262 HISTORY OF THE THEOEY OF NUMBERS. [CHAP, vm roots incongruent modulo p(p — 1), where 5 ranges over all divisors >1 of 6. If ind a=0, the number of such roots is p — 1-f N, where now 5 ranges over the divisors >1 of p-1. A. Chatelet180 noted that divergences between congruences and equations are removed by not limiting attention to the given congruence /(z) = 0 of degree n, but considering simultaneously all the polynomials g(x) derived from f(x) by a Tschirnhausen transformation ky = <t>(x), where k is an integer and <t> has integral coefficients and is of degree n—1. *M. Tihanyi1800 proved a simple congruence. R. Kantor181 discussed the number of incongruent values modulo m taken by a polynomial in n variables, and especially for aa?+...+d modulo p*, generalizing von Sterneck.140 The solvability of x3+9z+6==0 and x3+y(y+I)=Q (mod p) has been treated.182 A. Cunningham183 announced the completion, in conjunction with Woodall and Creak, of tables of least solutions (x, a) of the congruences (mod pk< 10000), r = 2, 10;*/ = 3, 5, 7, 11. T. A. Pierce184 gave two proofs that /(as) =0 (mod p) has a real root if and only if the odd prime p divides 11(1 — a^""1), where a* ranges over the roots of the equation /(x) = 0. Christie185 stated that £p(f+l) = l (mod p) if £= 2 sin 18° and p is any odd prime. Cunningham gave a proof and a generalization. *G. Rados186 found the congruence of degree r having as its roots the r distinct roots 5^0 of a given congruence of degree p — 2 modulo p, a prime. 180Comptes Rendus Paris, 158, 1914, 250-3. wMatb. 6s Phys. Lapok, Budapest, 23, 1914, 57-60. 181Monatshefte Math. Phys., 26, 1915, 24-39. l82Wiskundige Opgaven, 12, 1915, 211-2, 215-7. 188Messenger Math., 45, 1915-6, 69. ""Annals of Math., (2), 18, 1916, 53-64. itfMath. Quest. Educ. Times, 71, 1899, 82-3. 18flMath. 6s Terme's firtesito, 33, 1915, 702-10.