A computational approach for solving 𝑦²=1^{𝑘}+2^{𝑘}+…+𝑥^{𝑘}

Jacobson, M., Jr.; Pintér, Á.; Walsh, P.

doi:10.1090/S0025-5718-03-01465-0

A computational approach for solving $y^2=1^k+2^k+\dotsb +x^k$
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by M. J. Jacobson Jr., Á. Pintér and P. G. Walsh PDF

Math. Comp. 72 (2003), 2099-2110 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
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