. Carnegie Institution of Washington publication. CHAP. XXI] TWO EQUAL SUMS OF TWO CUBES. 553 there exists the simpler solution x = 6, 2/ = 5. To treat (4), he set (5) A=p+q, B = p-q, C = r-s, D = r+s. Thus (6) p(p2+3(?2)=5(s2+3r2). Taking p = ax-+-3by, q = bx — ay, s we have Hence our equation becomes p(ax+3by} =y(3cy — dx), whence x= — 3n6j8+3nc7, y Writing X, ju = Sac ±360^ ad +36d, we get The abbreviatons 0, 7, X, n were not used by Euler ; but their introduction49 enables us to point out the identity which underlies his solution. In it is the final factor which vanishes, and this in view
. Carnegie Institution of Washington publication. CHAP. XXI] TWO EQUAL SUMS OF TWO CUBES. 553 there exists the simpler solution x = 6, 2/ = 5. To treat (4), he set (5) A=p+q, B = p-q, C = r-s, D = r+s. Thus (6) p(p2+3(?2)=5(s2+3r2). Taking p = ax-+-3by, q = bx — ay, s we have Hence our equation becomes p(ax+3by} =y(3cy — dx), whence x= — 3n6j8+3nc7, y Writing X, ju = Sac ±360^ ad +36d, we get The abbreviatons 0, 7, X, n were not used by Euler ; but their introduction49 enables us to point out the identity which underlies his solution. In it is the final factor which vanishes, and this in view of the identit which in turn follows from. Euler noted (p. 206) that we may solve similarly Z:r = Xp, where Tr = mp2-{-nq2, p = mr2-\-ns2, while I, X are any linear functions of p, q, r, s, by setting p = nfx+gy, q = mfy — gx, r Then Hence x[y is ratonal. Euler50 treated (4) by setting, without loss of generality, A = (m — ri)p+q2, B=(m+ri)p — q2, C = pz—(m+ri)q, D = p2+(m-ri)q. Then (A+B)(A2-AB+B2) = (D-C)(D2+DC+C2} becomes, after division by 2m(p3-g3), ra2+3n2 = 3pg. Thus m = 3k, where pq = n?+3k2. But he had proved in the same paper that every divisor of n2+3/c2, in which n and k are relatively prime, is of like form. Thus while n is ac^FSbd or its negative. "L. E. Dickson, Amer. Math. Monthly, 18, 1911, 110-111. 60NoviComm. Acad. Petrop., 8, annees 1760-1, 1763, 105; Comm. Arith., I, 287; Opera Omnia, (1), II, Please note that these images are extracted from scanned page images that may have been digitally enhanced for readability - coloration and appearance of these illustrations may not perfectly resemble the original Carnegie Institution of Washington. Washington, Carnegie Institution of Washington
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