222 HISTORY OF THE THEORY OF NUMBERS. [CHAP.VII Hence =t=E and ^N are the roots of .#2=—1 (mod p) according as p = 8m+l or &m+5. U. Concina224 proved the first result by Legendre.154 A. Cunningham225 tabulated the roots of 2/4=s±2, 22/45=±l (mod p)r for each prime p< 1000. Cunningham226 listed the roots of t/'==*=l (mod p"), where l = qpa, p being an odd prime g!9, p*<104, a=l and often also a = 2, q a factor ofp —1. A. Gerardin and L. ValrofF7 solved 2?/4=l (mod p), 1000<p<5300. Cunningham228 announced the completion of tables giving all proper roots of 2/m=l (mod pk) for m odd ^ 15, and of ym= — 1 (mod pk) for m even ^ 14. These tables have since been completed up to pk < 100000 and are now nearly all in type. T. G. Creak228 announced the completion of like tables for w = 16 to 50; 52, 54, 56, 63, 64, 72, 75, and 108<p*<104. H. C. Pocklington229 noted that if p is a prime 8w+5 and a2m+l= -1, x2=o_ (mod p) has the roots =t=K4a)m+1. He showed how to use (t+ u-\/D)n to solve z2= —D (mod p = 4fc+l), and treated z3=a. *J. Maximoff230 treated binomial congruences and primitive roots. *G. Rados231 gave a new proof of known criteria for the solvability of z2—D (mod p). He232 gave a new exposition of the theory of binomial congruences without using indices. Congruences xp~l==l (mod pn) are treated in Chapter IV. Euler4'7 of Ch. XVI solved z2= -1 (mod p). Lazzarini172 of Ch. I erred on the number of roots of 22= —3 (mod ri). Many papers in Ch. XX treat xk=x (mod 10n). The following papers from the first part of Ch. VII treat also binomial congruences: Euler,2 Lagrange,3 Poinsot,11 Cauchy,14 Lebesgue,59 Epstein,112 Korkine.118 224Periodico di Mat., 28, 1913, 212-6. 22SMessenger Math., 43, 1913-4, 52-3. mlbid., 148-163. Cf. Cunningham.201 227Sphinx-Oedipe, 1913, 34; 1914, 18-37, 73. 228Messenger Math., 45, 1915-6, 69. ^Proc. Cambridge Phil. Soc., 19, 1917, 57-9. 23°Bull. Soc. Phys.-Math. Kasan, (2), XXI. M1Math. £s Term6s firtesito, 33, 1915, 758-62. mlbid., 34, 1916, 641-55.