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



F. Borletti 131 proved that, if n is a prime >2, x n +y n =z n has no positive 

 integral solutions if z is a prime, and x 2n y 2n =z 2n has no integral solution if 

 one of the unknowns is a prime; x n y n = 2 an is impossible if n> 1, and x, y 

 are odd and relatively prime. 



E. Lucas 132 proved Cauchy's 29 result. Set q = a? -\-db-\-b 2 , 



r = ab(a+b), S B = ( o +&)+(_ )+(-&). 



Then S n +3 = qS n+ i-\-rS n . Hence, by Waring's formula, S n is divisible by 

 (fr if n = 6m+l; by q, and not by r, if n = 6m+2; by r, and not by q, if 

 n = 6w+3; by q 2 , and not by r, if w = 6m+4; by gr if 7i = 6ra+5; by neither 

 5 nor riin = 6m. As a generalization, if p is a prime, 



is divisible by Q = 1+aH ----- }-x p ~ l if n is odd and prime to p, and by Q 2 if 

 w = 2p+l. For p arbitrary, let <f>(x)=Q be the equation for the primitive 

 pth roots of unity. Then without details it is stated that 



is divisible by 0(x) for n odd and prime to p. [Apparently the term $ An 

 should be added, and X taken to be the degree of #(x), which degree is the 

 number of integers <p and prime to p.^ 



L. Gegenbauer 133 proved that, if a is a positive integer with at least one 

 odd factor >1, and q is a prime, x a -{-y a = q n has positive integral solutions 

 only when q = 2, n = aa+l, x = y = 2 a , or a = g = 3, w = 2+3a, z = 2-3 a , y = 3. 

 Hence 3 2 is the only power of an odd prime representable as a sum of the 

 ath powers of two relatively prime integers. A special case of this gives 

 the seventh empirical theorem of Catalan. 122a It is proved that if q is a 

 prime, x a+1 q n = l is possible only for x = 2, n = l, a+1 a prime, or x = 3, 

 a = l, q = 2, n = 3. Hence a prime other than 2 n 1 is not followed by a 

 power, while 3 2 is the only power followed by a power of a prime. These 

 imply the third, fourth, fifth and sixth empirical theorems of Catalan. 



A. Rieke 134 attempted to prove x p -\-y p = z p impossible if p is an odd 

 prime >3. He proved and used (6). From an equation of degree 

 t=(p l)/2 in a quantity m admitted to be doubtless irrational, he drew 

 (p. 241) the meaningless conclusion " that m l has the factor p, and m the 

 factor p llt , and indeed for all values of m." 



D. Varisco 135 failed to prove Fermat's last theorem since he concluded 

 (p. 375) that there is a unique set of solutions <7i = 0, etc., of 



Xi ffi = 2ud, Xi<ii ffd= 77, ff\ = 2udi, <Tidi\d = tj } 



whereas the four equations are linearly dependent and have further sets 

 of solutions. The fault seemed irreparable to 0. Landsberg. 136 



131 Reale 1st. Lombardo, Rendiconti, (2), 20, 1887, 222-4. 



132 Assoc. frang. av. sc., 1888, II, 29-31; ThSorie des nombres, 1891, 276. 

 138 Sitzungsber. Akad. Wiss. Wien (Math.), 97, Ha, 1888, 271-6. 



134 Zeitschrift Math. Phys., 34, 1889, 238-248. Errors noted by a "reader," 37, 1892, 57, 



and Rothholz. 140 



136 Giornale di Mat., 27, 1889, 371-380. 

 186 Ibid., 28, 1890, 52. 



