other half non-residues of p. Thus in the case p = 1 1 every number not divisible by 1 1 is, to this modulus, = one of the series 1, 2, 3, 4, 5; whence the square of any such number is = one of the series 1, 4, 9, 16, 25, or say the series 1, 4, 9, 5, 3 ; that is, we have residues 1, 3, 4, 5, . . . 9, . non-residues . 2, . . . 6, 7, 8, . 10 . Calling as usual the residues a and the non- residues 6, we have in this case ^pr (2ft - 2) = yy (33 - 22), = 1, a posi tive integer ; this is a property true for any prime number of the form 4n + 3, but for a prime number of the form 4rt + 1 we have 26 2a = ; ths demonstration belongs to a higher part of the theory. It is easily shown that the product of two residues or of two non-residues is a residue ; but the product of a residue and a non-residue is a non-residue. 24. The law of reciprocity Legendre s symbol. The question presents itself, given that P is a residue or a non-residue of Q, can we thence infer whether Q is a residue or a non-residue of PI In particular, if P, Q, are the odd primes/), q, for instance, given that 13 = ^(17), can we thence infer that 17 = J?(13), or that 17 = M(13) ? The answer is contained in the following theorem : if p, q, are odd primes each or one of them of the form 4+l, then p, q, are each of them a residue or each of them a non-residue of the other ; but, if p, q, are each of them of the form 4n. + 3, then according as p is a residue or a non- residue of q we have q a non-residue or a residue of p. The theorem is conveniently expressed by means of Legendre s symbol, viz., p being a positive odd prime, and Q any positive or negative number not divisible by p, then ( J denotes + 1 or - 1 according as Q is or is not a residue of p ; if, as before, q is (as p) a positive odd prime, then the foregoing theorem is The denominator symbol may be negative, say it is p, we then have as a definition ( ) = ( ) observe that -Q f Q ~P/ P f ) is not = ( -^- ) and we have further the theorems P J -pJ viz., 1 is a residue or a non-residue of p according as p = 1 or = 3 (mod. 4), and 2 is a residue or a non-residue of p according as p = 1 or 7, or = 3 or 5 (mod. 8). If, as (P f P r ) = + 1 and ( -^ } = + 1, these may be i/ Z / written We have also, what is in fact a theorem given at the end of No. 23, p p p The further definition is sometimes convenient ( ) = 0, when p divides Q. The law of reciprocity as contained in the theorem is a fundamental theorem in the whole theory ; it was enunciated by Legendre, but first proved by Gauss, who gave no less than six demonstrations of it. 25. Jacobi s generalized symbol. Jacobi defined this as follows : the symbol ( - ^ - ) , where p. p p". .
ppp .../
are positive odd primes equal or unequal, and Q is any positive or negative odd number prime lopp p" . . . , denotes + 1 or - 1 according to the definition _ pp p". p p p" the symbols on the right-hand side being Legendre s symbols ; but the definition may be regarded as extending to the^case where Q is not prime topp p" . . . , then we have Q divisible by some factor p, and by the definition of Legendre s symbol in this case we have ( = hence pj m the case in question of Q not being prime to pp p" . . the value of Jacobi s symbol is = 0. We may further extend the definition of the symbol to the case where the numerator and the denominator of the symbol are both or one of them even, and present the definition in the most general form, as follows : suppose that, p, p , 21", being positive or negative even or odd primes, equal or unequal, and similarly q, q , q", . . . being positive or negative even or odd primes, equal or unequal, we have P=pp p" . . . and Q = qq q" . . . , then the symbol will denote +1, - 1, or 0, according to the definition the symbols on the right hand being Legendre s symbols. If P and Q are not prime to each other, then for some pair of factors p and q we have p q, and the corre sponding Legendrian symbol ( ) is = 0, whence in this / n P / case I 1-0. / 0 It is important to remark that ( ~ J = + 1 is not a sufficient condition in order that Q maybe a residue of P; if P= 2 a pp p" . . , p,p ,p , . . being positive odd primes, then, in order that Q may be a residue of P, it must be a residue of each of the prime factors p, p , p", . . , that is, we must have
- p / p / p"
equations as there are unequal factors p, p , p , . . of the modulus P. Ordinary Theory, Second Part, Theory of Forms. 26. Binary quadratic (or quadric) forms transforma tion and equivalence. We consider a form ax 2 + 2bxy + cy 2 , = (a, b, c} (x, y) 2 , or when, as usual, only the coefficients are attended to, = (a, b, c). The coefficients (a, ft, c) and the variables (x, y) are taken to be positive or negative integers, not excluding zero. The discriminant ac - b 2 taken negatively, that is, ft 2 - ac, is said to be the determinant of the form, and we thus distinguish between forms of a positive and of a negative determinant. Considering new variables, ax + j3y, yx + 8y, where a, /?, y, 8, are positive or negative integers, not excluding zero, we have identically (a, b, c)(ax + f3y, yx + dy) 2 = (a , b , c )(x, y) 2 , where a = (a, b, c)(a, y) 2 , = aa 2 + / 2bay + cy 2 , b = (a, b, c)(a, y)(/3, 8), = aaf3 + b(aS + /3y) + cyS, c = (a, b, c)(p, d) 2 , = afi 2 + 26/35 + cS- ; and thence b 2 - a c (ad - Py)~ (b 2 - ac). The form (a, b , c) is in this case said to be contained in the form (a, ft, c) ; and a condition for this is obviously that the determinant D of the contained form shall be equal to the determinant D of the containing form multi plied by a square number ; in particular, the determinants must be of the same sign. If the determinants are equal, then (a8-/3y) 2 =l, that is, a5-/?y=l. Assuming in this case that the transformation exists, and writing a8 (By = e, and writing also
