ON THE THEORY OF NUMBERS. 251 



p,=2k lP2 +e 2 p 3 

 p,=2k.,p 3 +€, Pl ... 



pp=2f</*P(* + i + ep+ 1, 



in which e.„ e, . . . e^+i denote positive or negative units, and p x , p„ p t 



which are all positive and uneven, form a descending series. Let a denote 

 the number of the quantities p r + e r pr+\ in which both p r and e, jj r +\ are of 



the form4» + 3 ; then ( — )=(— l)'- The demonstration of this result flows 



immediately from the definition of Jacobi's symbol of reciprocity. 



A numerical example is added (see Disq. Arith. Art. 328) from which the 

 reader will perceive the utility of these researches in their practical applica- 

 tion to congruences. 



Let the proposed congruence be x 2 = —286, mod 4272943, where 4272943 

 is a prime number. 



/ 286\ 



We have to investigate the value of the symbol | J, in which p is 



written for 4272943. Now (=M)=(zl)x(?) x(^)=-(^), be- 

 cause ( — )= — 1, and |_\= + l,_p being of the form 8«— 1. Tofindthe 



value of J Y we hav e 



143=0 X 4272943 + 143 f 

 4272943 = 29880 X 1 43 + 1 03 f 

 143=2x103—63 

 103=2x63-23 



63=2x23 + 17 



23=2x17-11 



17=2xll-5f 



11=2x5 + 1 



The obelisk (f) denotes that the equation to which it is affixed is one of 



those enumerated in * Hence (_lg^) = (-l) 3 = -l, and (^ggg) 



= + ],or the proposed congruence is resoluble. Its roots (as determined 

 by Gauss) are +1493445. 



24. Biquadratic Residues. — Reverting to the general theory alluded to in 

 Art. 12, we see that, when p is a prime of the form 4re + l, the congruence 

 a;* — 1=0, mod p, admits four incongruous solutions ; these are +1, — l,and 

 the two roots of the congruence a; 2 + l = 0, modj», which we shall denote by 

 +/and — /, or by/and/ 3 , so that the four roots of x*— 1=0 are l,f, —1, 

 and/ 3 . Further, if k be any number prime to p, k satisfies one or other of 

 the four congruences — 



(i.) ***^2Si, m0 djfc (iii.) ***- ,} e9~l, modjfc 



(ii.) k*"- 1 ^/, modp. (iv.) ft**- y =3/ 8 , modp. 



We see therefore that the p— 1 residues of p divide themselves into four 

 classes, comprising each \{p— 1) numbers, according as they satisfy the 

 1st, 2nd, 3rd, or 4th of these congruences. The first class comprises those 



