356 HlSTOEY OP THE THEORY OF NUMBERS. [CHAP. XII 



and 1393/985 to V2 and an approximation to Vlo. Jean Plana 69 gave 

 reasons to show that Girard there in effect reduced VA to a continued 

 fraction. 



A solution 70 of 44000z 2 + 1 = D is x = 40482981221781. 



Euler 71 published his formula (11), and treated ax 2 + bx + c = y z , 

 given the solution x = n, y = m. Set x = n + pz, y = m + vz. Then 

 (i> 2 afjf)z 2a(jin 2vm + 6ju. If a is positive and not a square, we 

 can make y 2 a/r = 1 and obtain integral solutions, and then a third 

 set, etc.; if the general set is (xi, yi), we have 



x i+ 2 = 2(v z + an*)x i+ i Xi + 2&/x 2 , y i+2 = 2(> 2 + a^}y i+l y t . 

 But we may obtain solutions not having v 2 an 2 = 1 ; setting 



v 2 + a/j? 2/j.v 



P = * q = v*-an*' 



we obtain the first formulas in Euler's 65 earlier paper. Euler proved that 

 if an odd prime, not dividing a, is of the form 6 2 a 2 , it is of one of the 

 linear forms 4cm + r 2 , 4cm + r 2 a, where r ranges over the odd and even 

 numbers < a and prime to a, respectively. He conjectured, conversely, 

 that if A is a prime or product of primes of these linear forms, then 

 A = x 2 ay 2 is solvable in integers [[not always true, Lagrange 76 ]. 



Euler 72 again repeated his initial formulas and added that, if P, Q, R 

 are the values of y in three successive sets of solutions, R = 2qQ P, while 

 the general set of solutions is said to be [after correction of signs] 



n> I s b T ~~ s 



r, s = ( 



where ju is an integer. The method published by Wallis to find integral 

 solutions of x 2 = ly 2 + 1, where I is positive and not a square, can be more 

 conveniently exhibited by means of the continued fraction for VL If 

 x = p, y = q is a solution, it is stated that p/q > V and that p/q gives so 

 close an approximation to VI that a closer one cannot be found without 

 using larger numbers. After developing Vs into a continued fraction for 

 z = 13, 61, 67, he took a general z and set 



r ill 



vz = v + - . T , - , 



a + b + c+ , 



where v is the largest integer < Vz, and a, b, c, are found as follows. 



In Vz = v + I/a?, x = l/( Vz v) = ( Vz + v)/<x, where a = z y 2 ; hence let 



69 Reflexions nouvelles sur deux mdmoires de Lagrange 74 . . . , Turin, 1S59, 24 pp; Memorie 



R. Accad. Torino, (2), 20, 1863, 87-108. 



70 Ladies' Diary, 1759, pp. 39-41, Quest. 443. The Diarian Repository, or Math. Register 



... by a Society of Mathematicians, London, 1774, 677-9. C. Button's Diarian 

 Miscellany, 3, 1775, 81-83. T. Leybourn's Math. Quest, proposed in Ladies' Diary, 

 2, 1817, 162-4. 



71 Novi Comm. Acad. Petrop., 9, 1762-3 (1759), 3; Comm. Arith. Coll., I, 297-315; Op. 



Om., (1), II, 576. 



"Novi Comm. Acad. Petrop., 11, 1765 (1759), 28; Comm. Arith. Coll., I, 316-336; Op. 

 Om., (1), III, 73. 



