CHAPTER XII. 



PELL EQUATION; ax 2 + bx + c MADE A SQUARE. 



The very important equation x 2 Dy 2 = 1, which has long borne the 

 name of Pell, due to a confusion originating with Euler, should have been 

 designated as Fermat's equation (cf. papers 41, 62-64). 



There appeared in India and Greece as early as 400 B.C. approxima- 

 tions a/b to V2 such that a 2 26 2 = 1, and similarly for other square 

 roots, the derivation of successive approximations being in effect a method 

 of solving the Pell equation. For example, Baudhayana, the Hindu author 

 of the oldest of the works, Sulva-sutras, gave the approximations 17/12 

 and 577/408 to V2. Note that 

 17 -1 ^77 



ll + 20702 = 408' i? 2 - 2 ' 12 ^ 1 ' 577* - 2-408' = 1. 



Proclus 1 (410-485 A.D.) noted that the Pythagoreans made the fol- 

 lowing construction: On the prolongation of the side AB of a square 

 with the diagonal BE lay off BC = AB, CD = BE. Then 



AD 2 + CD 2 = 2AJ5 2 + 2BD 2 . 

 But CD 2 = BE 2 = 2AB 2 . Hence 



AD 2 = 2BD 2 = FD 2 , FD = AD = 2AB + EB. 



Also BD = AB + EB. Write s t , s 2 , 

 di, d z , - - - for the diagonals BE, FD, 



for the sides AB, BD, , and 

 Then 



s n -j- d n , Wn+i 2s n ~r d n - 



Now let si = 1 and replace di = V2 by the integral approximation 5i = 1, 

 and employ our recursion formulae with d n replaced by 5 n . We get 



S2 = Si + Si = 2, 5 2 = 2s i + Si = 3, 



Ss = s 2 -f- 5 2 = 5, 63 = 2s 2 + 5 2 = 7, -. 



Then S, S B give a solution of 6 2 2s 2 = ( l) n . 



1 In Platonia rem publicam commentarii, ed., G. Kroll, 2, 1901, 24-9; excurs II (by F. 

 Hultsch), 393-400. 



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