286 History of the Theory of Numbers. [Chap, x 
S'X(i))r(^2) =^"d(^^' 2{W!'' = T(m^), 
Sr(0^(5) =S{^(d))''T(52), 2r(OX(5) =2|0(^) j^ 
where, in 2", e ranges over the biquadrate divisors of m. 
Liouville^^ gave the formula 
X{T{d)V={Xr{d)]', 
which implies that if 2m (m odd) has no factor of the form 4)u+3 and if we 
find the number of decompositions of each of its even factors as a sum of 
two odd squares, the sum of the cubes of the numbers of decompositions 
found will equal thesquare of their sum. Thus, for m = 25, 
50=l2+72 = 72+l2 = 52+5^ 10 = 32+12 = 12+32, 2 = 1 + 1, 
whence 3H2Hl' = 62. 
Liouville2^ stated that, if a, 6, . . . are relatively prime in pairs, 
a^iah. . .)=o'n(oVn(?>)- • •, 
while if p, 9, . . . are distinct primes, 
He stated the formulas 
2(r^((i)</)(5) =m<T,_,{m), 2ciV,(5) =2d''(T^(5), 
2X(d)r(d2)(r^(5) =2d''r(5)X(5), 2dV^(6) =2c/''r(d), 
2dX(d) =252v,(d), 2dV3,(5) =2c^V2,(d), 
2dX+,(d)(7,(5) =2(iV,+,(d)(r,(5), 2X(d)(T,(5) =S'(^)' 
2T(d2'')(r,(5) =2^^(5)7(5"), 2{^(rf)} V,(5) =2d''r(52'), 
and various special cases of them. To the seventh of these Liouville^" later 
gave several forms, one being the case p = of 
2d''-V.+X^)(r,+,(5)=2d''-X+,((i)(r,+,(5), 
and proved (p. 84) the known theorem that a{m) is odd if and only if m is a 
square or the double of a square [cf. Bouniakowsky,^^ end]. He proved that 
(t{N) = 2 (mod 4) if and only if N is the product of a prime 4X + 1, raised to 
the power 4Z + 1 (Z^O), by a square or by the double of a square not divis- 
"Jour. de Math6matiques, (2), 2, 1857, 393-6; Comptes Rendus Paris, 44, 1857, 753, 
^^Ibid., 425-432, fourth article of his series, 
"/bid., (2), 3, 1858, 63. 
