266 HISTORY OF THE THEORY OF NUMBERS. [CHAP, vn 



E. Catalan 39 noted that the excess of the number of even values of 

 x + y + z in 



(&c I) 2 + (Qy d= I) 2 + (62 d= I) 2 = 3(2n + I) 2 



over the number of odd values of x + y + z is (2n + 1)( l) n . There 

 are at least [_(2n + l)/6] decompositions of 3(2n + I) 2 into a SI. The 

 sextuple 40 of an odd square is a sum of three squares, two of which are of 

 the form (6/* I) 2 and the third is 4(6fc I) 2 . The excess of the number 

 of even values of x in 



4z 2 + 4i/ 2 + (2z + I) 2 = (2n + I) 2 



over the number of odd values is {(2n + 1)( l) n l}/4. If a prime p 

 is not a El, then p 2 is a SO. 



Catalan stated and V. A. Lebesgue 41 proved that the square of a US is 

 a SI, since (3) for 5 = becomes 



(4) (a 2 + /3 2 + 7 2 ) 2 = ( 2 + 2 - T 2 ) 2 + (2 7 ) 2 + (2 7 ) 2 . 



This formula was employed by Euler 308 of Ch. XXII. 

 J. Neuberg 42 also gave (4). 

 Catalan 43 gave the identity 



(a 2 + 6 2 + c 2 + ab + be + ac) 2 



= (a + c) 2 (a + 6) 2 + (6 + c) 2 (a + 6) 2 + (c 2 + ac + be - a6) 2 



and by a change of notation deduced 



{2<jr(/ + h) } 2 + (2//i - 20 2 ) 2 

 P + |20(/ - h) j 2 + (/ 2 - 20 2 + h*y. 



Catalan 44 stated empirically that the triple of any odd square not divis- 

 ible by 5 is a sum of squares of three prunes other than 2 and 3. 



G. H. Halphen 45 proved that every prune 8m + 3 is a S) by means of 

 his 104 recursion formula (Ch. VI) for the sum s(x) of the divisors of x whose 

 complementary divisors are odd. Let x be not a square, El or SI ; then 

 no one of the arguments x n 2 is a El, so that s(x) is divisible by 8. Let 

 also x be a prime, so that s(x) = x + 1. Hence a prime not a E] or IS 

 is of the form 8m 1. 



U. Dainelli 46 derived by integration the case c = of Catalan's 43 formula 



(a 2 + ab + 6 2 ) 2 = (a6) 2 + {a(a + 6)} 2 + [b(a + 6)} 2 . 

 S. R6alis 47 noted that kz z = z\ + z\ + z\ if 



k = A 2 + B* + C 2 , Z = a 2 + /S 2 + T 2 , *i = &(P + 7 2 - a 2 ) - 2a(B0 + Cy), 



T 2 ) ~ 2/3(CT + Aa), Z 3 = C(a 2 + ff 2 - 7 2 ) - 27(^0: + Bfl. 



19 Recherches sur quelques produits ind6finis, M6m. Acad. Roy. Belgique, 40, 1873, 61-191; 

 extract in Nouv. Ann. Math., (2), 13, 1874, 518-523. 



40 Repeated by Catalan, Nouv. Ann. Math., (2), 14, 1875, 428. 



41 Nouv. Ann. Math., (2), 13, 1874, 64, 111. 



42 Nouv. Corresp. Math., 1, 1874-5, 195-6. 



43 Ibid., 153; 2, 1876, 117. 



44 Nouv. Corresp. Math., 3, 1877, 29. 



45 Bull. Soc. Math. France, 6, 1877-8, 180. 



46 Giornale di Mat., 15, 1877, 378. 



47 Nouv. Corresp. Math., 4, 1878, 325. Cf. Malfatti 19 of Ch. VIII. 



