160 REPORT — 1860. 



is important, because, when the modulus is a very great number, we may not 

 be able to tell whether it is prime or composite, and, if composite, what the 

 primes are of which it is composed, although, when the prime divisors of a 

 composite modulus are known, it is simplest first to solve the congruence for 

 each of them separately, and afterwards (by a method to which we shall 

 hereafter refer) to deduce from these solutions the solution for the given 

 composite modulus. To apply the first of Gauss's methods, the congruence 

 is written in the form ?- 2 = D + Py, P denoting the modulus. If in the formula 

 V=D + Py we substitute for y in succession all integral values which satisfy 



the inequality — „ < y<i P— tj> and select those values of V which are per- 

 fect squares, their roots (taken positively and negatively) will give us all the 

 solutions of the congruence. We should thus have I|P or 1 -f-I \ P trials to 

 make, I denoting the greatest integer contained in the fraction before which it 

 is placed. If, however, we take any number E, greater than 2, and prime to 

 P (it is simplest to take for E a prime, or power of a prime), of which the 

 quadratic non-residues are a, b, c,..., and then determine the values of a, /3, y, 

 ... in the congruences a=D + «P, mod E, 6=D-f/3P, mod E, &c, -we shall 

 find that every value of y contained in one of the linear forms mE + a, 

 w*E+/3, &c, gives rise to a value of V which is a quadratic non-residue of 

 E, and which cannot, therefore, be a perfect square ; so that we may at once 

 exclude these values of y from the series of numbers to be tried. A second 

 excludent E' may then be taken, and by its aid another set of linear forms 

 may be determined, such that no value of y contained in them can satisfy 

 the congruence. Thus the number of trials may be diminished as far as 

 we please. The application of this method is still further facilitated by the 

 circumstance that it is not necessary actually to solve the congruences 

 fl=D+aP, mod E, . . . but only the single congruence D + Py=0, mod E 

 (Disq. Arith. art. 322). Gauss's second method depends on the theory of 

 quadratic forms; it supposes that the congruence is written in the form 

 ^-(-DssO, mod P. By a tentative process (abbreviated, as in the first 

 method, by the use of excludents) Gauss obtains all possible prime representa- 

 tions of P by the quadratic forms of determinant — D; whence the com- 

 plete solution of the congruence r 2 -fD=0, mod P, is immediately deduced. 

 This method involves the construction of a complete system of quadratic 

 forms of determinant — D, or, if the prime factors of D be known, of one 

 genus of forms of that system ; it becomes therefore more difficult of appli- 

 cation as D increases, whereas the first method is not affected by the increase 

 of D. The second method, however, especially recommends itself when P is 

 a very great number ; in fact, if we do not employ any excludent, the number 

 of trials required by the first method varies (approximately, and when P is 

 a great number) as P, whereas, on the same j>upposition, the number of trials 

 required by the second method varies as VDx vP. 



M. Desmarest (in his Theorie des Nombres) has proposed a method less 

 scientific in its character than those of Gauss, but sometimes easily applicable 

 in practice. He has shown that if the congruence ? >2 + D=0, mod P, be re- 

 soluble, we can always satisfy the equation mP=x 2 + Dy i with a value of 



p 

 m inferior to —- + 3, and of y not superior to 3. The demonstration of this 



16 

 theorem is not very satisfactory, and the number of trials that it still leaves 



is very great, viz. 3 ( l-^- + 3 ). 



The application of Gauss's second method is rendered somewhat more uni- 



