CHAP. X] QUADRATIC CONGRUENCE IN n UNKNOWNS. 327 



H. Minkowski 10 found the number /{w; N] of sets of solutions of 



n 



f 2-j QikXiXk = in (mod N). 

 If 



N 

 m=l 



then 



so that the problem remains to find/(w; N) whose determination depends 

 upon that of 2p m/ , where x\, -, x n range each over a complete set of 

 residues modulo N. The problem is reduced to the case of a power of 

 prime modulus. The paper is too complicated to admit of a brief report. 

 L. Gegenbauer 11 considered / = a\x\ + + a n xl, with r of the a's 

 quadratic residues of the odd prime p. Let <r' n (r) be the number of sets of 

 solutions of / = (mod p}, and cr n (r) the number of those in which no x = 0. 

 Let s' and s be the corresponding numbers for / = 1 (to which we may 

 reduce /i = b 4 s by multiplication). For r > 0, 



<r'n(r) = ff' n -i(r - 1) + (p - l)l_i(r - 1), <r^(0) = a' n (ri), 



*n(r) = (P - l)*n-l(r - 1), <r(0) = <7 n (ll), 



with more complicated recursion formulas for s' n (r), s n (r), which with 

 ff'i(r) = 1, <7i(r) 0, Si(r) = sl(r) = 1 + (2r 1)( 1/p) determine the s 

 and <r as by Jordan. 4 



K. Zsigmondy 12 proved the final results of Lebesgue. 1 



P. Bachmann 13 gave an exposition of the subject. 



L. E. Dickson 14 gave a generalization of Jordan's 4 - 7 work to any finite 

 field and a derivation of canonical forms. 



R. Le Vmvasseur 15 discussed/ = u (mod p), where p is a prime and 



/ = ax z + bxy + a'y z + ex + c'y + d, A = 4aa'd + bcc' ac' 2 a'c 2 db 2 , 



8 = 4aa f 6 2 . 



If 5 is a quadratic non-residue of p, f = A/5 has one and but one solution ; 

 for u 2$s A/5, f = u has p + 1 solutions. If 5 is a quadratic residue of p, 

 f = A/5 has 2p 1 solutions, / = u ^ A/5 has p 1 solutions. If 5 = 0, 

 / = u has p solutions. 



J. Klotz 16 found the number of sets of solutions of the general quadratic 

 congruence hi any algebraic field. 



10 Mem. presented a TAcad. Sc. Inst. France, (2), 29, 1884, No. 2, Arts. 7, 8, 9; Acta Math., 



7, 1885, 201-258, espec., pp. 210-37. Geeamm. Abh., 1, 1911, 3, 157. 



11 Sitzungsber. Akad. Wiss. Wien (Math.), 99, Ha, 1890, 795-9. 

 u Monatshefte Math. Phys., 8, 1897, 38. 



13 Arith. der Quadrat. Formen, 1, 1898, 478-515. 



"Linear Groups, 1901, 46-9, 158, 197-9, 205-6; Madison Colloquium Lectures, Amer. 

 Math. Soc., 1914. Cf. J. E. McAtee, Amer. Jour. Math., 41, 1919, 225-42, on Jordan.* 

 Mem. Acad. Sc. Toulouse, (10), 3, 1903, 44-8. 

 " Vierteljahrsschrift d. naturf. Gesell. Zurich, 58, 1913, 239-68. 



