774 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xxvi 



P. Montel 275 proved that if m, n, p are integers for which 1/m-f 1/n 

 <1, it is impossible to find three integral functions of a variable such that 

 x m -\-y n +z p = Q; in particular, x m -\-y m -}-z m ^Q if ra>3. 



P. Kokott 276 proved that x u -\-y n +z n = Q is impossible in integers prime 

 to 11, using residues modulo 11 of symmetric functions of x, y, z. 



W. Mantel 277 proved that if n>3 and p are primes, x n -\-y n +z n =Q 

 (mod p) is impossible in integers prime to p unless p= (Qkn n 3)/(n 3). 



E. T. Bell stated and F. Irwin 278 proved that if x n y n is a prime 2 a r+l 

 for r a prime > 2 and n > 2, then n = 3, x = 2, y = l. 



A. Gerardin 279 proved that 10^+1=2" is impossible in integers if n>l. 



H. H. Mitchell 280 treated the solution of co; A +l = cfo/ in a Galois field. 



A. J. Kempner 281 gave a simple proof that a 2n 1 = 26" has only the 

 integral solutions o = l, b = [Liouville, 32 Lind 224 ]. 



A. Korselt 282 proved, without using integrals as had R. Liouville, 105 that 

 m +?/ n +2 r =0 is not solvable in relatively prime integral rational functions 

 of a variable t if each exponent exceeds 2 or if one exponent is 2 and the 

 others exceed 3, the case 2820 z 3 +2/ 5 +z 2 =0 not being decided. In all the 

 remaining cases, the initial equation is solvable Cf. Velmine, 177 Montel. 275 



* J. Schur 283 gave a simpler proof of Dickson's 199 theorem. 



L. Aubry 284 proved that a-10 k +l^z n if 0<a<10, fc>l, and n is a 

 prime >1. 



E. Maillet 285 considered a m +b m = c m for m = n/p, where n, p are relatively 

 prime positive integers and p>l. It has integral solutions each =(=0 if 

 and only if 





has integral solutions each =|=0 such that a\, &i, Ci are prime to p and rela- 

 tively prime in pairs, while a 2 , b z , c 2 are relatively prime in pairs and have 

 no prime factors other than those of p. The last equation can be given a 

 similar form in a\, b\, c\, a\, b\, c\, which are relatively prime in pairs, while 

 any prime factor X of a\, bl or c\ is a divisor of p such that m^l/(X 1). 

 In particular, if m> l/(ju 1), where ju is the least prime factor of p, Fermat's 

 equation with the exponent m is equivalent to one with the exponent n. 

 This is also the case if one of o 2 , 62, c 2 , a\, 6j, c\ is an exact pih power and 

 hence if p has at most two distinct prime factors. Corresponding results 

 hold for a mi +6 m> = c m8 , with any fractional exponents, and with a, 6, c rela- 

 tively prime in pairs. 



276 Annales sc. P<5cole norm, sup., (3), 33, 1916, 298-9. 

 276 Archiv Math. Phys., (3), 24, 1916, 90-1. 



277 Wiskundige Opgaven, 12, 1916, 213-4. 

 1 Amer. Math. Monthly, 23, 1916, 394. 



279 L'intermMiaire des math., 23, 1916, 214-5; Sphinx-Oedipe, 1917. 



280 Trans. Amer. Math. Soc., 17, 1916, 164-177; Annals of Math., 18, 1917, 120-131. 



281 Archiv Math. Phys., (3), 25, 1916-7, 242-3. 



282 Ibid., 89-93. 



1820 This equation is satisfied by the fundamental invariants of the icosaeder group, ibid.. 27, 

 1918, 181-3. 



83 Jahresber. d. Deutschen Math.-Vereinigung, 25, 1916, 114-7. 



14 L'interme'diaire des math., 24, 1917, 16-17. 

 286 Bull. Soc. Math. France, 45, 1917, 26-36. 



