Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



A computational approach for solving $y^2=1^k+2^k+\dotsb+x^k$

Authors: M. J. Jacobson Jr., Á. Pintér and P. G. Walsh
Journal: Math. Comp. 72 (2003), 2099-2110
MSC (2000): Primary 11D25, 11J86
Published electronically: May 1, 2003
MathSciNet review: 1986826
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We present a computational approach for finding all integral solutions of the equation $y^2=1^k+2^k+\dotsb+x^k$ for even values of $k$. By reducing this problem to that of finding integral solutions of a certain class of quartic equations closely related to the Pell equations, we are able to apply the powerful computational machinery related to quadratic number fields. Using our approach, we determine all integral solutions for $2\le k\le 70$ assuming the Generalized Riemann Hypothesis, and for $2\le k\le 58$ unconditionally.

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

  • 1. M. A. Bennett and P. G. Walsh, The Diophantine equation $b^2X^4-dY^2=1$, Proc. A.M.S., 127 (1999), 3481-3491. MR 2000b:11025
  • 2. B. Brindza, On some generalizations of the diophantine equation $1^k+2^k+\dotsb+x^k=y^z$, Acta Arith. 44 (1984), 99-107. MR 86j:11029
  • 3. B. Brindza, Power values of the sum $1^k+2^k+\dotsb+x^k$, in ``Number Theory'' (Budapest 1987), Colloq. Math. Soc. János Bolyai, vol. 51, North-Holland, Amsterdam, 1990, 595-603. MR 91g:11027
  • 4. B. Brindza and Á. Pintér, On the number of solutions of the equation $1^k+2^k+\dotsb+(x-1)^k=y^z$, Publ. Math. Debrecen 56/3-4 (2000), 271-277. MR 2001i:11032
  • 5. J. Buchmann, C. Thiel, and H. C. Williams, Short representation of quadratic integers, in ``Computational Algebra and Number Theory, Mathematics and its Applications'' 325, Kluwer, Dordrecht, 1995, 159-185. MR 96c:11144
  • 6. J. H. E. Cohn, The Diophantine equation $x^4-Dy^2=1$ II. Acta Arith. 78 (1997), 401-403. MR 98e:11033
  • 7. K. Dilcher, On a Diophantine equation involving quadratic characters, Compositio Math. 57 (1986), 383-403. MR 87e:11046
  • 8. K. Dilcher, Zeros of Bernouli, generalized Bernouli, and Euler polynomials, Mem. A.M.S., 73 no. 386 (1988) (94 pages). MR 89h:30005
  • 9. J. Gebel, A. Pethö, and H. Zimmer, Computing integral points on elliptic curves, Acta Arith. 68 (1994), 171-192. MR 95i:11020
  • 10. M. Voorhoeve, K. Györy and R. Tijdeman, On the diophantine equation $1^k+2^k+\dotsb+x^k+R(x)=y^z$, Acta Math. 143 (1979), 1-8; Corr. 159 (1987), 151-152. MR 80e:10020 & MR 88i:11018
  • 11. M. J. Jacobson, Jr., Subexponential Class Group Computation in Quadratic Orders, Ph.D. thesis, Technische Universität Darmstadt, Darmstadt, Germany, 1999.
  • 12. M. J. Jacobson, Jr., R. F. Lukes, and H. C. Williams, An investigation of bounds for the regulator of quadratic fields, Experiment. Math. 4 (1995), no. 3, 211-225. MR 97d:11173
  • 13. M. J. Jacobson, Jr. and H. C. Williams, The size of the fundamental solutions of consecutive Pell equations, Experiment. Math. 9 (2000), no. 4, 631-640. MR 2002f:11142
  • 14. M. J. Jacobson, Jr. and H. C. Williams, Modular arithmetic on elements of small norm in quadratic fields, Submitted to Designs, Codes, and Cryptography.
  • 15. D. H. Lehmer, An extended theory of Lucas functions, Ann. Math. 31 (1930), 419-448.
  • 16. H. W. Lenstra, Jr., On the calculation of regulators and class numbers of quadratic fields, London Math. Soc. Lecture Note Series 56 (1982), 123-150. MR 86g:11080
  • 17. The LiDIA Group, LiDia: a C++ library for computational number theory, Software, Technische Univesität Darmstadt, Germany, 1997, see
  • 18. É. Lucas, Solution de la question 1180, Nouv. Ann. Math. (2) 16 (1877), 429-432.
  • 19. Á. Pintér, On a conjecture of Schaffer concerning the power values of power sums, preprint, (2000).
  • 20. J. J. Schäffer, The equation $1^p+2^p+3^p+\dotsb+n^p=m^q$, Acta Math. 95 (1956), 155-189. MR 17:1187a
  • 21. T. N. Shorey and R. Tijdeman, Exponential Diophantine Equations, Cambridge University Press, 87, New York, 1986. MR 88h:11002
  • 22. J. Urbanowicz, On the equation $f(1)1^k+f(2)2^k+\dotsb+f(x)x^k+R(x)=by^z$, Acta Arith. 51 (1988), 349-368. MR 90b:11025
  • 23. H. C. Williams, Édouard Lucas and Primality Testing, Canadian Mathematical Society Series of Monographs and Advanced Texts, vol. 22, John Wiley & Sons, New York, 1998. MR 2000b:11139

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 11D25, 11J86

Retrieve articles in all journals with MSC (2000): 11D25, 11J86

Additional Information

M. J. Jacobson Jr.
Affiliation: Department of Computer Science, University of Calgary, 2500 University Drive NW, Calgary, Alberta, T2N 1N4 Canada

Á. Pintér
Affiliation: Institute for Mathematics, University of Debrecen, P.O. Box 12, H-4010 Debrecen, Hungary

P. G. Walsh
Affiliation: Department of Mathematics, University of Ottawa, 585 King Edward St., Ottawa, Ontario, K1N 6N5 Canada

Keywords: Diophantine equations, elliptic curves, quadratic fields
Received by editor(s): June 5, 2001
Published electronically: May 1, 2003
Additional Notes: The first and third authors are supported by the Natural Sciences and Engineering Research Council of Canada
The second author is supported by the Hungarian National Foundation for Scientific Research, grants T29330, F34891, and FKFP-066-2001
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society