682 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xxm 



P. F. Teilhet 63 stated for m = 3 and several proved that, if m is a prime, 

 n(n+l)(n+2) =mA 2 is impossible. 



A. Gerardin 64 remarked that if l-{-x\ = y 2 has solutions other than 

 x=4, 5, 7; ?/ = 5, 11, 71, then y has at least 20 digits. 



SUM OF nTH POWERS AN nTH POWER. 



Euler (Ch. XXII, paper 187 and the one preceding it) expressed his 

 belief that no sum of four fifth powers is a fifth power. 



E. Collins 65 noted that if N= 1 +n+n 2 + +n fc ~ 1 is divisible by a prime 

 p, then p=l (mod &), since n k = (n l)N+l. Henceforth, let this N be a 

 prime. Then, if A is any integer not divisible by N, A q is congruent to a 

 power of n modulo N, where q=(N l)/k, since A q is a root of x k =l 

 (mod N), and its roots are powers of n. Hence if '+ +a* = A q , while 

 01, -, a n are not divisible by the prime N, the difference of some two 

 of the al is divisible by N. For example, if n = 2, k = 3, then N = 7, q = 2, 

 whence if a sum of two squares (each prime to 7) is a square, their difference 

 is divisible by 7. Again, let N = 1 + 5 + 5 2 = 3 1 ; then q = 10 and, if a sum of 

 five tenth powers (not divisible by 31) be a tenth power, a difference of 

 two of the powers is divisible by 31. He verified that q > n except when 

 & = 2, or A; = 3, n = 2. He conjectured that a sum of n numbers each an eth 

 power is not an eth power if n < e. 



F. Paulet 66 announced that no nth power is a sum of nth powers if n>2. 

 A committee reported adversely, citing the known formula 6 3 = 3 3 +4 3 -f5 3 . 



0. Schier 67 made an erroneous discussion of x n +y n +z n = u n . First, let 

 n be an odd prime. Then x-\-y-\-z = u-\-nd. Subtract its nth power from 

 the given equation. The new left member has the factor y-\-z which is 

 said to be divisible by the factor n of the new right member. This admitted, 

 the given equation would be impossible for n a prime >3 and hence for 

 any n > 3. Only special sets of solutions are found for n = 3 and n = 2. 



A. Martin 68 found by tentative methods (Hart 115 and Martin 119 of Ch. 

 XXI) 



45+5 5 +6 5 +7 5 +9 5 -fH 5 = 12 5 , 5 5 + 10 5 + 11 5 + 16 5 + 19 5 +29 5 = 30 5 , 



100 



! 3 +3 3 +4 3 +5 3 +8 3 = 9 3 , 



100 



]T& 4 -l 4 -2 4 -3 4 -4 4 -8 4 -10 4 -14 4 -24 4 -42 4 -72 4 = 



Barbette 198 of Ch. XXII noted that the first result is the only one in- 

 volving fifth powers each ^12 5 . 



Martin 69 would by trial express l n +2 n -|- -\-x n b n as a sum of distinct 

 nth powers each ^x n . For n = 5, x = ll, b = 12, we get his 68 first result. 



63 L'interme'diaire des math., 11, 1904, 68, 182-4. 



64 Nouv. Ann. Math., (4), 6, 1906, 223. 



65 M<m. Acad. Sc. St. Petersbourg, 8, ann<5es 1817 et 1818, 1822, 242-6. 

 66 Comptes Rendus Paris, 12, 1841, 120, 211. 



87 Sitzungsber. Akad. Wiss. Wien (Math.), 82, II, 1881, 883-892. 



68 Bull. Phil. Soc. Wash., 10, 1887, 107; in Smithsonian Miscel. Coll., 33, 1888. 



69 Math. Quest. Educ. Times, 50, 1889, 74-5. 



