ON THE THEORY OF NUMBERS. 159 



ad 3 + 3bd 2 +3cd+d=0, modp, 



p denoting a prime greater than 3, which does not divide the discriminant of 

 the congruence; i.e., the number 



D= -a 2 d 2 + 6abcd-iac 3 -4db 3 + 3b 2 c 2 ; 

 and in connexion with the congruence consider the allied system of functions* 

 U=(a,b,c,d) (x,y) 3 , 

 H=(ac—b 2 , \(ad—bc), bd—c 2 ) (x,y) 2 , 



<b=(—a 2 d+3abc-2b 3 , -abd+2ac a +b 2 c, acd-<2b 2 d+bc 2 , 

 ad 2 -3bcd+2c 3 ) (x,y) 3 , 

 which are connected by the equation 



<J> 2 + Dw 2 =-4H 3 ; 

 let also u and <j> denote the values of U and $ corresponding to any given 

 values of x and y, which do not render H=0, modp. Then, if (I_\=_i, 



the congruence has always one and only one real root; if (1— )= + !, it has 



either three real roots, or none : viz., if (?^ + w A = + 1, it has three; 

 V P h 



if/ 2 ^" 1 " ~ — l\=p ) or =p 2 , it has none. The interpretation of the 



cubic symbol of reciprocity will present no difficulty if we observe that V — D, 

 modp, is a real integer if p=3n+l, i.e. if ( ) = 1> an< ^ tnat > ^ p=3n— 1, 



i.e. if (-— j = — 1, we have >J~—b=*/~^3x Vp>=(p— p 2 )V|D, modp, 



so that V — D, mod p, is a complex integer involving p. It will however be 

 observed that the application of the criterion requires in either case the solu- 

 tion of a quadratic congruence, »- 2 = — D, modp, or r 2 =^D, mod p. 



Similar, but of course less simple, criteria for the resolubility or irresolu- 

 bility of biquadratic congruences may be deduced from the known formulae 

 for the solution of biquadratic equations. 



68. Quadratic Congruences — Indirect Methods of Solution. — The general 

 form of a quadratic congruence is ax 2 -{-2bx-\-c=0, modp, — p denoting an 

 uneven prime modulus, and a a number prime top. It may be immediately 

 reduced to the binomial form r 2 =D, mod p, by putting r=ax+b, Dss6 2 

 — ac, mod p. The number of its solutions is 2, 0, or 1, according as D is a 

 quadratic residue or non-residue of p, or is divisible by p, and is therefore 



in every case expressed by the formula l+( — ). 



Ifp=4w + i3, and I — ) = 1> tne congruence r 2 — D=0, modp, is satisfied 



by ?-=D" +1 , and r=— D n+1 , and is in fact resoluble by the direct method 

 of art. 65. But no direct method, applicable to the case when p = 4w+l, 

 is at present known. Two tentative methods are proposed in the sixth sec- 

 tion of the Disquisitiones Arithmeticae. They are both applicable to con- 

 gruences with composite as well as with prime modules. This circumstance 



* See a note by Mr. Cayley in Crelle's Journal, vol. 1. p. 285. 



