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On searching for solutions of the Diophantine equation $x^3 + y^3 +2z^3 = n$


Author: Kenji Koyama
Journal: Math. Comp. 69 (2000), 1735-1742
MSC (1991): Primary 11D25
DOI: https://doi.org/10.1090/S0025-5718-00-01202-3
Published electronically: February 21, 2000
MathSciNet review: 1680899
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Abstract:

We propose an efficient search algorithm to solve the equation $x^3+y^3+ 2z^3=n$ for a fixed value of $n>0$. By parametrizing $\vert z\vert$, this algorithm obtains $\vert x\vert$ and $\vert y\vert$ (if they exist) by solving a quadratic equation derived from divisors of $2\vert z\vert^3 \pm n$. Thanks to the use of several efficient number-theoretic sieves, the new algorithm is much faster on average than previous straightforward algorithms. We performed a computer search for six values of $n$ below 1000 for which no solution had previously been found. We found three new integer solutions for $n=183, 491$ and 931 in the range of $\vert z\vert \le 5 \cdot 10^7$.


References [Enhancements On Off] (What's this?)

  • 1. J. H. E. Cohn, private communication (1995).
  • 2. V.A. Demjanenko, On sums of four cubes (Russian), Izv.Vyssh.Uchebn. Zaved. Matematika, 1966 no. 5 (54) 64-69. MR 34:2525
  • 3. R. K. Guy, Unsolved Problems in Number Theory, First Edition, Springer, New York, 1981. MR 83k:10002
  • 4. R. K. Guy, Unsolved Problems in Number Theory, Second Edition, Springer, New York, 1994. MR 96e:11002
  • 5. D. R. Heath-Brown, W. M. Lioen and H. J. J. te Riele, On solving the Diophantine equation $x^3+y^3+z^3=k$ on a vector computer, Math. Comp. 61 (1993), 235-244. MR 94f:11132
  • 6. K. Koyama, Tables of solutions of the Diophantine equation $x^3+y^3+ z^3=n$, Math. Comp. 62 (1994), 941-942.
  • 7. K. Koyama, Y. Tsuruoka and H. Sekigawa, On searching for solutions of the Diophantine equation $x^3+ y^3+z^3=n$, Math. Comp. 66 (1997), 841-851. MR 97m:11041
  • 8. R. F. Lukes, private communication (1995).
  • 9. L. J. Mordell, Diophantine Equations, Academic Press, New York, 1969. MR 40:2600
  • 10. H. Sekigawa and K. Koyama, Nonexistence conditions of a solution for the congruence $x_1^k+\cdots+x_s^k\equiv N\pmod{p^n}$, to appear in Math. Comp., 68 (1999), 1283-1297. MR 99i:11024

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Additional Information

Kenji Koyama
Affiliation: NTT Communication Science Laboratories, 2-4 Hikaridai, Seika-cho, Soraku-gun, Kyoto 619-02 Japan
Email: koyama@cslab.kecl.ntt.co.jp

DOI: https://doi.org/10.1090/S0025-5718-00-01202-3
Keywords: Diophantine equation, cubic, number-theoretic sieves, search algorithm, computer search
Received by editor(s): October 7, 1996
Received by editor(s) in revised form: January 18, 1999
Published electronically: February 21, 2000
Article copyright: © Copyright 2000 American Mathematical Society

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