566 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xxi 



fied if a = 6, /=!, g = 9, u = 15, giving 12 3 +17 3 +19 3 = 4-15 3 . This contra- 

 dicts the statement by E. Turriere 133 that x 3 -\-y 3 +z 3 = nt z is impossible if 

 7i^4 or 5 (mod 9). 



A. S. Werebrusow 134 gave two equal sums of four cubes. 



SUM OF THREE CUBES MADE A SQUARE. 



V. Bouniakowsky 135 used fx(x+b)dx to get the identity 



Set 2x-b = (x+b)\ 3 , 2z+36 = /x 2 . Then 



YS i ya_^2 Y_ 3X y_ 2 ~ X3 



*> A ~8-X 3 ' ~8-X 3 ' 



Multiply by (8-X 3 ) 3 . Thus 



(3X) 3 +(2-X 3 ) 3 +(X 3 +l) 3 = [3( 

 E. Catalan, 136 by use of the toroid, obtained the identity 



which gives an infinitude of, but not all, solutions of 



E. Lucas 137 deduced from formulas of Cauchy 287 the generalization 



of Catalan's 136 identity. 



A. Desboves 138 gave a new proof of the last identity. 



A. S. Werebrusow 139 derived from one solution a, b, c, d the second solution 



We may start from the solution (n 2 ) 3 = (n 3 ) 2 . 

 A. Gerardin 140 gave the identities 



where t = 4a 3 6+4& 4 (c 3 +d 3 ). 



Gerardin 141 tabulated solutions of 



BINARY CUBIC FORM MADE A CUBE. 

 Fermat 142 solved Ax*+Bx' 2 +Cx+D = z* if D = d z by setting 



or if A =a 3 by setting z = ax+BI(3a 2 ). while if both D = d* and A =a 3 there 



133 L'enseignement math., 18, 1916, 421. 



134 L'intermediaire des math., 25, 1918, 75-6. 



136 Bull. Ac. Sc. St. Petersbourg, Phys, Math., 11, 1853, 72. 



138 Bull. Acad. Roy. de Belgique, (2), 22, 1866, 29; Mdlanges Math., 1868, 58; Nouv. 



Corresp. Math., 1, 1874-5, 153, foot-note. 



137 Bull. Bibl. Storia Sc. Mat. Fis., 10, 1877, 176. 

 8 Nouv. Ann. Math., (2), 18, 1879, 409. 



139 L'interm6diaire des math., 15, 1908, 136-7. 



140 Sphinx-Oedipe, 8, 1913, 29. 



141 L'intermediaire des math., 23, 1916, 9-10. 



143 J. de Billy's Inventum novum, III, 27-30, Oeuvres de Fermat, III, 386-8. 



