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On the Positive Integral Solutions of the Diophantine Equation x3 + by + 1 − xyz = 0

Luca, Florian, and Togbe, Alain, (2008) On the Positive Integral Solutions of the Diophantine Equation x3 + by + 1 − xyz = 0. Bulletin of the Malaysian Mathematical Sciences Society, 31 (2). pp. 129-134. ISSN 0126-6705

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Official URL: http://math.usm.my/bulletin/pdf/v31n2/v31n2p2.pdf

Affiliations

Instituto de Matem´aticas UNAM, Mexico
Purdue University North Central. Dept. of Mathematics

Abstract

In this paper, we prove a weaker form of a conjecture of Mohanty and Ramasamy [6] concerning the number of positive integer solutions (x, y, z) of the title equation.

Item Type:Journal
Keywords:Diophantine equations, cubic equations, counting solutions of Diophantine equations.
Subjects:Q Science, Computer Science
ID Code:3962

[1] V. Ennola, A note on a divisor problem, Ann. Univ. Turku. Ser. A I 118(1968), 11 pp. [2] P. Erd¨os, On the sum

Px k=1 d(f(k)), J. London Math. Soc. 27 (1952), 7–15.

[3] F. Delmer, Sur la somme de diviseurs P kx {d[f(k)]}s, C. R. Acad. Sci. Paris S´er. A-B 272 (1971), A849–A852.

[4] H. W. Lenstra, Jr., Divisors in residue classes, Math. Comp. 42 (1984), no. 165, 331–340.

[5] S. P. Mohanty, On the Diophantine equation x3 + y + 1 − xyz = 0, Math. Student 45 (1977),no. 4, 13–16 (1979).

[6] Mohanty, S.P. and Ramasamy, A.M.S. On the positive integral solutions of the Diophantine equation x3 +by+1−xyz = 0 (b > 0), Bulletin of the Malaysian Mathematical Society. (2) 7 (1984), no. 1, 23–28.

[7] M. Nair, Multiplicative functions of polynomial values in short intervals, Acta Arith. 62 (1992), no. 3, 257–269.

[8] A. M. S. Ramasamy and S. P. Mohanty, On the positive integral solutions of the Diophantine equation ax3 + by + c − xyz = 0, J. Indian Math. Soc. (N.S.) 62 (1996), no. 1-4, 210–214.

[9] W. R. Utz, Positive solutions of the Diophantine equation x3 +2y +1−xyz = 0, Internat. J. Math. Math. Sci. 5 (1982), no. 2, 311–314.

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