238 HISTORY OF THE THEORY OF NUMBERS. [CHAP. VI 



P. Volpicelli 64 noted that, if z = a] + bj (j = 1, -, m), (1) shows that 

 z 2 is a sum of two squares in m(m 1) ways, not necessarily distinct. If 

 z = m 2 + n 2 = p 2 + g 2 , then 



z = 



n - q 



To show that a number having a prime factor p = 4n + 3 is not a sum of 

 two relatively prime squares, raise a 2 = pq b 2 to the power 2n + 1, 

 whence s = a p ~ l + 6 P-1 is a multiple of p, whereas s = 2 (mod p) by 

 Fermat's theorem. In attempting to prove that every prime p = 4n + 1 

 is a O, he employed relatively prune integers x, y, not divisible by p and 

 one even. By Fermat's theorem, x* n y* n = pQ. Since every odd num- 

 ber can be expressed as a difference of two squares, he claimed that we can 

 satisfy x 2n y 2n = Q, whence p = (x n ) 2 + (y n } z . By use of (1), a product 

 of k distinct primes of the form 4n + 1 is a sum of two squares in 2 k ~ l 

 ways, and only in that many ways. Several examples illustrate the method 

 to express A as a El by use of the continued fraction for ^A. The nth 

 power of a C2 is a ED in n{2 or (n + l)/2 ways, according as n is even or 

 odd. 



Volpicelli 65 considered the number v of ways of expressing z as a El, 

 when each prime factor of z is a 121. When z is a product of k distinct 

 primes, v = 2 fc ~ 1 . When just two of these k primes have exponents m 

 and m f , his three formulas can be combined into the single one v = 2"~ 1+M+ ' I ' > 

 where fj, = m/2 or (m + l)/2 according as w is even or odd, and similarly 

 for //. When the roots of the two squares are given double signs, the 

 number is 4v. 



Volpicelli 66 considered Gauss' 37 theorem on the number v of the ways 

 of expressing P = a a tf as a 121, when a, b, - are distinct primes of 

 the form 4n + 1. Let N = (a + l)(/5 + 1) be the number of divisors 

 of P. Let N' be the number of ways of expressing P as a product of two 

 factors A, B. Then N' = (N + l)/2 or N/2 according as a, 13, are all 

 even or not all even. If P is a product of two distinct factors > 1 each 

 expressible as a El, the product theorem (1) yields two expressions for 

 P as El, and conversely. Thus if P is not a square, v = N' = N/2. 

 If P is a square, v I = N' 2, v = (N l)/2, whereas Gauss gave 

 v = (N + l)/2. [It is merely a question as to the inclusion or exclusion 

 of P = P + 0, cf. Genocchi. 75 ] The special cases in which P is a power 

 of a prime or a product of distinct primes are treated (pp. 71-81). He 67 

 insisted until 76 1854 that there is a misprint in Gauss' formula. 



M Raccolta di Lettere . . . Fis. ed Mat. (Palomba), Roma, 5, 1849, 263, 313, 392, 402. 



88 Giornale Arcadico di Sc., Let. ed Arti, Roma, 119, 1849-50, 20-26; Annali di Sc. Mat. e 



Fis., 1, 1850, 156. 



66 Atti Accad. Pont. Nuovi Lincei, 4, 1850-1, 22-31. Same by Volpicelli. 67 

 "Nouv. Ann. Math., 9, 1850, 305-8; Annali di Sc. Mat. e Fis., 1, 1850, 527-531; 2, 1851, 



61-4. 



