540 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xx 



Then 



a+b V^= VN (cos 0+i sin 0), A = N A/2 cos X0, JB= {AT A/2 sin X0}/ V^. 



Hence B is a minimum = for a rational value of X approximately equal 

 to 7T&/0, where & is an integer. 



Euler 66 made z 2 +7 a biquadrate. For x = (7p 2 q 2 )/(2pq), it is the 

 square of (<? 2 +7p 2 )/(2pg). To make the latter a square, take q = pz, whence 

 we are to make 2z(7+2 2 ) = D. Since an evident solution is 2 = 1, set 

 2 = l+2/. We get 16+20y+Qy 2 +2y 3 , which is the square of 4+5?//2 for 

 2,= 1/8. 



A. M. Legendre 67 treated Ly 2 +Myz+Nz z = bP, where P is a product of 

 powers of several variables, in particular, x k . 



G. L. Dirichlet 68 recalled that if I is an odd prime not dividing a and if 

 d 2 ae z j=l it is known that d 2 ae 2 = l n holds for the numbers d, e given by 

 d+eVa = (5+eVa) n . It is proved that d, e are relatively prime. If also 

 dlaet = kl n , where di, e\ are relatively prime, and k is odd and prime to al, 

 we can find solutions of t 2 au 2 = k such that 



for a suitable choice of signs. Application is made to show that, if P, Q 

 are relatively prime, the most general manner of making P 2 5Q 2 a fifth 

 power, odd and not divisible by 5, is to set 



where M, JV are relatively prime, one even and M not divisible by 5. If 

 P, Q are relatively prime, both odd, and Q is divisible by 5, the most general 

 way to make P 2 5Q- = 4z 5 is to set 



where <, \f/ are relatively prime, both odd, and < is prime to 5. 



Cauchy's papers on the representation of p k or 4p fc , where p is a prime, 

 by x 2 -\-ny 2 will be considered under binary quadratic forms. Luce 127 of 

 Ch. XII discussed x 2 ny 2 = z\ 



F. Landry 69 obtained a new kind of continued fraction from 



A=a-+r, 



If m/n is a convergent of order u, m 2 An 2 =( l) v r u . Hence to solve 

 x 2 Ay 2 = z m , take as z any integer for which A = a 2 z. 



V. A. Lebesgue 70 recalled the fact that, if a is an odd prime and A is 

 an odd integer dividing t 2 +a, but not divisible by a, A = x 2 +ay 2 holds for 

 an infinitude of values /*, when x, y are relatively prime. The least /j. is 



66 Algebra, St. Petersburg, 2, 1770, 160; French transl., Lyon, 2, 1774, pp. 191-3; Opera 



Omnia, (1), I, 413. 



67 Th6orie des nombres, 1798, 435-40; ed. 2, 1808, 374-9; ed. 3, 1830, II, 43-49; German 



transl. by Maser, II, 43-50. 



68 Jour, fur Math., 3, 1828, 354; Werke, I, 21. 



69 Cinquieme m6moire sur la thdorie des nombres, Paris, July, 1856. 



70 Jour, de Math., (2), 6, 1861, 239-240. 



