## 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.

## Additional Information

**Kiran S. Kedlaya**- Affiliation: Department of Mathematics Princeton University Princeton, New Jersey 08544
- MR Author ID: 349028
- ORCID: 0000-0001-8700-8758
- Email: kkedlaya@math.princeton.edu
- Received by editor(s): January 11, 1995
- Received by editor(s) in revised form: November 4, 1996
- Additional Notes: This work was done during a summer internship at the Supercomputing Research Center (now Center for Computing Studies), Bowie, MD, in the summer of 1992.
- © Copyright 1998 American Mathematical Society
- Journal: Math. Comp.
**67**(1998), 833-842 - MSC (1991): Primary 11Y50; Secondary 11D09, 11D25
- DOI: https://doi.org/10.1090/S0025-5718-98-00918-1
- MathSciNet review: 1443123