ON TUE THEORY OP NUMBERS. 317 



IfU;^^,be the first number of its series which is divisible by q, wc shall have 

 either \'=\, or2\'=\. In either case, the only numbers U which are divisible 



by J, are those whose indices are divisible by \'; and the formula T,„a', — ^ 



comprises all the solutions of the equation T^—'Dq-\J-=m''. Thus, in 

 solving the equation T"*— DU^=m-, wc can always substitute for D its 

 quotient when divided by its greatest square divisor. (See Lagrange, Ad- 

 ditions to Euler's Algebra, art. 78. Gauss, Disq. Arith.art.201,Obs. 3 and 4.) 



We may add, that if §' be a prime (an uneven prime when ?«=2), and if 

 q^ and qt^ be the highest powers of q, dividing U^^ and n respectively, 

 ^ic+t* will be the highest power of q dividing U„a.« (Dirichlet, in Liouville's 

 Journal, New Series, vol. i. p. 76.) 



(vi.) Tiie methods of Lagrange and Gauss are applicable to the equation 

 T^— DU^=4, only when D ees 1 , mod 4- ; because they suppose the existence of 

 an improperly primitive form of det. D. In all other cases the equation 

 T-— DU^=4 may be divided by 4, and reduced to the formT'— DU^=1 : viz. 

 if D ^ss 0, mod 4, T is even ; and if D es 2, or ss 3, mod 4, T and U are both 

 even. A similar reduction takes place if D^l, mod 8; the equation 

 T"— DU'=4 admitting in that case only even solutions. But if D :^ 5, mod 8, 

 T' — DU'=4 may or may not have uneven solutions; and no criterion is known 

 for distinguishing a priori these two cases. If T^ — DU^=4 admit of uneven 

 solutions, its least solution [Tj, Ui] will be uneven ; its even solutions will be 

 comprised in the formula [T3,,, Ua^], and consequently [J-Ts,!, iUs,.] will 

 represent the solutions of T^ — DU"=1. 



(vii.) The equations T^-DU-=-4, T--DU'=-1 are not resoluble for 

 all values of D, but only for those values for which — 1 is capable of represen- 

 tation by the principal form of det. D. Whenever the period of integral quo- 

 tients in the development of VD consists of an uneven number of terms, 

 these equations will be resoluble, and conversely. This will always happen 

 when D is a prime number of the form 4« + l, and may happen in many other 

 cases, but never can happen when D is divisible by any prime of the form 

 4« + 3. IfT-— DU-= — 1 be resoluble and [Tj,Ui] be its least solution, the 

 formula [Ton+i, U2n+i] contains all its solutions, and [T2», Uok] all the solu- 

 tions of T'— DU'=1. If, in addition to the supposition thatT'— DU'= — 1 

 is resoluble, we suppose that T" — DU" = 4 admits of uneven solutions, 

 T^ — DU'= — 4 will also admit of uneven solutions; and if [Tj,Uj] be its least 

 solution, [T2«+i, U2n+i], [Tan, U2«]. [5 T6„+3> 2 Uen+a]? [i Tea, g T6«] 

 will represent all the solutions of T^ — DU^= — 4, ^4, = — 1, and =:], 

 respectively. It is evident that these considerations will frequently serve to 

 abbreviate the process of finding the least solution of T^ — DU^=1. (See a 

 memoir of Euler's in the Comment. Arith. vol. ii. p. 35.) 



(viii.) The "Canon Pellianus" of Degen (Havniae 1817) contains a Table, 

 giving for every not square value of D less than 1000, the least solution of 

 the equation T-— DU-=1, together with the development of i\/T> in a. con- 

 tinued fraction. Its arrangement will be seen in the following specimens :— 



357 



97 



18, 1,8,(2) 

 1, 33, 4, 17 

 180 

 3401 



9, 1,5, 1,1,(1,1) 

 1,16,3,11,8,(9,9) 

 6377352, 

 62809633. 



