CHAP, xxil] ax 4 + by 4 = cz 2 . 627 



S. Re*alis 60 noted that if a 4 -20 4 = 7 2 , then x 4 -2y 4 =z 2 for 



2/ = 4(147a 3 -226/3 3 )-27aj3(5a+64 j 8)+77(108a+113|3). 

 For a = 7 = l, = 0, x :y : 2 = 113 : 84 : 7967. For a = 3, = 2, 7 = 7, 

 re = 57123, # = 6214, 2 = 3262580153. 



A. G6rardin 61 treated the last problem, assuming that also a second 

 solution A 4 - 25 4 = C 2 is known. Set 



Then 



2 ) -S 2 -2yC}u 



Equating to zero the coefficient of u 2 , we get S and u. Taking A = 3, 

 B = - 2, C = - 7, we obtain Realis' solution. Starting with 3 4 - 2 2 4 = 7 2 , set 



and annul the coefficient of m 2 ; we get g and m in terms of x, y and hence 

 a solution of the sixth degree. Modifying the last method, we again get 

 Realis' solution. 



A. Cunningham 62 noted that the solution of (2) by Lebesgue 56 and Lucas 57 

 appears to be complete and to indicate that the only integral solutions of 

 x 2 -2y 4 =-l are (1, 1) and (239, 13). But Euler's 53 solution of (2) yields 

 only half the solutions. 



L. C. Walker 63 quoted Fermat's last two integers (1), whose sum is a 

 square and sum of squares is a biquadrate. 



4 + by 4 MADE A SQUARE OR MULTIPLE OF A SQUARE. 



The cases x 4 y 4 , 2x 4 7/ 4 , x 4 2y 4 , x 4 -\-Sy 4 have been treated above. 

 For x 4 h 2 y 4 , see Congruent Numbers in Ch. XVI, especially papers 43, 54. 



G. W. Leibniz 64 treated before 1678 the problem to find an integer x such 

 that x+a/x = y 2 , where a is a given integer and y is to be rational. Set 

 a = bc, x = bz, where c and z are relatively prime integers. Set y = v/w, a 

 fraction in its lowest terms. Then bz 2 +c = zv 2 /w 2 , so that z is divisible by w 2 . 

 Similarly, since cw 2 fz is an integer v 2 bzw 2 , w 2 is divisible by z. Hence 

 z = w 2 and bw 4 -\-c = v 2 . Since c is the product of vw 2 V6, it exceeds each 

 of the factors and hence their difference, whence c 2 >4bw 4 . The resulting 

 tentative process to solve x-{-a/x = y 2 is to express a as a product be of two 

 integers, choose an integer w such that 46w 4 <c 2 and test the value x = bw 2 

 (or what is equivalent, see if bw 4 +c is a square). 



60 Nouv. Corresp. Math., 6, 1880, 478-9. 



61 Sphinx-Oedipe, 6, 1911, 87-8. 



62 Math. Quest. Educ. Times, (2), 14, 1908, 76-8. 

 63 Amer. Math. Monthly, 11, 1904, 39. 



64 Math. Schriften (ed., C. I. Gerhardt), 7, 1863, 114-9. 



