pocklington--^ noted that if p is a prime 8m+5 and a}"'^^=-l, 
x^=a (mod p) has the roots =»=^(4a)'"'^^ He showed how to use {t-\- 
u\/DY to solve a;-= —D (mod p=4A-+l), and treated a:^=a. 
*J. Maximoff^^° treated binomial congruences and primitive roots. 
*G. Rados-^^ gave a new proof of known criteria for the solvability of 
x- = D (mod p). He-^^ gave a new exposition of the theory of binomial 
congruences without using indices. 
Congruences ^''"^^l (mod p") are treated in Chapter IV. Euler**'' 
of Ch. XVI solved x-=— 1 (mod p). Lazzarini"^ of Ch. I erred on the 
number of roots of 2-= —3 (mod n). Many papers in Ch. XX treat x*=a; 
(mod 10"). The following papers from the first part of Ch. VII treat also 
binomial congruences: Euler,^ Lagrange,^ Poinsot," Cauchy,^^ Lebesgue," 
Epstein,i^2 Korkine."^ 
=«*Periodico di Mat., 28, 1913, 212-6. 
22*Messenger Math., 43, 1913-4, 52-3. 
2"/Wd., 148-163. Cf. Cunningham .201 
227Sphinx-0edipe, 1913, 34; 1914, 18-37, 73. 
228Messenger Math., 45, 1915-6, 69. 
2^Proc. Cambridge Phil. Soc, 19, 1917, 57-9. 
2^Bull. Soc. Phys.-Math. Kasan, (2), XXI. 
23iMath. 4s Term^s Ertesito, 33, 1915, 758-62. 
^Ibid., 34, 1916, 641-55. 
