686 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xxm 



is always a square, but never a biquadrate. 



Moret-Blanc 95 found the re's for which (Lucas 91 ) 



Hence (3u 2 l)/2 = v 2 or (3u 2y) 2 Q(v u} 2 = l, whose solutions are given 

 by the convergents of odd rank in the continued fraction for V6. 



E. Catalan 96 noted that, if p is an odd prime and j is an odd integer 

 ^p 1, the sum of the %(p l)th powers of j integers relatively prime to p 

 is not divisible by p. 



A. Berger 97 proved that, if s, m, n, gi, , g s are positive integers, and 

 \I/(ri) is the number of positive integral solutions of giX-\ '+g,x? = n t 



L. Gegenbauer 98 proved a generalization of Catalan's 96 theorem. If X 

 is one of the numbers 2, 3, 4, and if p is a prime =1 (mod X), and r an 

 integer prime to X and <p l/t , where t is the largest integer ==(X+l)/2, 

 then the sum of the (p l)/Xth powers of r integers relatively prime to p 

 is not divisible by p. 



H. Fortey" found that 1 5 H ----- \-n 5 =H for n = l, 13, 133, 1321, , 

 by use of Zy 2 - 2z 2 = 1 . Cf . Moret-Blanc. 95 



E. Lemoine 100 said that A is decomposed into maximum nth powers 

 if A = a"-\ ----- |-G, where a", a, aj|, are the largest nth powers ^A, 

 A a", A a" a", , respectively. Similarly, consider the decomposition 

 A = oi\ 0:2+0:3 - -dzo, where i is the least integer ^ ^JA and 72 1 the 

 remainder a" A, a 2 is the least integer ^ ^Ki and 72 2 the remainder, 3 

 the least integer ^ ^ 2 , etc., and call y p the least number requiring p 

 powers. Then, for w = 2, 71 = !, 72 = 3, T3 = 6 = 3 2 -2 2 +l 2 , 7p+i = H+l. 

 For n = 3, he 101 gave elsewhere the possible forms of the final power 0%. 



L. Aubry 102 proved that P+3 3 5 3 + +(4?z I) 3 is never a square, 

 cube or biquadrate. 



Welsch and E. Miot 103 noted cases in which a n +(a+l) n H ----- f-(a+&) n 

 is of the form I 2 m 2 and hence is a sum of consecutive odd numbers of 

 which the least is 2m+l. 



C. Bisman 104 noted that a sum of like even powers of n 2 +4 numbers 

 can be expressed as the algebraic sum of n 2 +5 squares of which only one 

 is taken negatively. 



96 Nouv. Ann. Math., (2), 20, 1881, 212. 



* M<m. Soc. R. Sc. de LiSge, (2), 13, 1880, 291. Cf. Gegenbauer. 98 



97 Ofversigt K. Vetenskaps-Akad. Forhand., Stockholm, 43, 1886, 355-66. 

 9 Sitzungsber. Akad. Wiss. Wien (Math.), 95, II, 1887, 838-842. 



99 Math. Quest. Educ. Times, 48, 1888, 30-31. 



100 Assoc. frang., 25, 1896, II, 73-7. For n = 2, see papers 20, 21 of Ch. IX. 



101 L'interme'diaire des math., 1, 1894, 232. 



102 Sphinx-Oedipe, 6, 1911, 38-9. E. Lucas, Nouv. Corresp. Math., 5, 1879, 112, had asked 



for solutions. 



103 L'interme'diaire des math., 20, 1913, 47-48. 

 1M Mathesis, (4), 3, 1913, 257-9. 



