366 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xn 



even (since the right members would be multiples of 4), and hence both 

 are odd, whence g 2 =h 2 = l (mod 8), and the right members would be 

 multiples of 4. Hence the only possibility is the case 2 = 2Ag~ 2h 2 , so 

 that h 2 Ag z = 1 is solvable. Besides the remaining two theorems for 

 primes 8n+3, 8n 1, cited above, Legendre proved that one of 



is solvable if M and N are primes of the form 4n+3. Given a positive 

 integer A not a square, it is always possible to decompose it into two factors 

 M, N, such that one of Mx 2 Ny* = l, Mx z Ny 2 =2 is solvable when 

 the signs are suitably chosen. When x 2 Ay 2 = 1 is solvable, A is a sum 

 of two squares. Cf. Arndt. 124 In Table XII, he gave the least positive 

 solutions of m?an 2 = 1, when it is solvable, and of m 2 cw 2 = +1 in the 

 contrary case, for 2 ^a^ 1003, a not a square [errata, Cunningham, 259 ' 309 

 Richaud, 198 Whitford 4 (p. 97), Gerardin 311 ], but with no indication as to 

 which equation has the solution listed. It was reprinted (with fewer errata) 

 as Table X in ed. 3, 1, 1830, and abridged to a ^135 in ed. 2, 1808. 



J. Tessanek 89 considered (a 2 +6)n 2 +l = D, say (an+p) 2 . Set n = 

 Then p satisfies a quadratic. Write b a = h, 2a+l b = g. Then 



Replace p by q+r and solve for q in terms of r. Thus 



g'q= 

 where 



h' = g-h = 3a-2b+l, g' = 



y 



Replace q by r+s and solve for r in terms of s. Thus 



0"r = 

 where 



9 



Replace r by s+t. Then 



According to the method of Pell, 62 " 4 one ultimately obtains an equation in 

 which the number g under the radical is + 1. To find values of n for various 

 a's, set g = l or g" = l, etc., whence b = 2a; or 3fr = 4a+l, s = 0, r = q l,p = 2, 

 n = 3; etc. The terms free of a, b in 1, g, g", {/ (iv) , are 1, 1, 4, 25, , 

 i. e., the squares of 1, 1, 2, 5, 13, 34, , whose differences of second order 

 give the same series. Thus the scale of relation is u n+ i = 3u n Un-i, so 

 that the general term is expressible in terms of the roots of 1 32+2 2 = 0; 

 likewise for the coefficients of b, a. 



89 Abh. Bohmischen Gesell. Wise., Prag, 2, 1786, 160-171. 



