CHAP, xxiv] INTEGERS WITH EQUAL SUMS OF LIKE POWERS. 707 



A. Gerardin 12 noted thato; 3 +?/ 3 +2 3 = (z+l) 3 + (?/ 2) 3 +(z+l) 3 is equiva- 

 lent to A x +A z =(y-iy, where A x = x(x+l)/2. He took A 2 = l, 3, 6, 10, 15, 

 in turn and found the possible z's^llOO by use of a table of triangular 

 numbers. He found 13 solutions like 



1 3 +15 3 +12 3 = 2 3 +10 3 +16 3 , 1 + 15+12 = 2+10 + 16. 



The sum of the squares of 1, 15, 12 exceeds that of 2, 10, 16 by 10. Consider 

 two of our 13 solutions for which the ratio of the excesses mentioned is a 

 square m 2 ; multiply the numbers of the first solution by m and add to the 

 second solution ; in this way we get 



2,4,20,22,33 = 1,6, 16,26,32; 



1,4, 12, 13, 20 = 2, 3, 10, 16, 19; 

 3, 4, 15, 20, 23, 26 = 2, 5, 17, 18, 22, 27; 

 2, 6, 30, 46, 53, 73 = 3, 4, 34, 44, 51, 74; 

 2, 6, 44, 58, 63, 91 = 1, 8, 40, 60, 65, 90. 



Others follow by adding two of these. From x-\-y-\-z = x-\-2-\-y 4+2+2, 

 he got 



1, 19, 23, 24, 32, 48 = 3, 15, 20, 25, 40, 44. 



Gerardin 13 noted that 14, 23, 25, 138 = 7, 26, 30, 137, 



1, 0+3, 30+2, 40+4 = 2, 0+1, 30+4, 40+3, 



2, 12, 15, 35, 38, 48 = 3, 8, 20, 30, 42, 47, 



while x+h, y-\-p, z = x, y, z+h-\-p is impossible. [The last fact is a case of 

 Bastien's 48 evident theorem.] 

 H. B. Mathieu 14 noted that 



I, lman, l+(a l)ra n = l m n, I an, l+(a l)m. 



U. Bini 15 gave a +6, c, d = c+d, a, & if ab = cd. [Cunningham. 11 ]] 

 E. B. Escott 16 showed how to find all solutions of 



n n 



//-Y\ v ^ TI V ^ 2 



/ ) I > f 7^01 > *V - 



\*>) Zt^i ^yii / **'i- 



i-l i=l 



for 7i = 3. Set Xi = Xi+S, 2/,-=F;+, where 3S=Xi+x 2 +x z . But, if 

 is not divisible by 3, take S = So;,-, 3a: = Xi+S, 3y f = F<+. Thus 



Using these to eliminate X 3 and F 3 from 2Xi-XT 2 = SFiF 2 , we get 

 (3) Xl+X,X,+Xl= Yl+Y^+Yl 



Hence the problem reduces to solving (3). To find all its solutions, let 

 N be any number all of whose prime factors are of the form 6n+l or 3, 

 besides square factors common to Xi, X 2 , FI, F 2 . Then represent N in all 

 ways in the form 



12 Sphinx-Oedipe, 1906-7, 120-4. 



13 Ibid., 1907-8, 27, 94-5. Also, a case of (1). 



14 L'intermgdiaire des math., 14, 1907, 201. All the solutions, ibid., 50, 200-3, by the other 



writers are special cases of (1). 

 16 Ibid., 227. His other solution is equivalent to (1). 

 IUd., 15, 1908, 109-111. 



