CHAPTER XXI. 



EQUATIONS OF DEGREE THREE. 



IMPOSSIBILITY OF x*-\-y 3 = z 3 . 



According to Ben Alhocain a defective proof was proposed before 972 

 by the Arab Alkhodjandi. 1 



The Arab Beha-Eddin 2 (1547-1622) listed among the problems remaining 

 unsolved from former times that to divide a cube into two cubes. 



Fermat 3 stated that it is impossible to decompose a cube into two cubes. 



Fermat proposed the problem to find two cubes whose sum is a cube to 

 Sainte-Croix Sept., 1636 (Oeuvres de Fermat, II, 65; III, 287), to Frenicle 

 May(?), 1640 (Oeuvres, II, 195), to the mathematicians of England and 

 Holland Aug. 15, 1657 (Oeuvres, II, 346; III, 313). Oddly enough, 

 Frans van Schooten 4 proposed Feb. 17, 1657, the same problem to Fermat. 

 Fermat 5 insisted that the problem is impossible. 



Frenicle 6 proposed the equivalent problem to find r central hexagons, 

 with consecutive sides, whose sum is a cube. By a central hexagon of n 

 sides he meant the number 



The sum of H n , H n -i, , H n - r+ i is thus a cube z 3 if and only if 



n 3 =(n r) 3 +2 3 . 



J. Kersey 6a stated that J. Wallis proved that no rational cube equals a 

 sum of two rational cubes, but gave no reference. 



L. Euler 7 stated Aug. 4, 1753 that he had proved the problem impossible. 



Euler 8 gave the following proof, incomplete at one point. We may 

 assume that x and y are relatively prime and both odd. Set x+y = 2p, 

 xy = 2q. Then we are to prove that 2p(p 2 +3<? 2 ) is not a cube. Suppose 

 that it is a cube. First, let p be not divisible by 3. Then p/4 and 



1 F. Woepcke, Atti Accad. Pont. Nuovi Lincei, 14, 1860-1, 301. 



2 Essenz der Rechenkunst von Mohammed Beha-eddin ben Alhossain aus Amul, arabisch u. 



deutsch von G. H. F. Nesselmann, Berlin, 1843, p. 55. French transl. by A. Marre, 

 Nouv. Ann. Math., 5, 1846, 313, Prob. 4; ed. 2, Rome, 1864. Cf. A. Genocchi, Annali di 

 Sc. Mat. e Fis., 6, 1855, 301, 304. 



3 Observation 2 on Diophantus (quoted in full in Ch. XXVI on Fermat's last theorem). 



Oeuvres de Fermat, I, 291; III, 241. The problem was sent (1637?) by Fermat to 

 Mersenne to be proposed to St. Croix; cf. P. Tannery, Bull, des sc. math., (2), 7, 1883, 

 8, 121-3. 



4 Correspondance of Huygens, No. 378, Oeuvres completes de Chr. Huygens, 2, 1889, 17; 



Oeuvres de Fermat, 3, p. 558. 

 6 Oeuvres, II, 376, 433, letter to Digby, Apr. 7, 1658, to Carcavi, Aug. 1659. 



6 Solutio duorum problematum . . . 1657 [lost]; Oeuvres de Fermat, III, 605, 608. 

 6a The Elements of Algebra, London, Book III, 1674, 73. 



7 Corresp. Math. Phys. (ed., Fuss), 1, 1843, 618. Also stated in Novi Comm. Acad. Petrop., 



8, 1760-1, 105; Comm. Arith. Coll., I, 287, 296; Opera Omnia, (1), II, 557, 574. 



8 Algebra, 2, 1770, Ch. 15, art. 243, pp. 509-16; French transl., 2, 1774, pp. 343-51; Opera 



Omnia, (1), I, 484-9. Reproduced by A. M. Legendre, The'orie des nombres, 1798, 

 407-8; ed. 3, 1830, II, 7; transl. by Maser, II, 9. 



36 545 



