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

is always a square, but never a biquadrate.

Moret-Blanc95 found the x's for which (Lucas91)

Hence (3u2—1)/2 = ^ or (3u—2v)2—6(v—^)2=1, whose solutions are given by the convergents of odd rank hi the continued fraction for VS.

E. Catalan96 noted that, if p is an odd prime and j is an odd integer ^p—1, the sum of the J(p — l)th powers of j integers relatively prime to p is not divisible by p.

A. Berger97 proved that, if 5, m, n, gi, • • •, g, are positive integers, and $(ri) is the number of positive integral solutions of #iZil+ •

-to...„.)-"-r(1+1/m)t

L. Gegenbauer98 proved a generalization of Catalan's96 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 <pl/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 l54----+n5=D for n = l, 13, 133, 1321, ••-, by use of 3y2 - 2xz = 1. Cf. Moret-Blanc.95

E. Lemoine100 said that A is decomposed into maximum nth powers

if A = a?H-------ha;, where aj, aj, aj, • - • are the largest nth powers ^A,

A—aj, A—aJ—oj, • • -, respectively. Similarly, consider the decomposition

A = <XI—o?2+o£-----•=!=<& where «i is the least integer ^^3 and JKi the

remainder m—A, «2 is the least integer ^ZfRi and J?2 the remainder, d3 the least integer ^ A/S2j etc., and call yp the least number requiring p powers. Then, for n=2, 71=!, T2 = 3, T3 = 6 = 32~22+l2, Tp+i^b^+l-For n=3, he101 gave elsewhere the possible forms of the final power tip.

L. Aubry102 proved that -l3+33-53H-------h(4n-l)3 is never a square,

cube or biquadrate.

Welsch and E. Miot103 noted cases in which an+(a-f 1)*H-------\-(a-\-k)n

is of the form Z2—m2 and hence is a sum of consecutive odd numbers of which the least is 2m+l.

C. Bisman104 noted that a sum of like even powers of n2+4 numbers can be expressed as the algebraic sum of n2+5 squares of which only one is taken negatively.

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

« M&n. Soc. R. Sc. de Ltege, (2), 13, 1880, 291. Cf. Gegenbauer.98

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

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

•» Math. Quest. Educ. Times, 48, 1888, 30-31.

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

101 L'rnterm&liaire des math., 1, 1894, 232.

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

for solutions.

1ML'interm6diaire des math., 20,1913, 47-48. *«Mathesis, (4), 3, 1913, 257-9.