420 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xm 



Since a, b are relatively prime, we can make bman = l in an infinitude of 

 ways. Then x = 2h+l, y = 2(h+ni) + l, g = 2(M-m+w) + l. 



F. Woepcke 107 gave an analogous interpretation of Leonardo's method 

 and wondered why Leonardo preferred this ingenious method to the more 

 natural one of Diophantus] of substituting x = y-\-m, z = yn, and thus 

 finding x, y, z as rational functions of m, n, a, b. The last method and other 

 simple ones were used by C. L. A. Kunze. 108 



For a different presentation of Leonardo's method and a proof of the 

 equivalence of the problem with that of concordant forms, see Genocchi 87 

 of Ch. XVI. 



Matsunago, 109 in the first half of the eighteenth century, noted that 

 rx 2 +y~ = z 2 has the solution x-2mn } y = rm 2 n 2 , z = rm 2 +n 2 . If k l = t 2 , 

 kx 2 ly 2 = z 2 has the solution y = a.-\-tj3, z = l(3 at, provided x- = a 2 -\-lfi~, 

 which is of the preceding type. Again, (k 2 -\-l 2 )x 2 y 2 = z 2 for 



x = c, y = kalb, z = la I Fkb, a 2 -f& 2 = c 2 . 

 J. L. Lagrange 110 treated the solution of 



(2) Ar 2 = p 2 -Bq 2 



in integers. The cases A = D, B = D are easily treated (pp. 381-2) by the 

 methods of Diophantus. In (2) let p, q, r be integers, p and q relatively 

 prime, while A and B are integers neither a square nor divisible by a square, 

 and (as may be assumed) | A \ > \ B . A necessary condition is that 

 there exist an integer a such that a 2 B is divisible by A. This is shown 

 by multiplying (2) by pl-Bql, using 



(3) ( p i-Bq 2 *)(pt-Bq*) = (ppiBqq i y-B(pq l qp i y, 



and taking pg^ gpi=d=l, whence Ar 2 (plBq 2 l }=a 2 B. We may also 

 take | a | < | A |/2, since also (/xAa) 2 B is divisible by A. When such 

 an a exists, AA 1 = a 2 B, set ai = /Z]Aia, the integer jui and the sign 

 being chosen so that | i < | AI |/2. Then a\B is divisible by A^; 

 call the quotient A 2. In this manner we get a series of decreasing integers 

 \A | , | AI | , | At , -, and hence get \A n ^ \B \ . It suffices to stop 

 when A n is of the form a 2 (7, where C has no square factor and C \ == | B \ . 

 Multiply together the equations 



AA l = a 2 -B, -, A^ l A n = c^ l . l -B 



and use (3). Hence AA\ -Al^A n = P 2 -BQ\ Multiply by (2). We get 

 Cql = p? Br\, where qi = A A i A n -\ar. Hence 



(4) Brl = pl-Cql 



Conversely if (4) is solvable, (2) is solvable. Treating (4) as we did (2), 

 we get Crl = pl Dql, etc. Since A |, B , | C , form a decreas- 

 ing series, we finally get a term 1. If it be 1, we proceed and get 

 + 1. The resulting equation Vz 2 = x 2 y 2 is easily solved in integers. Let 



107 Jour, de Math., 20, 1855, 59. 



108 Ueber einige Aufg. Dioph. Analysis, Weimar, 1862, 14-15. 



109 Y. Mikami, Abh. Gesch. Math. Wiss., 30, 1912, 231-2. 



110 M6m. Acad. Berlin, 23, annexe 1767, 1769, 385-406; Oeuvres, II, 384-399. 



