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



L. Calzolari 82 attempted a proof, starting as before. 74 



P. G. Tait 83 stated that if x m = y m -\-z m has integral solutions when m is 



an odd prime, then x=y=l t 2=0 (mod m). 



H. F. Talbot 84 noted that Barlow 15 made the same error in his proof of the 



impossibility of x n y n =z n as for the case n = 3 (p. 139), where he stated 



that, if r, s, t are relatively prime in pairs, 



p s 2 9r 2 



----- =|= O 



sr tr st 



since each fraction is in its lowest terms and each denominator has a factor 

 not common with the other denominators, and hence the algebraic sum of 

 the fractions is not an integer (by the false Cor. 2, Art. 13). On the con- 

 trary, we have 



2-3 3-5 2-5" 



A. Genocchi 85 abbreviated Lame's 28 proof for n = 7. Let x, y, z be roots 

 of V 3 pv z -\-qv pq-\-r = 0. Then x 7 -\-y 7 +z 7 = Q is equivalent to 



After excluding the trivial case p = Q, we may change q to p 2 q, r to p 3 r, 

 and get 7r 2 7r(l q+q 2 ) = 1. The radical in the expression for the root 

 r must be rational. Thus (1 g+<? 2 ) 2 /4 1/7 is a square. Set 2q l=s/t. 

 Then 



Proof of the impossibility of the latter is not given. 



Gaudin 86 attempted to prove that, if n is an odd prime, (x-\-h) n x n z n 

 is impossible in rational numbers. Treating xfh as a new variable, we are 

 led to the case h = l. To avoid the author's complicated formulas, take 

 n = 5. Then 



is of the form 1CM+1. Since z 5 is of that form, z = 10s+l and 



2 5 = 5-10s{10s[10s(10s-2s+l+2)+2]+l}+l, 



which is said never to equal the first expression. His remaining two argu- 

 ments are trivial. 



I. Todhunter 87 proved Cauchy's 29 theorem and that, if q=x*+xy+y z , 

 b=xy(x+y), 



q m (m-r-l)(m-r-2) -(m-3r + l) r 



g b ' 



2m m (2r)! 



82 Annali di Mat., 6, 1864, 280-6. 



83 Proc. Roy. Soc. Edinburgh, 5, 1863-4, 181. 



84 Trans. Roy. Soc. Edinburgh, 23, 1864, 45-52. 



85 Annali di Mat., 6, 1864, 287-8. 



86 Comptes Rendus Paris, 59, 1864, 1036-8. 



87 Theory of Equations, 1861, 173-6; ed. 2, 1867, 189; 1888, 185, 188-9. 



