246

HISTORY or THE THEORY OF NUMBERS.

[CHAP. VI

If the path is symmetrical, there are further conditions. He gave a history of the subject.

N. V. Bougaief109 applied elliptic functions to the decomposition of numbers into squares (with relation to Jacobi's47 Fundamenta Nova).

E. Fauquembergue119 noted that a cube + 1 is never a sum of squares of two consecutive integers,

E. Cesaro111 considered the function \p(n) = 2/(a), where a ranges over all the positive integers for which n — a2 is a square. Then

For/(rc) = 1, ^(n) is the number of positive integral solutions of x2 + y* = n; then 2^0) equals nx/4 asymptotically, whence the number of ways of decomposing a number into a sum of two squares is in mean ir/4.

T. J. Stieltjesm states that if f(ri) is the number of solutions of a;2 + if- = n, and if 41 is the largest odd integer ^ Vn, then

/(I)

. """««•

£J(Vn~+~4 — 1)]. If

is the sum of the odd - 1),

where, in the last, divisors of x,

1), + 0(2) + - . . + 0(n)

are expressed as sums of greatest integers.

T. Pepin113 proved that, if m is an odd number not a square,

m<r(m) = 2]£ {2 + (- l)™-»}(5tt2 - m)X(m - n2),

where X(k) is the sum of the odd divisors of k and <r(k) is the sum of all the divisors of k. Let m be a prune 4Z + 1. Hence

1 = 2(2(V - m)<r(m - 4/x2) (mod 2).

Thus among the differences m — 4//2 occur an odd number of squares, so that m is a S3.

»M Math. Soc. Moscow, 11, 1883, 200-312, 415-456, 515-602; 12, 1885, 1-21.

»« Nouv. Ann. Math., (3), 2, 1883, 430.

m M&n. Soc. Roy. Sc. de Li^ge, (2), 10, 1883, No. 6, pp. 192-4, 224.

"* Comptes Rendus Paris, 97, 1883, 889-891.

u> Atti Accad. Pont. Nuovi Lincei, 37, 1883-4, 41.