168 HISTORY OF THE THEORY OF NUMBERS. [CHAP. IV 



P. Volpicelli 24 noted that z = a 2 + 6 2 = a 2 + /3 2 imply that 

 x = (act =F 6/3), y = =b (a/3 db 6a) 



are solutions of x 2 + y 2 = z 2 and stated incorrectly that they give all the solu- 

 tions, whereas formulas (1) do not. As to J. Liouville's 25 remark that, for z 

 given, x 2 + y 2 = z 2 has relatively prime solutions if and only if z is a product 

 of primes 4n + 1, the solutions x 1020, y = 425, z = 5-13-17 are not 

 relatively prime. 



Volpicelli 26 distinguished k types of solutions of x 2 + y 2 = z 2 , where 

 z = hi- -hk, hj = a* + bj. The k solutions of the first type areg(aj 6j), 

 2qcijbj, where g = z//i/. The &(& 1) solutions of the second type are 



q{(a\ - bl)(aj - 6j) db 4a i a j b l b j }, 0{2(a f a y b i b j }(a j b i =F a&)}, 



where <? = z/(hihj), the quantities aj 2 , 2/2 in brackets being such that 

 2 + yl = h*hj. From 



a + yl = (x\ + yl)l(a* t - tf) 2 



we obtain the 4(a) solutions 023, 92/3 of the third type, where q = s/(hihjht). 

 Thus the total number of solutions is 



=i 1 -2- -s 



Volpicelli 27 noted that all solutions of x 2 + y 2 = 2 2 depend on the solu- 

 tions of a; 2 + y 2 = z* (j = 1, -,&), where i, -, z k = z are the products 

 of the factors of z taken 1, 2, -, k at a time. For z 2 = (a 2 + 6 2 ) fc , a 

 solution is 



z = a k - g)a fc ~ 2 & 2 + (4)a fc ~ 4 Z> 4 - - - -, y = (\}a k ~ l b - a fc - 3 6 3 + -. 



For, if (a + ib) k = A + i5, (a 2 + 6 2 )* = A 2 + 2 , which was verified with- 

 out using i = V^- 1. Also a 2 6 2 is a factor of B if k = 4h, but is a 

 factor of 4 if k = 4/i + 2. 



C. A. W. Berkhan 28 gave nineteen methods of finding two numbers the 

 sum of whose squares is a square, with references on several proofs. 



E. de Jonquieres 29 discussed Volpicelli's 26 topic. 



A. J. F. Meyl 30 noted that, according to an argument by de Jonquieres, 29 



(x + 3) 2 + (x + 4) 2 = l(y + I) 2 + (y + 2) 2 ] 2 



has only the solutions x + 3 = 3 or 4, whereas z + 3 = 0or 1 also. 



C. de Polignac 31 used a rectangular lattice to prove that (1) gives all 

 integral solutions of x 2 + y~ = z 2 . 



24 Giornale Arcadico di Sc. Let. ed Arti, Rome, 119, 1849-50, 27. Annali di Sc. Mat. Fis., 1, 



1850, 159-166, 369, 443. 

 26 Comptes Rendus Paris, 28, 1849, 687. 



26 Atti Accad. Pont. Nuovi Lincei, 4, 1850-1, 124-140, 346-377, 508-510. 

 Ibid., 5, 1851-2, 315-352; Comptes Rendus Paris, 36, 1853, 443-5. Extract in Annali di 



Sc. Mat. Fis., 3, 1852, 130-3; 4, 1853, 286-297. 



28 Die merkwiirdigen Eigenschaften der Pythag. Zahlen, Eisleben, 1853. 



29 Nouv. Ann. Math., (2), 17, 1878, 241-7, 289. Cf. papers 26-31 of Ch. XVII. 



30 Ibid., (2), 18, 1879, 332-3. 



31 Bull. Math. Soc. France, 6, 1877-8, 162. 



