CHAPTER XXIV. 



SETS OF INTEGERS WITH EQUAL SUMS OF LIKE POWERS. 



If t % (a+b+c), a, b, c and t a, t b,tc have the same sum and same 

 sum of squares; this double property shall be denoted by 



(1) a,b, c = ta, tb, tc, ^ = |(a+6+c). 



The separation of two sets of numbers by the symbol = shall denote that 

 they have the same sum of A'th powers for k 1, , n. 

 Chr. Goldbach 1 noted that 



a+/3+5, a+7+5, P+y+8, 8 = a+d, f3+8, y+d, a +0+ 7 +5. 



L. Euler 2 remarked that a, b, c, a+b+c = a-\-b, a+c, b+c. This is the 

 case 5 = of Goldbach's result, but it implies the latter since (Frolov 7 ) each 

 number may be increased by any constant 5. 



If2a jy b e c hosen so that N, Nai, , Na t have the same sum as 

 n, n+ai, - -, n+a t , then the sum of the squares of the former numbers 

 equals that of the latter. 



E. Prouhet 3 noted that 1, , 27 can be separated into three sets, 

 two of which are 1, 6, 8, 12, 14, 16, 20, 22, 27 and 2, 4, 9, 10, 15, 17, 21, 23, 

 25, such that the sum and sum of squares of the numbers in any set are 

 the same as for the other sets. As a generalization, it is stated that there 

 are n m numbers separable into n sets each of n m ~ l terms such that the sum 

 of the kih powers of the terms is the same for all the sets when k<m. 



F. Pollock 4 noted the fact, equivalent to (1), that 



p, p+a, p+2a+3n = pn, p-\-a+2n, p+2a+2n. 

 F. Proth 5 noted that 



and the numbers derived by interchanging b and c have the same sum and 

 sum of squares. 



E. Cesaro 6 proved that if o, , k form a rearrangement of 1, , 9 

 and 



a, b } c, d = d, e, /, g=g, h, k, a, 



then a = 2, 6 = 4, c = 9, d = 5, e = l, /=6, g = S, h = 3, k = 7. Note that the 

 three sets of four numbers each may be placed on the sides of a triangle, 

 with a, d, g at the vertices. 



1 Corresp. Math. Phys. (ed., Fuss), 1, 1843, 526, letter to Euler, July 18, 1750. 



2 Ibid., 549, letter to Goldbach, Sept. 4, 1751. Special case by Nicholson 80 of Ch. XXIII. 

 20 New Series of Math. Repository (ed., T. Leybourn), 3, 1814, I, 75-77. 



3 Comptes Rendus Paris, 33, 1851, 225. 



4 Phil. Trans. Roy. Soc. London, 151, 1861, 414. 

 8 Nouv. Corresp. Math., 4, 1878, 377-8. 



8 Ibid., 293-5. Question by F. Proth. 



46 705 



