264 HISTORY OF THE THEORY OF NUMBERS. [CHAP, vn 



show that there exists a positive ternary quadratic form F of determinant 

 + 1 whose first coefficient is a. Indeed, such a form is equivalent to 

 3.2 _j_ yi _|_ ^ so that the latter can be transformed into F by a substitution 

 of determinant unity; thus a is the sum of the squares of three of the 

 coefficients (having no common factor) of the substitution. Now the 

 ternary form 



ax 2 - + by 2 + cz 2 + 2a'yz + 2xz (A = be - a' 2 ) 



has the determinant +1 if b = aA 1. The form is positive if A > 0. 

 It suffices to show that a positive value of A can be found for which A is 

 a quadratic residue of 6, so that c and a' may be determined to satisfy 

 a' 2 - fa = - A. For a = 4& + 2, we take A odd. Then 6=1 (mod 4). 

 We seek a suitable prime b. Since, for Jacobi symbols, 



- G) - (f) - () -" 



A must be of the form 4t + 1, whence b = 4at + a 1. The latter is the 

 general term of an arithmetical progression, containing primes. For 

 a = Sk + 1, we take A = St + 3, and seek a prime p for which 2p = b. 

 Since 2p = aA 1, p = 1 (mod 4), 



There exists a prime in the progression p = 4aZ + |(3a 1). A like 

 result follows for a = Sk + 3, A = St + 1, and for a = Sk + 5, A = 8t + 3. 

 H. Burhenne 29 noted that x* + y* + z 2 = (a 2 + 6 2 + c 2 )s 2 if 



s = m 2 + n 2 + p z 

 and 



x = 2ml as, y = 2nl bs, z = 2pl cs, I = am + bn + cp. 



H. Faure 30 noted that no number m 2 (8x + 7) is a SJ . 



V. A. Lebesgue 31 proved that every odd number p is of the form 

 x z H- ?/ 2 + 2z 2 , where x, ?/, z are integers with no common factor. The 

 method is that of Dirichlet. 28 It follows that 



2p = (x + yY + (x- y} z + (2z) 2 . 



J. Liouville 32 denoted the number of sets of integral solutions of 

 X 2 + y 2 _j_ 2 2 = M by_^Gu). Set w = 2m, m odd, a > 0. Let co be the 

 greatest integer ^ Vn. Then 



S (^s 4 + Bs 2 + C}t(n - s 2 ) = (3An z + QBn + 24C)<r(w) 



i 



(s = 0, 1, .-., ), 

 where a(m) is the sum of the divisors of m. 



" Archiv Math. Phys., 20, 1853, 466-8. 

 80 Nouv. Ann. Math., 12, 1853, 336. 



31 Jour, de Math., (2), 2, 1857, 149-152. 



32 Jour, de Math., (2), 5, 1860, 141-2. 



