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



If m is a prime 6k+l, (a+1)- 1 ^!, a m ~ l ^l (mod ra 2 ) do not hold simul- 

 taneously. If w is a prime, the integers u, not divisible by m, which satisfy 



(u m +l) m -u m *=l (mod m 2 ) 



are of the form u = am 1 . 



P. F. Teilhet 159 found A for which x n Ay n =l by taking x = y n +l, or, 

 when n is even, x = y n 1. H. Brocard (pp. 116-7) found special solutions 

 when n 3, n = 5. T. Pepin (pp. 281-3) noted that we may apply to 

 x n Ay n the method of Lagrange in his Additions to Euler's Algebra to 

 find the minima of any homogeneous polynomial in x, y. 



W. L. A. Tafelmacher 160 treated x*-\-y 3 = z 2 and proved x 6 -\-y 6 = z 6 to be 

 impossible. 



H. Tarry 161 mentioned a mechanical device of double-entry tables for 

 solving indeterminate equations, in particular, x m -\-y m = z n . 



F. Lucas 162 used Cauchy's 29 theorem to prove that, if x, y are relatively 

 prime and m is an odd prime, when x-\-y is prime to m it is prime to 



Q=(x m +y m )l(x+y), 



but when x-\-y is divisible by m, m(x j ry] is prime to Q/m. From this he 

 deduced Legendre's formulas (1) and (3). 



Axel Thue 163 noted that, if L, M, N are functions of x such that 

 L n M n = N n for all values of x, where n>2, then aL = bM = cN, where 

 a, b, c are constants. If A n B n = C n , then 



If p n q n = r n , then x s y* = z 3 (pqr) n for 

 x = p 3n +3p 2n q n Qp n q 2n +q Zn , 



Z = 



E. Maillet 164 considered, for a, b, c, x, y, z integers not divisible by the 

 odd prime X, the equation 



A necessary condition for solutions is that the congruence 



a +6 7? *'= c ( a +/3, 7 )" (mod X m ) 



have a solution 77 such that 0<?7<X, a+/3r?^0 (mod X), where ac = a, 

 /3c = 6 (mod X). This is applied to show that x x +2/ x = 2 x is impossible for 

 X = 197, hence extending Legendre's limit to X<223. By the method of 

 Kummer it is shown that, if X is a prime > 3, 



is impossible in complex integers, formed from a Xth root of unity, relatively 

 prime by twos and prime to X, if X'" 1 is the highest power of X dividing the 



169 L'intermediaire des math., 3, 1896, 116. 



160 Anales de la Universidad de Chile, 97, 1897, 63-80. 



161 Assoc. frang. av. sc., 26, 1897, I, 177 (five lines). 



162 Bull. Soc. Math. France, 25, 1897, 33-35. Extract in Sphinx-Oedipe, 4, 1909, 190. 



163 Archiv for Math, og Natur., Kristiania, 19, 1897, No. 4, pp. 9-15. 



164 ASBOC. franc, av. sc., 26, 1897, II, 156-168. 



