CHAP, vni] MISCELLANEOUS RESULTS ON CONGRUENCES. 261 E. Landau172 proved that, if /(x) = 0 is an equation with integral coefficients and at least one root of odd multiplicity, there exist an infinitude of primes p = 4n — 1 such that /(re) = 0 (mod p) has a root. R. D. von Sterneck173 found the number of combinations of the ith class (with or without repetition) of the numbers prime to p of a complete set of residues modulo p* whose sum is congruent to a given integer modulo p*, p being a prime. E. Piccioli174 gave known theorems on adding and multiplying congruences. C. Jordan175 found the number of sets of integers aih for which the determinant \aa\ of order n is congruent to a given integer modulo M. C. Krediet176 gave theorems on congruences of degree n for a prime modulus analogous to those for an algebraic equation of degree n, including the question of multiple roots. The determination of roots is often simplified by seeking first the roots which are quadratic residues and then those which are non-residues. The exposition is not clear or simple. G. Rados177 proved that, if p is a prime, f(x) =a0xp~2+ . . . +ap_2=0, g(x) = have a common root if and only if each E; - • - +&P-2=0 (mod p) (mod p), where For 0=/', let <£(w) become DQup~2+ . . . +ZV_2; thus /(a) =0 (mod p) has a multiple root if and only if each D*=0 (mod p). Each of these theorems is extended to three congruences. Finally, if f(x) and f'(x) are relatively prime algebraically, there is only a finite number of primes p for which the number of roots of /=0 (mod ph) exceeds the degree of /. G. Frattini178 proved that if p and q are prunes, q a divisor of p — 1, every homogeneous symmetric congruence in q variables is solvable modulo p by values of the variables distinct from each other and from zero except when the degree of the congruence is divisible by q. C. Grotzsch179 noted that if a is a root of xx=a (mod p), where a is prime to p, then z= a (mod p2-~p) is a root, and proved that if 6 is the g. c. d. of ind a and p — 1 and if ind a>0, it has exactly 172Handbuch. . .Verteilung der Primzahlen, 1, 1909, 440. 173Sitzungsber. Ak. Wiss Wien (Math.), 118, 1909, Ha, 119-132. 17«I1 Pitagora, Palermo, 16, 1909-10, 125-7. 178Jour. deMath., (6), 7, 1911, 409-416. 176Wiskundig Tijdschrift, Haarlem, 7, 1911, 193-202 (Dutch). 177Ann. sc. e*cole norm, sup., (3), 30, 1913, 395-412. 178Periodico di Mat., 29, 1913, 49-53. l7»Archiv Math. Phys., (3), 22, 1914, 49-53.