CHAP. IV] SOLUTION OF x--\-y 2 = z 2 . 169 



C. M. Piuma 32 quoted the known result that all relatively prime integral 

 solutions of x- -f y~ = z- are given by 



m n m 2 + n 2 



x = mn, y = - - - , z = 



, 



where m and n are relatively prime odd integers, and proved conversely 

 that then these three expressions are relatively prime in pairs, by showing 

 by use of congruences that no two are divisible by the same power of a prime. 



D. S. Hart 33 proved for n ^ 4 that, if z is a product of n primes each a 

 sum of two squares 02, z 1 is a El in (3 n l)/2 ways [Volpicelli 26 ]. 



L. E. Dickson 34 obtained, as a solution equivalent to (1), r + s, r + t, 

 r + s + t, where r 2 = 2st is a square. The same rule was given later by 

 P. G. Egidi, 35 D. Gambioli, 36 A. Bottari, 39 and H. Schotten. 36 " 



Graeber 37 noted that if the point of tangency of a circle inscribed in a 

 right triangle divides the hypotenuse z into the segments k and m, while 

 n and m are the corresponding segments of leg y, then 



(k + m) 2 = (m + n) 2 + ( n H- k)"> k = (n 2 + mn)/(m n). 



Thus x, y, z are proportional to (1). The sides if integral are shown by a 

 long proof to be (1). 



L. Kronecker 38 proved that all positive integral solutions of x~ + y 2 = z 2 

 are given without duplication by 



x = 2pqt, y = t(p 2 - g 2 ), z = t(p 2 + g 2 ), p > q > 0, t > 0, 



p and q being relatively prime and not both odd. The reason why every 

 solution is obtained once and but once is due to the fact that the circle 

 2 + if = 1 is of genus zero, all its points being expressible rationally in 



T = tan co/2 : 



1 - r 2 2r 



A. Bottari 39 proved that all integral solutions of x 2 + y 2 = z 2 are given 

 by x = u -\- w, y = v -\- w, z = u + v + w, where u = p z k, v = 2 2s ~ 1 g 2 A;, 

 w = 2 a pqk, p and q being relatively prime odd integers. Thus xy is not 

 a square. 



P. Cattaneo 40 gave a simple proof of Bottari's theorem. 



P. Reutzel 41 noted that, if a > 2, we can solve c 2 6 2 = a 2 . Set 

 c = b + v. Then 6 = (a 2 v 2 )/(2v) is an integer if v = 1 and a is odd, 

 or if y = 2 and a is even. We may take v to be any divisor a/n of a; then 

 b = (n 2 - l)y/2, c = (n 2 + 1)0/2. 



32 Giornale di Mat., 19, 1881, 311-5. 



33 Math. Quest. Educ. Times, 39, 1883, 47-8. 



34 Amer. Math. Monthly, 1, 1894. 8. 



35 Atti Accad. Pont. Nuovi Lincei, 50, 1897, 103. 



36 Periodico di Mat., 16, 1901, 151-5. 



360 Zeitschrift Math. Naturw. Unterricht, 47, 1916, 181-2. 



37 Archiv Math. Phys., (2), 17, 1900, 36. 



38 Vorlesungen iiber Zahlentheorie, 1, 1901, 31-35. 



39 Periodico di Mat., 23, 1908, 104-110. Cf. Dickson. 34 



40 Ibid., 218. 



41 Zeitschrift Vermessungswesen d. Deutschen Geometervereins, Stuttgart, 38, 1909, 208-11. 



