CHAP. XX] BINARY QUADRATIC FORM MADE AN nTii POWER. 541 



said to be even if A is a quadratic non-residue of a or if A = 4n+3, a = 4/j-f- 1. 

 When /i = 2y, y is odd. Then A" x = p~, A"+x = aq~, where p 2 and aq z 

 are relatively prime. Hence 2 A" = p "-\-aq~ and v is a minimum. 



L. Ottinger 71 tabulated solutions of x 2 y 2 = z n , n = 2, 3, 4, and gave the 

 identity 



{(4m 2 2w+r 2 ) 2 -8m 4 } 2 - {4m(md=r)(2w 2 2mr+r 2 ) } 2 = (2?/irr 2 ) 4 . 



T. Pepin 72 proved that, if c is positive and such that there is a single 

 class of positive odd quadratic forms of determinant c (as for c= 1, 2, 3, 

 4, 7), the most general manner of solving x*+cy~ = z m , where x, y are to be 

 relatively prime integers and z odd, is to set 



x = P, y=Q, z = 



where p, q are any relatively prime integers for which z is odd. Hence we 

 can justify the method of Euler for c = 1 or 2. Next (pp. 333-8), let n be 

 a positive integer such that all the quadratic forms of determinant n are 

 distributed into various genera each composed of a single class; then all 

 relatively prime solutions of x 2 +ny z = z~ m+1 , with z odd, are obtained from 



(1) 



where p, q are relatively prime. But for x--\-ny 2 = z 2m , z odd, we use (1) 

 with the exponent m, and employ the complete solution 



p ~ 



- k ' k ' k ' 



of p 2 -\-nq z =z 2 , where for a, b are to be taken all the decompositions of n 

 into two relatively prime factors, except that when k = 2, n = St, the two 

 factors shall have 2 as their g.c.d. For ax 2 +cy z =z m , a>\, c>l (pp. 

 339-343), when ac is one of the numbers for which the number of classes of 

 quadratic forms of determinant ac equals the number of genera, there is 

 no solution in integers H=0 if m is even; while if m is odd we get all relatively 

 prime solutions with z odd from 



where p, q are relatively prime. Thus 2x 2 +3 and 2+3?/ 2 can not equal 

 cubes. 



Pepin 15 proved that z 5 +a is not a square if a = 32(2d-f I) 5 56 2 , where b 

 is prime to 10 and has no prime factor 20Z+11. If d = 0, 6 = 1, 3, then 

 a = 27, -13. 



M. d'Ocagne 73 solved x-ky 2 = z n in positive integers by use of 



*(,An)= 



i=0 



71 Archiv Math. Phys., 49, 1869, 211-222. 



72 Jour, de Math., (3), 1, 1875, 325. Results for m = 3 cited under Pepin. 10 



73 Comptes Rendus Paris, 99, 1884, 1112. 



