A conjecture of Erdös on 3-powerful numbers

Erdös conjectured that the Diophantine equation x + y = z has infinitely many solutions in pairwise coprime 3-powerful integers, i.e., positive integers n for which p | n implies p3 | n. This was recently proved by Nitaj who, however, was unable to verify the further conjecture that this could be done infinitely often with integers x, y and z none of which is a perfect cube. This is now demonstrated. Theorem. The conjecture will follow if even one such solution can be found.Proof. Let a + b = c be one solution. Then for these values of a, b and c theDiophantine equationaX+bY 3 = cZ has the solution [1, 1, 1] in integers [X,Y, Z]with aX , bY and cZ pairwise coprime and hence has infinitely many. This is aspecial case of a well-known theorem [2], but is easily proved for our case. Forstarting from one such solution [X,Y, Z], we find that another is given by X ′ =X(bY 3 + cZ), Y ′ = −Y (aX + cZ), Z ′ = Z(aX − bY ), as is easily verified.Here we find that (a, Y ′) = (a, Y (aX + cZ)) = (a, cY Z) = 1 and so aX ′, bY ′and cZ ′ are pairwise coprime provided X ′ and Y ′ are; this is not always true, forwe find that (X′, Y ′) = (bY 3 + cZ, aX + cZ) = (aX + 2bY , 2aX + bY ) = 3or 1 according as 3 does or does not divide aX − bY . However, dividing out bythis common factor if it occurs, we obtain a new solution with |X′Y ′Z ′| ≥ 2|XY Z|,for since the three nonzero integers X′/X , Y ′/Y and Z ′/Z have sum zero, theirproduct must be at least 2 in absolute value.For any such solution, x = aX, y = bY 3 and z = cZ provides a solution of theoriginal equation, and none of x, y and z will be a cube if none of a, b and c is.The result therefore follows on observing thatX = 9712247684771506604963490444281,Y = 32295800804958334401937923416351,Z = 27474621855216870941749052236511,is a solution of the equation 32X + 49Y 3 = 81Z, for which 7 | Y . References[1] P. Erdös, Problems and results on consecutive integers, Eureka 38 (1975–76), 3–8.[2] L. J. Mordell, Diophantine equations, Academic Press, London and New York, 1969, p. 78.MR 40:2600 Received by the editor May 15, 1996 and, in revised form, September 13, 1996.1991 Mathematics Subject Classification. Primary 11P05. c©1998 American Mathematical Society439 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 440J. H. E. COHN [3] A. Nitaj, On a conjecture of Erdös on 3-powerful numbers, Bull. London Math. Soc. 27 (1995),317–318. MR 96b:11045Department of Mathematics, Royal Holloway University of London, Egham, SurreyTW20 0EX, United KingdomE-mail address: [email protected] License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use