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286                  HISTORY OF THE THEORY OF NUMBERS.
where, in S", e ranges over the biquadrate divisors of m. Liouville28 gave the formula
which implies that if 2m (m odd) has no factor of the form 4ju+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 77^= 25,
50=l2+72 = 72+l2=52+52,          10=32+l2 = l2+32,          2=1+1,
whence 33+23+l3 = 62.
Liouville29 stated that, if a, 6, ... are relatively prime in pairs,
(rn(cib...)=(rn(a)(rn(&)..., while if p, q, . . . are distinct primes,
He stated the formulas
and various special cases of them.   To the seventh of these Liouville30 later gave several forms, one being the case p = 0 of
and proved (p. 84) the known theorem that cr(m) is odd if and only if m is a square or the double of a square [cf. Bouniakowsky,19 end]. He proved that <r(AO=2 (mod 4) if and only if 2V 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 MathSmatiques, (2), 2, 1857, 393-6; Comptes Rendus Paris, 44, 1857, 753. **Ibid., 425-432, fourth article of his series. 30/6id., (2), 3, 1858, 63.