262 History of the Theory of Numbers. [Chap, viii 
iV = 0(p-l)+2^^) 
roots incongruent modulo p(p — 1), where 5 ranges over all divisors > 1 of ^. 
If ind a = 0, the number of such roots is p — l-\-N, where now 5 ranges over 
the di\asors >1 of p — 1. 
A. Chdtelet^^° noted that divergences between congruences and equa- 
tions are removed by not Umiting attention to the given congruence fix) = 
of degree n, but considering simultaneously all the polynomials g{x) derived 
from f{x) by a Tschirnhausen transformation ky — (i>{x), where k is an 
integer and </> has integral coefficients and is of degree n — 1. 
*M. Tihanyi^^"" proved a simple congruence. 
R. Kantor^^^ discussed the number of incongruent values modulo m 
taken by a polynomial in n variables, and especially for ax^-\- . . .+d 
modulo p', generalizing von Sterneck.^"*" 
The solvabiUty of x^+9a;+6=0 and x^-\-y{y-\-l) = (mod p) has been 
treated.^«2 
A. Cunningham^^ announced the completion, in conjunction with 
Woodall and Creak, of tables of least solutions {x, a) of the congruences 
T^=^y% rV==*=l (mod p*< 10000), r = 2, 10; ?/ = 3, 5, 7, 11. 
T. A. Pierce^^^ gave two proofs that /(a;) = (mod p) has a real root if 
and only if the odd prime p divides 11(1— a^^"^), where a^ ranges over the 
roots of the equation f{x) = 0. 
Christie^^ stated that P(F+1) = 1 (mod p) if t= 2 sin 18° and p is any 
odd prime. Cunningham gave a proof and a generalization. 
*G. Rados^^^ found the congruence of degree r having as its roots the 
r distinct roots ?^0 of a given congruence of degree p — 2 modulo p, a 
prime. 
i8'€omptes Rendus Paris, 158, 1914, 250-3. 
"""Math, es Phys. Lapok, Budapest, 23, 1914, 57-60. 
i8iMonatshefte Math. Phys., 26, 1915, 24-39. J 
'»2Wiskundige Opgaven, 12, 1915, 211-2, 215-7. 
'"Messenger Math., 45, 1915-6, 69. 
'"Annals of Math., (2), 18, 1916, 53-64. 
i«*Math. Quest. Educ. Times, 71, 1899, 82-3. 
'»Math. is Term6s firtesito, 33, 1915, 702-10. 
