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

HTML articles powered by AMS MathViewer

- by M. J. Jacobson Jr., Á. Pintér and P. G. Walsh;
- Math. Comp.
**72**(2003), 2099-2110 - DOI: https://doi.org/10.1090/S0025-5718-03-01465-0
- Published electronically: May 1, 2003
- PDF | Request permission

## 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

- Michael A. Bennett and Gary Walsh,
*The Diophantine equation $b^2X^4-dY^2=1$*, Proc. Amer. Math. Soc.**127**(1999), no. 12, 3481–3491. MR**1625772**, DOI 10.1090/S0002-9939-99-05041-8 - B. Brindza,
*On some generalizations of the Diophantine equation $1^k+2^k+\cdots +x^k=y^z$*, Acta Arith.**44**(1984), no. 2, 99–107. MR**774093**, DOI 10.4064/aa-44-2-99-107 - B. Brindza,
*Power values of sums $1^k+2^k+\cdots +x^k$*, Number theory, Vol. II (Budapest, 1987) Colloq. Math. Soc. János Bolyai, vol. 51, North-Holland, Amsterdam, 1990, pp. 595–611. MR**1058236** - B. Brindza and Á. Pintér,
*On the number of solutions of the equation $1^k+2^k+\cdots +(x-1)^k=y^z$*, Publ. Math. Debrecen**56**(2000), no. 3-4, 271–277. Dedicated to Professor Kálmán Győry on the occasion of his 60th birthday. MR**1765981**, DOI 10.5486/pmd.2000.2354 - Johannes Buchmann, Christoph Thiel, and Hugh Williams,
*Short representation of quadratic integers*, Computational algebra and number theory (Sydney, 1992) Math. Appl., vol. 325, Kluwer Acad. Publ., Dordrecht, 1995, pp. 159–185. MR**1344929** - J. H. E. Cohn,
*The Diophantine equation $x^4-Dy^2=1$. II*, Acta Arith.**78**(1997), no. 4, 401–403. MR**1438594**, DOI 10.4064/aa-78-4-401-403 - Karl Dilcher,
*On a Diophantine equation involving quadratic characters*, Compositio Math.**57**(1986), no. 3, 383–403. MR**829328** - Karl Dilcher,
*Zeros of Bernoulli, generalized Bernoulli and Euler polynomials*, Mem. Amer. Math. Soc.**73**(1988), no. 386, iv+94. MR**938890**, DOI 10.1090/memo/0386 - J. Gebel, A. Pethő, and H. G. Zimmer,
*Computing integral points on elliptic curves*, Acta Arith.**68**(1994), no. 2, 171–192. MR**1305199**, DOI 10.4064/aa-68-2-171-192 - Dunham Jackson,
*A class of orthogonal functions on plane curves*, Ann. of Math. (2)**40**(1939), 521–532. MR**80**, DOI 10.2307/1968936 - M. J. Jacobson, Jr.,
*Subexponential Class Group Computation in Quadratic Orders*, Ph.D. thesis, Technische Universität Darmstadt, Darmstadt, Germany, 1999. - Michael J. Jacobson Jr., Richard F. Lukes, and Hugh C. Williams,
*An investigation of bounds for the regulator of quadratic fields*, Experiment. Math.**4**(1995), no. 3, 211–225. MR**1387478** - Michael J. Jacobson Jr. and Hugh C. Williams,
*The size of the fundamental solutions of consecutive Pell equations*, Experiment. Math.**9**(2000), no. 4, 631–640. MR**1806298** - M. J. Jacobson, Jr. and H. C. Williams,
*Modular arithmetic on elements of small norm in quadratic fields*, Submitted to Designs, Codes, and Cryptography. - D. H. Lehmer,
*An extended theory of Lucas functions*, Ann. Math.**31**(1930), 419–448. - H. W. Lenstra Jr.,
*On the calculation of regulators and class numbers of quadratic fields*, Number theory days, 1980 (Exeter, 1980) London Math. Soc. Lecture Note Ser., vol. 56, Cambridge Univ. Press, Cambridge, 1982, pp. 123–150. MR**697260** - The LiDIA Group,
*LiDia: a C++ library for computational number theory*, Software, Technische Univesität Darmstadt, Germany, 1997, see http://www.informatik.tu-darmstadt.de/TI/LiDIA. - É. Lucas,
*Solution de la question 1180*, Nouv. Ann. Math. (2)**16**(1877), 429–432. - Á. Pintér,
*On a conjecture of Schaffer concerning the power values of power sums*, preprint, (2000). - Saunders MacLane,
*Steinitz field towers for modular fields*, Trans. Amer. Math. Soc.**46**(1939), 23–45. MR**17**, DOI 10.1090/S0002-9947-1939-0000017-3 - T. N. Shorey and R. Tijdeman,
*Exponential Diophantine equations*, Cambridge Tracts in Mathematics, vol. 87, Cambridge University Press, Cambridge, 1986. MR**891406**, DOI 10.1017/CBO9780511566042 - Jerzy Urbanowicz,
*On the equation $f(1)1^k+f(2)2^k+\cdots +f(x)x^k+R(x)=by^z$*, Acta Arith.**51**(1988), no. 4, 349–368. MR**971086**, DOI 10.4064/aa-51-4-349-368 - Hugh C. Williams,
*Édouard Lucas and primality testing*, Canadian Mathematical Society Series of Monographs and Advanced Texts, vol. 22, John Wiley & Sons, Inc., New York, 1998. A Wiley-Interscience Publication. MR**1632793**

## Bibliographic Information

**M. J. Jacobson Jr.**- Affiliation: Department of Computer Science, University of Calgary, 2500 University Drive NW, Calgary, Alberta, T2N 1N4 Canada
- Email: jacobs@cpsc.ucalgary.ca
**Á. Pintér**- Affiliation: Institute for Mathematics, University of Debrecen, P.O. Box 12, H-4010 Debrecen, Hungary
- Email: pinterak@freemail.hu
**P. G. Walsh**- Affiliation: Department of Mathematics, University of Ottawa, 585 King Edward St., Ottawa, Ontario, K1N 6N5 Canada
- Email: gwalsh@mathstat.uottawa.ca
- 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 - © Copyright 2003 American Mathematical Society
- Journal: Math. Comp.
**72**(2003), 2099-2110 - MSC (2000): Primary 11D25, 11J86
- DOI: https://doi.org/10.1090/S0025-5718-03-01465-0
- MathSciNet review: 1986826