CHAP. XII] PELL EQUATION, ax*+bx+c=E3. 343 



By (6), p = 5q, z = 30 y, y = 11(297? -f 7), whence lly = 80/3 - 19067g. 

 Then (7) gives 30y = (1505g + 18) 13/2, q = 2r, = 25. Comparing the 

 resulting 7 with the earlier 7, we get a linear equation in 5, r, whence 



r = 4657w, 5 = 1359235^. 

 By substitutions, we get m = 93140^, whence 



W = 207329640**, w = 144127200w 



X = 149210280^, x = 97864920w 



Y = 147161200u, y = 70316400^ 



Z = 82987740w, z = 108784260^. 



For u = 4, we get the numbers in the scholium; but they satisfy neither 

 (8) nor (9), since neither W + X nor S(Y + Z) + 1 is a square. 



Returning to (10'), we note that the greatest common divisor of the 

 numerical factors is 20, whence u = v{20, where v is an integer. Then 



W + X = 4 957 4657y, v = 957 4657n 2 , 

 since W + X is to be a square. Then Y + Z = (P + fy/2 gives 



(2t + I) 2 = 8(7 + Z) + 1 = an 2 + 1, a = 410286423278424. 



Since a is positive and not a square it is possible to choose an integer n 

 so that cm 2 + 1 = D by Euler. 81 If the resulting square is even, we can 

 deduce one making an z -f- 1 an odd square (Euler, 83 86, 88). 



J. J. I. Hoffmann 7 said the problem was due to a much later computer. 



J. Struve 8 gave a 36 page discussion making no advance over Leiste. 



Gottfried Hermann 9 made an interpretation which led, not to (8) and 

 (9), but to W + X = a square whose side is of the form a 2 (a 6), 

 Y + Z = A, W + X + Y + Z = Ai. Thus if we take the numbers (10), 

 we must make 



3828m = {^(a - b}} 2 , 2471m = c( + ^ , 6299m = ^ j" ^ . 



i Zi 



He stated on the authority of K. B. Mollweide that C. F. Gauss had com- 

 pletely solved the problem under the earlier interpretation, but had not 

 published the solution. 



J. Fr. Wurm, 10 in a review of Hermann's paper, replaced (p. 201) condi- 

 tion (8) by the condition that W + X shall be a product of two approxi- 

 mately equal factors. Without returning to this condition, he passed to (9) : 



Y + Z = 2471m = 247M51* = A. 



7 Ueber die Arith. der Griechen, Mainz, 1817, Introd., p. xvi (transl. of Delambre). 



8 Altes griechisches Epigramm, mathematischen Inhalts, von Lessing erst einmal zum Drucke 



befordert, jetzt neu abgedruckt und mathematisch und kritisch behandelt von Dr. J. 

 Struve und Dr. K. L. Struve, Vater und Sohn. Altona, 1821, 47 pp. 



9 Ad memoriam Kregelio-Sternbachianam in and. jur. die 17 Julii 1828 : De Archimedis 



Problemate Bovino, Universitats programm, Leipzig, 1828. Reprinted in Godofredi 

 Hermanni, Opvscvla, Lipsiae, 4, 1831, iii-v, 228-238. 



10 Jahrbiicher fur Philologie u. Paedagogik (ed., J. C. Jahn), 14, 1830, 194-202. 



