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.