112 History of the Theory of Numbers. [Chap, iv 
±xi modulo p2 of CO" is a root, that (^xg) of xi^ is a second root, that ( ^Xg) 
of X1X2 is a third root, etc., until the root =tx, is reached, where s = (p-l)/2. 
The remaining roots are p^-x,(i = l,. . ., s). He proved that 
ixi...xy={-l)-^ (modp2). 
Hence Xi. . .x,= ±l if p = 4n+l. 
W. Meissner^^ WTote /i^ for the residue <p'" of /i^""' modulo p"*. When 
h varies from 1 to p-1, we get p-1 roots h^ of xP-^=1 (mod p"*). The 
product of the roots given by h = l,.. ., (p-l)/2, is =(-1)' or (-l)V 
(mod p'"), according as p = 4m-1 or 4n+l, where z is the number of pairs 
of integers <p/2 whose product is = -1 (mod p), and c is the smaller of 
the two roots of x^=-l (mod p). No number <p which belongs to one 
of the exponents 2, 3, 4, 6, modulo p, can be a root of x^^=l (mod p^). 
A root of the latter is given for each prime p< 300, and a root modulo p^ for 
each p<200; also the exponent to which each root belongs. 
N. Nielsen^^ noted that, if we select 2r distinct integers a„ &, (s = 1, . . . ,r) 
from 1, . . ., p-1, such that a,+b, = p, then 
"^ = i-mi-pA), A^ ^ = k^lg^-g^^'j (modp). 
Proof is given of various results by Lerch,-^ also of simple relations between 
Qa and BernouUian numbers, and of the final formula by Plana,^ here attrib- 
uted to Euler.^^ 
H. S. Vandiver^- p roved that there are not fewer than [Vp] and not 
more than p — (l + \/2p— 5)/2 incongruent least positive residues of 
1, 2^\..., (p-l)^\ modulo f. 
N. Nielsen^^ noted that, if a is not di\'isible by the odd prime p, 
a — I (p-3)/2 2 
9a=-^+ S^ 2j^-ir'i5.(a^-'^-^-l) (modp), j 
gi+?2+...+gp-i=(-l)"-'5„+--l (modp2), n=ip-l)/2. 
V 
W. Meissner^ gave various expressions for ^2 and ^3. 
A. G^rardin^^ found all primes p<2000, including those of the form 
2"—!, for which 52 is sjTnmetrical when written to the base 2. 
H. S. Vandiver^^ proved that 52—0 (mod p^) if and only if 
He gave various expressions for (n* — l)/m. 
"Sitzungsber. Berlin IMath. Gesell., 13, 1914, 96-107. 
"Ann. sc. I'^cole norm, sup., (3), 31, 1914, 171-9. 
"Euler, Institutiones Calculi Diff., 1755, 406. Proof, Math. Quest. Educ. Times, 48, 1888, 48. 
«BuU. AmQT. Math. Soc, 22, 1915, 61-7. 
«K)versigt Danske Vidensk. SeLsk. ForhandUnger, 1915, 518-9, 177-180; cf. Lerch's»» N. 
♦^Mitt. Math. Gesell. Hamburg, 5, 1915, 172-6, 180. 
«Xouv. Ann. Math., (4), 17, 1917, 102-8. 
"Annals of Math., 18, 1917, 112. 
