Solving constrained Pell equations

Kedlaya, Kiran

doi:10.1090/S0025-5718-98-00918-1

Solving constrained Pell equations
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by Kiran S. Kedlaya PDF

Math. Comp. 67 (1998), 833-842 Request permission

Abstract:

Consider the system of Diophantine equations $x^2 - ay^2 = b$, $P(x,y) = z^{2}$, where $P$ is a given integer polynomial. Historically, such systems have been analyzed by using Baker’s method to produce an upper bound on the integer solutions. We present a general elementary approach, based on an idea of Cohn and the theory of the Pell equation, that solves many such systems. We apply the approach to the cases $P(x, y) = cy^2 + d$ and $P(x, y) = cx + d$, which arise when looking for integer points on an elliptic curve with a rational 2-torsion point.

References

W. S. Anglin, The square pyramid puzzle, Amer. Math. Monthly 97 (1990), no. 2, 120–124. MR 1041888, DOI 10.2307/2323911
A. Baker, Linear forms in the logarithms of algebraic numbers. IV, Mathematika 15 (1968), 204–216. MR 258756, DOI 10.1112/S0025579300002588
A. Baker and H. Davenport, The equations $3x^{2}-2=y^{2}$ and $8x^{2}-7=z^{2}$, Quart. J. Math. Oxford Ser. (2) 20 (1969), 129–137. MR 248079, DOI 10.1093/qmath/20.1.129
Ezra Brown, Sets in which $xy+k$ is always a square, Math. Comp. 45 (1985), no. 172, 613–620. MR 804949, DOI 10.1090/S0025-5718-1985-0804949-7
Duncan A. Buell, Binary quadratic forms, Springer-Verlag, New York, 1989. Classical theory and modern computations. MR 1012948, DOI 10.1007/978-1-4612-4542-1
J. H. E. Cohn, Lucas and Fibonacci numbers and some Diophantine equations, Proc. Glasgow Math. Assoc. 7 (1965), 24–28 (1965). MR 177944, DOI 10.1017/S2040618500035115
Charles M. Grinstead, On a method of solving a class of Diophantine equations, Math. Comp. 32 (1978), no. 143, 936–940. MR 491480, DOI 10.1090/S0025-5718-1978-0491480-0
John E. Hopcroft and Jeffrey D. Ullman, Formal languages and their relation to automata, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR 0237243
Loo Keng Hua, Introduction to number theory, Springer-Verlag, Berlin-New York, 1982. Translated from the Chinese by Peter Shiu. MR 665428, DOI 10.1007/978-3-642-68130-1
P. Kangasabapathy and Tharmambikai Ponnudurai, The simultaneous Diophantine equations $y^{2}-3x^{2}=-2$ and $z^{2}-8x^{2}=-7$, Quart. J. Math. Oxford Ser. (2) 26 (1975), no. 103, 275–278. MR 387182, DOI 10.1093/qmath/26.1.275
De Gang Ma, An elementary proof of the solutions to the Diophantine equation $6y^2=x(x+1)(2x+1)$, Sichuan Daxue Xuebao 4 (1985), 107–116 (Chinese, with English summary). MR 843513
D. McCarthy (ed.), Selected Papers of D. H. Lehmer, Charles Babbage Research Centre, Winnipeg, 1981.
S. P. Mohanty and A. M. S. Ramasamy, The simultaneous Diophantine equations $5y^{2}-20=X^{2}$ and $2y^{2}+1=Z^{2}$, J. Number Theory 18 (1984), no. 3, 356–359. MR 746870, DOI 10.1016/0022-314X(84)90068-4
K. Ono, Euler’s concordant forms, Acta Arith. 78 (1996), 101–123.
R. G. E. Pinch, Simultaneous Pellian equations, Math. Proc. Cambridge Philos. Soc. 103 (1988), no. 1, 35–46. MR 913448, DOI 10.1017/S0305004100064598
C. L. Siegel, Über einige Anwendungen diophantischer Approximationen, Abh. Preuss. Akad. Wiss. (1929), 1.
A. Thue, Über Annäherungenswerte algebraischen Zahlen, J. reine angew. Math. 135 (1909), 284–305.
P. G. Walsh, Elementary methods for solving simultaneous Pell equations, preprint.