CHAP. xii] PELL EQUATION, ax 2 +bx+c=n. 353 



of the integral quotients in the development* of T/U into a continued 

 fraction, where T = 649, U = 180, is the fundamental solution of 

 T 2 13 C7 2 = 1. But 51 Brouncker did not prove that his method will 

 always lead to a solution of T~ DU 2 = 1. 



Frenicle 52 cited his table 53 of solutions of x 2 Dy 2 = 1 for all values 

 of D up to 150 which are not squares and suggested that Wallis extend it 

 to 200 or at least solve it for D = 151, not to speak of D = 313 which is 

 perhaps beyond his ability. In reply, Brouncker 54 stated that within an 

 hour or two he had found by his method that 313a 2 I = b 2 for 

 a = 7170685, b = 126862368, whence x = 2ab is the desired solution. 



Wallis 55 gave the last solution and 151(140634693) 2 + 1 = (1728148040) 2 . 



Fermat 56 was at first satisfied with the solution of an 2 -f- 1 = D by 

 Brouncker and Wallis. Later, Fermat 57 stated that he had proved by 

 the method of descent the existence of an infinitude of solutions n of 

 an 2 -f 1 = D when a is any number not a square. He admitted that 

 Frenicle and Wallis had given various special solutions, though not a 

 proof and general construction. 



In an anonymous letter to Digby, either by Frenicle 58 or inspired by 

 him, it is stated that Wallis 47 affirmed that he could easily prove the exis- 

 tence of an infinitude of integral solutions of an 2 + 1 = D and implied 

 that the proof is expressly contained in that passage; "but our analysts 

 recognize no trace of proof there". 



N. Malebranche 59 (1638-1715), after stating that he had not seen the 

 work in the Commercium Epist. of Fermat and Wallis on Ax 2 + 1 = D, 

 remarked that we can find a solution if A = a 2 ka, k = 1, 2, or ^ (no 

 details given), or if the difference between A and some square t 2 divides 2t. 

 Thus, if A = 33 or 39, t = 6, A - t 2 = db 3, a divisor of 2t. We have 

 39z 2 + 1 = (6x + I) 2 , x = 4; 33x 2 + 1 = (6z - I) 2 , x = 4. He treated 

 by a tentative process the new types A = 13, 19, 21. For 13, multiply by 

 the squares 1, 4, 9, , until we get a product whose difference from the 

 square divides double the root of the same square; since 13-25 1 = 18 2 , 

 set 325z 2 + 1 = (ISx + I) 2 , whence x = 36. Again, 19-9 - 13 2 = 2, 

 whence I7lx 2 + 1 = (13x + I) 2 , x = 13. He noted that if Ax 2 + 1 = y\ 



_ 6 mg // cX // 



b + c / \ o/ / \ 



a 



-Q4.1 I i I 1 1 i I 1 = 649 



1 + 1 + 1 + 1 + 6 + 1 + 1 + 1 + 1 180 ' 



61 Also noted Sept. 6, 1658, by Chr. Huygens, Oeuvres completes, II, 1889, 211. 



62 Commercium, 821, letter XXVI to Digby, sent by the latter to Wallis Feb. 20, 1658; 



Oeuvres de Fermat, III, 530-3. 



63 Solutio duorum problematum . . . , 1657 (lost work). 



64 Commercium, 823, letter XXVII to Digby, March 23, 1658; Oeuvres de Fermat, III, 536-7. 



65 Letter XXIX to Brouncker, March 29, 1658; Oeuvres de Fermat, III, 542. 



56 Letters from Fermat, June, 1658, and Frenicle to Digby, Oeuvres, III, 314, 577; II, 402 

 (Latin). 



67 Oeuvres, II, 433, letter to Carcavi, Aug. 1659. 



68 Oeuvres de Fermat, III, 604-5 (French transl., 607-8). 



69 C. Henry, Bull. Bibl. Storia Sc. Mat. Fis., 12, 1879, 696-8. 

 24 



