646 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xxn 



* 



whence 



(4) 158 4 +59 4 = 133 4 +134 4 . 



For m=n = l, b=f/g, we get a=/ 2 /(/ 2 2 )- In the resulting fraction 

 for pfr, take p to be the product of the numerator by g. We obtain the 

 solution of (1). 



The case/ =2, = 1 gives p = 275, 5 = 928, r = 626, s = 550 ; whence 



2379 4 +27 4 = 729 4 +577 4 . 



From one set of solutions of (1) we obtain the second set 

 p'=p+q+r+s, q' = p+q r s, r' = p q+rs, s' = p q 



A. Desboves 169 noted that 1203 4 +76 4 = 1176 4 +653 4 . 



Desboves 170 wrote sfq = m in (1) and obtained p 3 +pg 2 m 3 # 2 r rar 3 = 0. 

 Regard m as a parameter. From the solution p = m, q = r = l, we derive 

 by Cauchy's formula the new solution 



p = 2ra(m 6 +10m 4 +ra 2 +4), g = (ra 2 +l)(-ra 4 +18m 2 -l), 



r = 2(4ra 6 +ra 4 -HOra 2 +l). 



Replace m by fig. The resulting solution is not new, as supposed by 

 Desboves, 171 but 172 is Euler's (5). For/=l, g = 3, we get (4). For/=l, 

 g = 2, we get Desboves' 169 numbers. 



A. Cunningham 173 discussed the solution of the problem and proved the 

 impossibility of 4 +2/ 4 = 4 +4?7 4 . 



R. Norrie, 174 starting with an evident solution of (1) took p = pxi s, 

 r = pX2 q; thus 



After making the coefficient of p zero by choice of x^/xi, we have only to 

 take p equal to the ratio of the coefficient of p 2 to that of p 3 . After reduc- 

 tions, we obtain Euler's (5). The same method applies also to 



A. S. Werebrusow 175 gave 239 4 +7 4 = 227 4 +157 4 and Euler's solution (4). 



T. Hayashi 176 reduced the problem to the solution of 3w 4 -fv 4 = t0 2 , from 

 one solution of which we obtain an infinitude (Desboves 77 ). 



F. Ferrari 177 expressed (4 2 +5 2 )(7 2 +8 2 )(4 2 +15 2 )(13 2 -f20 2 ) as a sum of two 

 squares in eight ways and noted that the squares are biquadrates in two 

 cases, giving Euler's (4). 



169 Nouv. Corresp. Math., 5, 1879, 279. 



170 Assoc. fran$., 9, 1880, 239-242. 



171 Nouv. Corresp. Math., 6, 1880, 32. 



172 Noted by E. Fauquembergue, Mathesis, 9, 1889, 241-2; reproduced in Sphinx-Oedipe, 5, 



1910, 93-4. 



173 Messenger Math., 38, 1908-9, 83-9. 



174 University of St. Andrews 500th Anniversary Mem. Vol., Edinburgh, 1911, 60-1 



175 L'interm6diaire des math., 20, 1913, 197; 19, 1912, 205. 

 "The Tohoku Math. Jour., 1, 1912, 143-5. 



177 Periodico di Mat., 28, 1913, 78. 



