CHAP. XXIII] SUM OF 71TH POWERS AN 71TH POWER. 683 



Martin and G. B. M. Zerr 70 multiplied the numbers 4, 5, , 12 in the 

 formula just cited by 42 4 and obtained six numbers whose sum is a fifth 

 power 42 5 and sum of fifth powers is a fifth power. 



Martin 71 multiplied his 68 first formula by 2 5 and replaced the new third 

 term 12 5 by its value to get a formula for 24 5 . There is an analogous longer 

 formula for 50 5 . Again, 



l e +2 6 +4 6 +5 6 +6 6 +7 6 +9 6 +12 6 



+ 13 6 +15 6 +16 6 +18 6 +20 6 +21 6 +22 6 +23 6 = 28 6 . 



Martin 72 found sets of fifth powers whose sum is a fifth power. 



G. de Rocquigny 73 proposed for solution (x r) m +x m +(x+r) m = y m . 

 H. Brocard 74 noted x = 4, r = l, y = Q [m = 3], and E. B. Escott 74 noted 

 x = l, r = 2, y = 3, for m any odd number. Cf. Gelin, 93 also Escott 261 of 

 Ch. XXI, and Bottari 190 of Ch. XXV. 



A. Martin 75 found sixth powers whose sum is a sixth power by the tenta- 

 tive method of expressing p 6 q 6 as a sum of distinct sixth powers =f=^ 6 , 

 or S b 6 as a sum of sixth powers ^in 6 , where $ = 1 6 + +n 6 . By each 

 method he found his 71 example, also that the sum of the sixth powers of 



1, 1, 2, 5, 9, 11, 12, 13, 15, 18, 21, 22, 23, 24 is 29 6 [false] and that of 1, 2, 



2, 4, 5, 6, 8, 9, 10, 12, 14, 15, 18, 19, 27, 33, 49 is 50 6 (each with one repeated 

 term). By combining these, he found eleven new sets of 29, 31 (seven), 32, 

 46, 47. He tabulated the values of n 6 and 1 6 H \-n* for n ^228. 



C. Bianca 76 noted that s = a?-f- +a p n+l is a pih power if 



01 : a 2 : : a n +i = b n : b n ~ l c : b n ~ 2 cd : b n ~ 3 cd 2 : : bcd n ~* : cd n -\ 



where b p +c p = d p . For, if ai = kb n , , then s = (kd n ) p . 



A. Martin 77 reported on sums of nth powers equal to an nth power. 



* N. Agronomof 78 proved that x z +l -\- - +* m+1 = is solvable in integers 

 if k = 4 n + 1 and n=m. He proved the identity 



Si - Z 2 + -..+(- 1 ) 2w+1 2W 2 = 0, 



where S/ denotes the sum of the (2w+l)-th powers of all the sums of 

 2m+2 parameters taken j at a time. A. Filippov 780 gave an account in 

 French of this paper, with details for the case m = 2. 



70 Math. Quest. Educ. Times, 55, 1891, 118. 



71 Quar. Jour. Math., 26, 1893, 225-7. 



72 Math. Papers Internat. Congress of 1893 at Chicago, 1896, 168-174. Republished, 



Math. Mag., 2, 1898, 201-8, with the following corrections: In Ex. 18, p. 173, insert 16 6 ; 

 on p. 169, fourth line up, delete one 3 s ; on p. 174, delete the final equation. In Part III 

 (combining earlier sets) he added a new set of n fifth powers for n = 17, 21, 24, 26, 28, 36, 

 42, 48, 52, 63, 67, 72 and three sets for n = 33. 



73 L'intermediaire des math., 9, 1902, 203. 



74 Ibid., 10, 1903, 131-3. 



75 Math. Mag., 2, 1904, 265-271. 



76 II Pitagora, Palermo, 13, 1906-7, 65-6. 



77 Proc. Fifth Intern. Congress of Math., 1912, I, 431-7. 



78 Izv. Fis. Mat. Obs. Kazan (Bull. Soc. Phys. Math. Kasan), 1914, 1915. 

 78 T6hoku Math. Jour., 15, 1919, 135-40. 



