ON THE THEORY OP NUMBERS. 241 



mod p. Again, for every quadratic residue the congruence a; 3 = a, mod p, is 

 resoluble; for every non-quadratic residue the congruence x 2 = b, mod p, 

 is irresoluble. The solution of almost every problem relating to the in- 

 determinate analysis of quadratic functions involves a congruence of the 

 simple form x° = A, mod p. It is therefore of great importance to 

 obtain a criterion which shall enable us to determine a priori whether a 

 given number is or is not a quadratic residue of a given prime. If we 

 have a Table of Indices for the given prime, we have only to see whether 

 the index of the given number is even or uneven ; if even, it is a qua- 

 dratic residue; if uneven, it is a quadratic non-residue. Or, again, we 

 may raise the given number a (by M. Crelle's method, or any other) to 



the power " - , and see whether the residue is +1 or — 1. It is usual to 



25 



denote the positive or negative unit which is the remainder of a 2 , mod p 

 by the symbol (~\ which is known as " Legendre's Symbol ;" so that in every 



case «~ ^V P m °dp> and (-)=-(- 1 or = — 1, according as a is or is not 

 a quadratic residue of p. It will be seen that we also have in every case the 

 equation ( ^± ) ( ^) = rhJh Y If a instead of being prime to p be divisible 



by p, it is convenient to attribute to ( -) the value zero. 



16. Legendre's Lata of Reciprocity. — The two methods alluded to for the 

 discrimination of quadratic and non-quadratic residues, or, which is the same 



thing, for the determination of the value of the symbol (-), are not satis- 

 factory, — the first because it supposes a reference to a Table of Indices («. e. 

 to a recorded solution of the problem it is proposed to solve), the second on 

 account of its inapplicability to high numbers. A very different solution of 

 the problem is supplied by a theorem which is known as " Legendre's Law 

 of Quadratic Reciprocity," and which is, without question, the most important 

 general truth in the science of integral numbers which has been discovered 

 since the time of Fermat. It has been called by Gauss " the gem of the higher 

 arithmetic," and is equally remarkable whether we consider the simplicity of 

 its enunciation, the difficulties which for a long time attended its demonstra- 

 tion, or the number and variety of the results which have been obtained by 

 its means. The theorem is as follows : — 

 " Ifp and q be two uneven prime numbers 



(f)=( _ 1)) .-„«.,-„ ( , ) ., i(i); 



to which we must add the complementary propositions relating to the resi- 

 dues— 1 and 2 



(= 1 )=(-» 00. (J)=(-0 * 0"). 



In (ii), p is supposed to be positive; in i}),p and q are supposed not to 

 be simultaneously negative. 



1859. r 



