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



composite; but erred in his proof that the least unknown is composite. 

 He gave abstracts of the papers by Calzolari, 72 Dirichlet, 23 Kummer, 63 and 

 Legendre, 17 a list (191-2) of references on Bernoullian numbers and ideal 

 complex numbers, and (189-191) a short proof of the impossibility of 

 x 5 -\-y 5 = z 5 . In an appendix (ibid., 17, 1902, 48-50) he quoted Kummer 49 

 and Liouville 46 on the insufficiency of the proofs by Lame, 45 and Cauchy. 54 " 56 



Soons 172 proved theorems stated by Catalan. 90 



P. Stackel 173 proved Abel's theorem as given by Lindemann. 170 



G. Candido 174 proved a theorem of Catalan. 121 



* D. Gambioli's 175 paper was not available for report. 



P. Whitworth 176 noted that if Sl/z = 0, Sz = l, then 2x n = x n +y n +z n 

 equals a series in xyz. 



P. V. Velmine 177 (W. P. Welmin) proved that, if m, n, k are integers >1, 

 there exist rational integral functions u, v, w of a variable which satisfy 

 u m -\-v n w k only for the cases i4 m d=w 2 = w 2 , u 3 -\-v 3 = w 2 , u*-}-v 3 = w z (when 

 the solution is easy), and i* 5 +v 3 = w 2 , the complicated formulas for whose 

 solution are not proved to give all solutions. Cf. Korselt. 282 



D. Mirimanoff 178 studied P(x) = (x+l) 1 x l 1 where I is a prime >3. 

 Since it is unaltered when x is replaced by 1x, a root a of P(aO=0 

 implies the roots 



(9) a, I/a, -1-a, -l/(l+a), -1-1/a, -/(!+), 



all of which are distinct unless a = or 1 or 2 +a+l = 0. Now P has 

 the factors x(x+l) and x z +x+l. Set 



P(x} 



where e = l if Z+l (mod 3), t = 2 if Z=l (mod 3). Then E(x)=Q has only 

 distinct imaginary roots which fall into sets of six. Thus E(x)=Uej(x) t 

 where each BJ(X) is of the form x 6 +l+3(x 5 +x}-\-t(x*+x 2 )-}-(2t 5)o: 3 , 

 where t is real. If E(x) has a factor which is irreducible in the domain of 

 rational numbers, the factor is a product of certain of the BJ(X). 



A. S. Werebrusow 179 denoted u?+uv v z by (u, v}. Then x 5 - y 5 = 

 becomes 



This decomposes into two equations, one being the second factor equated 

 to Aizl, the other being x-}-y = A z 5 , where A Ai = A, z Zi = z, and z\ is a 

 product of primes 5n+l. Multiplying (u, v) by 1 = 9 2 5-4 2 and its powers, 



172 Mathesis, (3), 2, 1902, 109. 

 178 Acta Math., 27, 1903, 125-8. 



174 La formula di Waring e sue notevoli applicazioni, Lecce, 1903, 20. 



175 II Pitagora, 10, 1903^, 11-13, 41-43. 



176 Matb. Quest. Educ. Times, (2), 4, 1903, 43. 



177 Mat. Sbornik (Math. Soc. Moscow), 24, 1903-4, 633-61, in answer to problem proposed 



by V. P. Ermakov, 20, 1898, 293-8. Cf. Jahrbuch Fortschritte Math., 29, 1898, 139; 

 35, 1904, 217. 



178 Nouv. Ann. Math., (4), 3, 1903, 385-97. 



179 L'interme'diaire des math., 11, 1904, 95-96; Math. Soc. Moscow (Mat. Sbornik), 25, 1905, 



466-473 (Russian). Cf. Jahrbuch Fortschritte Math., 36, 1905, 277-8. 



