ON THE THEORY OF NlJMBERS. 313 



x^-V-y'^=^2p, Jacobi considers two solutions, such as x^-\-y^=-'ip and 

 x^-k-y^=1lh to be identical, when x^=x.^, y^=y.i\ the number of solu- 

 tions is thus a fourth part of the number of representations.) In particular 

 every prime of the form4M + l (and the double of every such prime) is 

 capable of decomposition in one way, and one only, into two squares relatively 

 prime ; and, conversely, every uneven number capable of such decomposition 

 in one way only is a prime of the form \n-\-\. 



If D=— 2, oo'^-\-2y^ represents the only class of forms; and every uneven 

 number can be represented by x-+2y', in twice as many ways as it has divi- 

 sors of either of the forms 8?i + 1, or 8w + 3, in excess of divisors of the forms 

 8?i+5, or 8w + 7. (Dirichlet, loc. cit.) In particular every prime of either 

 of the forms Sn + 1 or 8n + 3 is decomposable in one way, and in one only, 

 into a square and the double of a square. 



Again, for each of the determinants — 3 and — 7, there is but one properly 

 and one improperly primitive class, which may be represented by the forms 

 (1,0, 3) and (2, 1,2); (1, 0, 7) and (2, 1, 4). Uneven numbers are there- 

 fore represented by x- + 3y~, in twice as many ways as they have divisors of 

 the form 3?i + l, in excess of divisors of the fotm 3n—l ; and by x^ + 7y^ in 

 twice as many ways as they have divisors of the forms 7« + l, 2, 4, in excess 

 of divisors of the forms 7n-\-3, 5, 6. Similarly, x'^+iy'^ represents the only 

 primitive class of det. — 4. 



For each of the eleven positive determinants of the first century 2, 5, 13, 

 17, 29, 41, 53, 61, 73, 89, 97, there is but one properly primitive class ; there 

 is also for the ten uneven determinants one improperly primitive class. Re- 

 presenting any one of these eleven numbers by D, by [T, U] the least solu- 

 tion of T^ — DU^=1, and by M an uneven positive number prime to D we 

 may enunciate the theorem, 



"The equation x' — D?/=M is capable of as many solutions in positive 

 numbers x and y, satisfying the conditions a; < TVM^> y^ UV^, as M 

 has divisors of which D is a quadratic residue in excess of divisors of which 

 D is a quadratic non-residue." 



Thus the number of solutions of the equation x^ — 2y'^M, where M is an 

 uneven number, and 0<:a;<.3 s/ M, 0<y<2 VM, is the excess of the divisors 

 of M of the forms 8n + l above its divisors of the fornis8« + 3. 



The conditions <:a;;^T V M, 0<y '^U V M, which are satisfied by one 

 representation, and only one, in each set, are obtained by considerations to 

 which we shall hereafter refer (art. 100). 



96. The Pellian Equation. — The two indeterminate equations, T^— DU^=1 

 and T'^— DU^=4, are, as we have seen, of primary importance in the theory 

 of quadratic forms of a positive and not square determinant. When the 

 complete solution of these equations is known, we can deduce, from a single 

 representation of a number by a form, every representation of the same set; 

 and, from a single transformation of either of two equivalent forms into the 

 other, every similar transformation. The same equations also present them- 

 selves in the solution in integral numbers of the general equation of the 

 second degree containing two indeterminates, and enable us in the principal 

 case in which it admits an infinite number of solutions to deduce them all 

 from a certain finite number. This fundamental importance of the equation 

 T^— DU"=1 was first recognized by Euler, who has left several memoirs 

 relating to it (see Comment. Arith., vol. i. pp. 4, 316; vol. ii. p. 35; also 

 Euler's Algebra, vol. ii. cap. vii.) ; but the equation itself had already given 

 rise to a discussion which forms a well-known passage in the scientific history 

 of the seventeenth century. Its solution was proposed by Fermat (see the 



