CHAP. XIII] OX 2 + by 2 + CZ 2 = 0. 427 



, 



noted that A =a 2 +6 2 , whence (x-\-bz)f(az+y) = (azy)/(x bz)mfn (say), 

 which determine x : y : z. 



P. F. Teilhet 134 stated that x 2 +y 2 = (m 2 +n 2 )z 2 implies 



z = K(A-+B-), x = mK(A 2 -B 2 )2nKAB, y = nK(A 2 -B 2 )^2mKAB. 



F. Ferrari 135 noted the solution, with K=l, given by Teilhet. 134 

 A. Gerardin 136 stated that the general solution of x 2 -\-2y 2 =H is 



z = 2Z 2 -m 2 -n 2 +2ra(3ft-40, y = 4:l 2 +2n 2 -2m 2 -2l(3n-m). 

 Gerardin 137 noted the identities 



{(p-q)b 2 -qy 2 +2bqy} 2 +pq{2b(y-b)} 2 ={(p+q}b 2 +qy 2 -2bqy} 2 , 

 (m 2 +n 2 )(mnx 2 -2z 2 ) 2 +2mn{mnx 2 +2z 2 -2xz(m+ri)} 2 



= { (m+w) (mnx 2 +2z 2 } ^mnxz } 2 , 

 another similar to the last and several for x 2 -\-Sy 2 z 2 . 



A. Thue 138 discussed the possibility of Ax 2 +By 2 = Cz 2 , where x, y, z 

 are relatively prime in pairs and z^y^x>0. We can determine integers 

 p, q, r without a common factor such that px+qy = rz, with p 2 , q 2 , r 2 all 

 <3z. Then 



(Bp 2 +Aq 2 )x 2 - 2Bprxz+(Br 2 - Cq 2 )z 2 = 0, 

 (Bp 2 + Aq 2 )y 2 - 2Aqryz + (Ar 2 - Cp~)z 2 = 0. 

 Hence 



ax = Cq 2 -Br 2 , by = Cp 2 -Ar 2 , cz = Bp 2 +Aq 2 , 



az + 2Bpr = cx, bz+2Aqr = cy, 



where a, b, c are integers. Let U be the greatest of A \ , B \ , C \ . 

 By the last five equations, | c \ < QU, \a < 12U, b < 12U. But 

 a, b, c satisfy the initial linear and quadratic equation. Thus the possi- 

 bility of the latter can be decided by a finite number of trials. 



L. Aubry 139 proved that if pA- = B 2 +rC 2 , where B and C are prime to A, 



then pX 2 = Y 2 +rZ 2 for X^2 V^/3 if r>0, and X^ -\Pr if r<0; if B and C 

 are prime to p, then also Y=Ba, Z=Ca (mod p). 



Several writers 140 solved 13x 2 +17y 2 = 230z 2 . 



C. Alasia 141 solved x 2 79y 2 = Wlz 2 [Plana 120 ] by several classic methods. 



G. Bonfantini 142 noted that the evident sufficient condition for integral 

 solutions of x 2J t-y 2 = mz~ is that m be a sum of two squares. To prove that 

 the condition is necessary, consider integers &,-, pi, qt such that 



By induction, k m = (t> 2 m +(t m +p l (f) m } 2 , where 



134 L'intermediaire des math., 12, 1905, 81. 



135 Suppl. al Periodico di Mat., 12, 1908-9, 34-5. 



136 Assoc. frang., 1908, 17. To make 109 m=0 replace I by l+2m, n by n+3m. 



137 Sphinx-Oedipe, 1907-8, 109-110. 



138 Skrifter Videnskapsselskapet, KJ-istiania, 1, 1911, No. 4, p. 18. 



139 Sphinx-Oedipe, 8, 1913, 150 (error in his 7, 1912, 81-2. 



140 Wiekundige Opgaven, 11, 1912-4, 281-6. 



141 Giornale di Mat., 53, 1915, 292-302. 



142 Suppl. al Periodico di Mat., 18, 1915, 81-6. For m=2, ibid., 17, 1914, 84-5. 



