CHAP. XXI] IMPOSSIBILITY OF x 3 -\-y 3 = z 3 . 549 



whence 6 and q must be cubes. It is stated that (cf. Euler 8 ) 



g=(a 2 +36 2 ) 3 = 

 In 



' (A+B) + (A-B) ' 18B 



6 = 2.4 or 185, according as x-\-y is not or is divisible by 3. But a and 36 

 are relatively prime and not both odd. Hence 6 is a cube only if a = 4k 3 , 

 a 3b = i 3 , a+3b=j 3 ; or b = 4k 3 , b a = i 3 , b+a=j s , in the respective cases. 

 In either case, j 3 -\-i 3 = (2k} 3 and i, j, k are smaller than x, y, z. He noted 

 numerical results like 



(7 3 +2 3 )(8 3 -7 3 ) =39 3 , (43 3 -36 3 )(54 3 -5 3 ) = (12 3 +1) 3 = (10 3 +9 3 ) 3 . 



P. G. Tait 25 noted that x 3 +y 3 = z 3 implies 



and said that this leads easily to a proof of the impossibility of integral 

 solutions of the former equation. Every cube is a difference of two squares 

 of which one is divisible by 9 since 



[/ i -i\~|2 r ( I^HS 

 -2 J~L 2 j- 



T. Pepin 26 proved the impossibility of x 3 -\-y 3 = z 3 . 



S. Giinther 27 showed how the square root occurring in the solution 

 x, y of x 3 -\-y 3 = a 3 , x-\-y z, can be replaced by a cube root which is " abso- 

 lutely irreducible." 



J. J. Sylvester 28 gave a proof of the impossibility. 



R. Perrin 29 showed how one (hypothetical) set of integral solutions of 

 a 3 +6 3 +c 3 = leads to a new set of integral solutions. 



Schuhmacher 30 stated that Euler 8 erred in affirming that p+q-f3 

 must be the cube of t-\-u^ 3, since it might be a x (+aw) 3 , where 3 = 1. 

 He argued that the first of Euler's two cases may be dispensed with. 



J. Sommer 31 proved Rummer's 63 result (Ch. XXVI) that x 3 +y 3 = z* 

 is not solvable in integral numbers of the domain defined by a cube root 

 of unity. 



H. Krey 32 made the impossibility proof by use of the theory of quadratic 

 forms. Set f(x, y} =x' 2 xy- ! ry 2 . Then 2f is an improperly primitive form 

 of determinant 3 and of class number 1. We can represent properly by/ 

 any positive odd number, not divisible by 3, all of whose prime factors p 

 have 3 as a quadratic residue. If (u, v) is a representation of m, and 

 (u', v') of m', then 



(uu'-\-vv f uv' , uu'-\-vv' vu'} 



26 Proc. Roy. Soc. Edinburgh, 7, 1869-70, 144 (in full). 



26 Jour, de Math., (2), 15, 1870, 225-6. 



27 Sitzungsber. Bohm. Ges. Wiss., Prag, 1878, 112-9. 



28 Amer. Jour. Math., 2, 1879, 393; Coll. Math. Papers, 3, 1909, 350. 



29 Bull. Math. Soc. France, 13, 1884-5, 194-7. Reprinted, Sphinx-Oedipe, 4, 1909, 187-9. 



30 Zeitschrift Math. Naturw. Unterricht, 25, 1894, 350. 



31 Vorlesungen iiber Zahlentheorie, 1907, 184-7. 



32 Math. Naturwiss. Blatter, 6, 1909, 179-180. 



