CHAP. XXIII] SUMS OF LlKE POWERS. 685 



V. Bouniakowsky 88 obtained the identity 



(10X 2 +z) 5 +(10X 2 -z) 5 +8(10X 2 ) 5 =(10 3 X 5 +10Xz 2 ) 2 



from f{(x-\-aY (x aY\dx by setting a = 10X 2 . 



E. Lucas 88a stated that the sum of the cubes of the first n (odd) integers 

 is never a cube, fifth or eighth power (cube, fourth or fifth power). The 

 sum of the cubes of three consecutive integers is never a square, cube or 

 fifth power, except for ! 3 +2 3 +3 3 = 6 2 , 3 3 +4 3 +5 3 = 6 3 [correction, Aubry 286 

 of Ch. XXIJ. The sum of the first n biquadrates is never a square, cube 

 or fifth power. The sum of the first n fifth powers is never a cube, fourth 

 or fifth power. 



E. Lucas 89 asked for what values of n the sum of the fifth powers of 

 the first n odd numbers is a square. The problem reduces to 



whose complete solution was given by L. Aubry. 90 



Lucas 91 asked for what n's the sum of the fifth or seventh powers of 1, 

 , n is a square. H. Brocard 91 noted that the sum of the fifth powers is 

 \n\n+ 1}H, where t = (2n 2 +2n - 1)/3. To make t = y*, we have 



which must have 9 as its final digit, whence y = lOml. He noted the 

 special solutions y = n=l', y = 11, n = 13. Cf. Moret-Blanc, 95 Fortey." 



H. Brocard 92 noted that the sum n 2 (2n 2 1) of the cubes of the first n 

 odd numbers is a square for n = l,5, 29, 169, 985, . As to Lucas' 880 theorem 

 that the sum s of the squares of the first n odd numbers is not a square, cube 

 or fifth power, he stated that this is evident since s= (2n l)(2n)(2n+l)/6. 

 Lucas (p. 247-8) noted that this proof would require extensive develop- 

 ments; if p is a product of three consecutive numbers, p/Q is not a square 

 if the first of the three numbers is odd, and also if it be even except for 

 2 3 4/6 = 2 2 , 48 49 50/6 = 140 2 . 



Abbe Gelin 93 proved that (x-l) 2n +x 2n +(x+l) 2n = y 2n is impossible 

 and that the sum of like even powers of 9 or 12 consecutive integers is never 

 an exact power (stated for 9 by Lucas, p. 248). The proof is by use of 

 various properties of 2(A r ), obtained by adding the digits of N, then adding 

 the digits of this sum, etc., until there results a sum with a single digit. 



E. Lucas stated and H. Brocard, Radicke and E. Cesaro 94 proved that 



{! 5 -3 5 +5 5 ----- (4x- 1) 5 }/ {1-3+5 ----- (4z-l) } 



88 Bull. Acad. Sc. St. Petersbourg (Phys.-Math.), 11, 1853, 65-74. Extract in Sphinx-Oedipe, 



5, 1910, 14-16. 



880 Recherches sur 1'analyse indeterminee, Moulins, 1873, 91-2. Extract from Bull. Soc. 

 d'Emulation du Departement de TAllier, 12, 1873, 531-2. 



89 Nouv. Corresp. Math., 2, 1876, 95. 



90 L'intermediaire des math., 18, 1911, 60-62. Cf. 16, 1909, 283. 



91 Nouv. Corresp. Math., 3, 1877, 119-120. Cf. 4, 1878, 167. 



92 Ibid., 3, 1877, 166-7. 



93 Ibid., 388-390 (extract from Les Mondes, July 14, 1877). 

 9 *Ibid., 5, 1879, 112, 213-5; 6, 1880, 467. 



