CHAP. XXI] DlOPHANTINE EQUATION OF DEGREE THREE. 595 



where w is a primitive nth root of unity. The product of two circulants of 

 order n is such a circulant. This is used to prove that 



x 3 -f ay 3 + a~z 3 Saxyz = v n 



has an infinitude of integral solutions for every pair of integers n, a. 



R. D. Carmichael 321 proved that every prime =|=3 is representable in 

 one and but one way by f=x 3 -\-y 3 -\-z 3 3xyz, where x, y, z are ^0. All 

 positive integers are representable by / with x, y, z each ^0, with the sole 

 exception of the integers divisible by 3, but not by 9. A prime 6n+l can 

 be represented in one and but one way by / with at least one variable 

 negative. 



A. Cunningham 322 considered primes of the preceding form /. Take 

 y = x+p, z = x+y. Then /= AB, A=3x+$+y, B = p 2 -py+y 2 . If B = l, 

 then /3 = 7=1, /=3#2. Since any prime p>3 is of the last form, we 

 get positive integers x, y such that / represents p. Next, let A = 1 ; if B 

 is prime it is of the forms 6co+l = & 2 +3Z 2 . 



E. Turriere 323 noted that the above form / represents the rational 

 number n when x = n, y = n + 1/3, z = n 1/3. Ifn=l (mod 3), it represents 

 n when x = y = (n 1)/3, 2 = (n-f 2)/3. 



MISCELLANEOUS SINGLE DIOPHANTINE EQUATIONS OF DEGREE THREE. 



Bhascara 323 " noted that the sum of the cubes of y, 2y, 3y, ty equals the 

 sum of their squares if lOOi/ 3 = 3(h/ 2 , whence ?/ = 3/10. 



T. Robinson 3236 found two cubes a; 3 , z; 3 ^ 3 and a square m 2 x 2 in arithmetical 

 progression, since 2v z x 3 = x*-\-m 2 x 2 determines x rationally. 



A. J. Lexell 323c noted that, if a cubic equation has rational roots, its 

 discriminant is a square. 



J. L. Lagrange 324 employed the " tangent method " to determine new 

 solutions of the cubic equation f(x, y)=0 from one set of solutions p, q. 

 Set x = p+t, y = q-\-u, and take 



dA dA 



Substituting the resulting expression for u into f(p-\-t, q+u)=Q, we may 

 delete the factor t 2 and thus express t, and hence u, as a rational function 

 of the partial derivatives of A. Cf. Lagrange 252 of Ch. XXII. 



To express ! 2 +2 3 as a sum of another square and cube, J. Cunliffe 325 

 took 9 = v 2 +(2-z) 3 , v = 2Lr 2 -6z-l, whence a; = 253/441. J. Whitley took 

 9 = z 3 +(3 nx} 2 , whence 2x+n 2 = V24n+n 4 , which equals 5+pq q 2 if 



521 Bull. Amer. Math. Soc., 22, 1915, 111-7. Cf. Carmichael. 90 



522 Math. Quest, and Solutions, 1, 1916, 14-15. 

 323 L'enseignement math., 18, 1916, 417-20. 



3230 Vija-ganita, 119. Algebra . . . from Sanscrit of Brahmegupta and Bhiiscara, tranal. 



by Colebrooke, 1817, 200. 

 3236 The Gentleman's Diary, or Math. Repository, London, No. 25, 1765; Davis' ed., 2, 1814, 



98. 



m " Euler's Opera postuma, 1, 1862, 504-6 (about 1770). 

 124 Nouv. mm. acad. Berlin, annee 1777, 1779, 153; Oeuvres, IV, 396. 

 126 The Gentleman's Math. Companion, London, 2, No. 13, 1810, 220-1. 



